Sheet Resistance: A Guide to Theory

This guide describes the theory behind sheet resistance, explaining what it is and how it can be measured using the four-point probe technique.

What is Sheet Resistance?

Sheet resistance is a common measurement used to characterise thin films of conducting and semiconducting materials. It is a measure of the lateral resistance through a thin square of material, i.e. the resistance between opposite sides of the square. The key advantage of sheet resistance over other resistance measurements is that it is independent of the size of the square. This enables an easy comparison between different samples. Another advantage is that it can be measured directly using a four-point probe.

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Sheet Resistance General Theory

Sheet resistance (RS) is commonly defined as the resistivity (ρ) of a material divided by its thickness (t):

Sheet Resistance Definition

The units of this equation resolve to ohms (Ω); however, it actually represents the resistance between opposite sides of a square of the material (rather than bulk resistance). As such the units Ω/□ (ohms per square) are commonly used.

Sheet Resistance Applications

Sheet resistance is a critical property for any thin film of material in which electrical charges are intended to travel along (rather than travel through). For example, transparent conductors used as electrodes in organic and perovskite photovoltaics require low sheet resistances to reduce energy loss during operation. Alternatively, a resistor may consist of a thin strip of material with a high sheet resistance to achieve a desired bulk resistance.

Measuring Sheet Resistance using a Four-Point Probe

A four-point probe consists of four electrical probes in a line, with equal spacing between each of the probes. It operates by applying a current (I) on the outer two probes and measuring the voltage drop between the inner two probes. As very little current flows between the inner probes (ideally zero) due to the use of a high-impedance voltmeter, the measured voltage drop (ΔV) is independent of any contact or cable resistance, meaning it results only from the resistance of the material being tested. The sheet resistance can then be calculated using the equation below:

Sheet Resistance Equation

It should be noted that this equation is only valid if: i) the material being tested is no thicker than 40% of the spacing between the probes, and ii) the lateral size of the sample is sufficiently large. If this is not the case, then geometric correction factors are needed to account for the size, shape, and thickness of the sample. The value of this factor is dependent on the geometry being used, and is covered in detail in the next section.

Geometric Correction Factors

Whilst the above equation for sheet resistance is independent of sample geometry, this only applies when the sample is significantly larger (typically having dimensions 40 times greater) than the spacing of the probes, and if the sample is thinner than 40% of the probe spacing. If this is not the case, the possible current paths between the probes are limited by the proximity to the edges of the sample, resulting in an overestimation of the sheet resistance. To account for this difference, a correction factor based upon the geometry of the sample is required.

All the correction factors in this guide were obtained from Haldor Topsøe, Geometric Factors in Four Point Resistivity Measurement, 1966.

Circular Samples

For a circular sample of diameter d, measured at the centre of the sample, the correction factor can be calculated using:

 Sheet Resistance Correction Factor

Where s is the distance between probes. For d >> s this equation tends to unity, enabling the use of the uncorrected equation.

Rectangular Samples

For a rectangular sample, the determination of the geometrical correction factor is slightly more complicated as there is no equation. Instead, a table of empirically-determined correction factors is used. The values in this table only apply when the probes make contact in the centre of the sample, and are aligned parallel to the sample's longest edge (l), as shown in Figure 1.

 Four point probe positioning
Figure 1: Illustration of probe positioning and dimensions for a rectangular sample.

As an example, suppose the rectangular sample shown in the figure above has a long edge of l = 20 mm and short edge of w = 10 mm, and the spacing of the probes being used is s = 2 mm. In this case, w = 2 and s = 5, so the table is searched for the correction factor which satisfies these two values, looking along the columns for w = 2 and the rows for s = 5, which is C = 0.7887. The measured sheet resistance is multiplied by this value to get the correct value for the sample.

s

w = 1

w = 2

w = 3

w = 4

1

 

 

0.2204

0.2205

1.25

 

 

0.2751

0.2751

1.5

 

0.3263

0.3286

0.3286

1.75

 

0.3794

0.3803

0.3803

2

 

0.4292

0.4297

0.4297

2.5

 

0.5192

0.5194

0.5194

3

0.5422

0.5957

0.5958

0.5958

4

0.6870

0.7115

0.7115

0.7115

5

0.7744

0.7887

0.7887

0.7887

7.5

0.8846

0.8905

0.8905

0.8905

10

0.9313

0.9345

0.9345

0.9345

15

0.9682

0.9696

0.9696

0.9696

20

0.9822

0.9830

0.9830

0.9830

40

0.9955

0.9957

0.9957

0.9957

∞ 

1

1

1

1

 

Obviously, not every sample will fall neatly into these categories. If this is the case, it is recommended that cubic spline interpolation is used to estimate the appropriate correction factor for the sample.

It is important to note that the correction factors for circular and rectangular samples detailed above only apply for measurements taken in the centre of sample. So if the measurement is not in the centre, different correction factors are needed.

Other Shapes and Probe Positions

For different sample shapes and for measurements not performed at the centre of the sample, alternative correction factors are required, most of which can be found by Haldor Topsøe, Geometric Factors in Four Point Resistivity Measurement, 1966, or F. M. Smits, Measurement of Sheet Resistivities with the Four-Point Probe, Bell Syst. Tech. J., May 1958, p. 711.

If the shape of the sample is irregular, consider whether it is closer to rectangular or circular and then estimate what size of that shape could fit within the sample.

Thick Samples

If the sample being tested is thicker than 40% of the probe spacing, an additional correction factor is required. The correction factor used is dependent upon the ratio of the sample thickness (t) to the probe spacing (s) and some of the possible values are listed in the table below:

t s

Correction Factor

0.4

0.9995

0.5

0.9974

0.5555

0.9948

0.6250

0.9898

0.7143

0.9798

0.8333

0.9600

1.0

0.9214

1.1111

0.8907

1.25

0.8490

1.4286

0.7938

1.6666

0.7225

2.0

0.6336

 

As with the rectangular samples, if t s does not equal one of the values given in the table, a cubic spline interpolation is recommended to estimate the appropriate correction factor for the sample.

Derivation of the Sheet Resistance Equation

In order to determine the how the sheet resistance of a thin film is measured using a four-point probe, a simplified scenario must first be evaluated. Imagine an arbitrarily sharp probe contacting and injecting current (through an applied voltage) into a semi-infinite volume (infinite in all directions except towards the probe) of a conductive material.

 Current injection into semi-infinite sample
Figure 2: A probe injecting current I into a semi-infinite volume of conductive material. The green hemisphere is a shell of the injected current with radius r.

 

The current travels outwards from the point of contact through concentric hemispherical shells of equipotential, each of which have current density (J) of:

 Sheet Resistance Derivation 1

Where r is the radial distance from the probe (2πr2 being the surface area of the hemisphere). By applying Ohm’s Law (E = ρJ) with the electric field across each shell equal to the voltage drop over the shell thickness, or -ΔV/Δr (this term is negative as voltage decreases with r), and with the thickness of the shell tending towards zero, the following equation is obtained:

 Sheet Resistance Derivation 2

This can be integrated between r and r’ to obtain:

 Sheet Resistance Derivation 3

By applying the boundary condition that V approaches zero as r approaches infinity, the equation simplifies to:

 Sheet Resistance Derivation 4

Now imagine that there are four arbitrarily sharp probes (labelled 1 to 4) in contact with the semi-infinite conducting material which are in a line with equal spacing (s) and set up so that current is injected through probe 1 and collected by probe 4.

 Four point probe schematic
Figure 3: Four probes with equal spacing (s) in contact with the semi-infinite volume of conductive material. The current (I) is injected through probe 1 and collected through probe 4.

 

If equivalent boundary conditions are assumed for each probe, the voltage at any point is equal to the sum of the voltage due to each probe separately, i.e.:

 Sheet Resistance Derivation 5

Where r1 and r4 are the radial distances from probe 1 and probe 4 respectively. Measurements of the voltage are then made between probes 2 and 3. Using the above equation, the voltage at probes 2 and 3 are:


 Sheet Resistance Derivation 6

Sheet Resistance Derivation 7

Hence, the change in voltage (ΔV) between probes 2 and 3 is:

 Sheet Resistance Derivation 8

Therefore, the resistivity between the probes is:

 Sheet Resistance Derivation 9

This expression only applies in the case of a semi-infinite volume, and does not apply in the case of a thin film. However, a new expression can be derived using a similar analysis. As before, imagine the arbitrarily sharp probe contacting and injecting current into a thin film of material with thickness t.

 Current injection into thin sample
Figure 4: A probe injecting current I into a thin film of conductive material with thickness t. The green cylinder is a shell of the injected current with radius r.

 

In this case, the current travels away from the probe (through the material) in short cylindrical shells of equipotential, each with a current density of:

 Sheet Resistance Derivation 10

By applying the same conditions for the electric field as previously (Ohm’s Law and shell thickness tending to zero), the electric field across each shell is:

 Sheet Resistance Derivation 11

The resistivity has already been defined as the sheet resistance multiplied by the thickness of the materials, so this can be replaced in the above equation to give:

 Sheet Resistance Derivation 12

This can be integrated between r and r’ to obtain:

 Sheet Resistance Derivation 13

Unlike before, it cannot be assumed that the voltage tends to zero as r approaches infinity as the natural logarithm of infinity is not zero. However, this does not impact the analysis as the difference in voltage at different points (ΔV) is the value measured by the four-point probe.

Now imagine the four-probe system in contact with a thin film, with an additional condition that the thickness of the film (t) is negligible compared to the probe spacing (s). For current being injected by probe 1 and collected by probe 4, the equation becomes:

 Sheet Resistance Derivation 14

The voltages measured at probes 2 and 3 are therefore:

 Sheet Resistance Derivation 15

Sheet Resistance Derivation 16

Hence, the change in voltage is:

 Sheet Resistance Derivation 17

Which can be rearranged to give:

 Sheet Resistance Derivation 18

Therefore, by measuring the change in voltage between the inner probes and the applied current between the outer probes, we can measure the sheet resistance of a sample.

References and Further Reading

(Please note that Ossila has no formal connection to any of the authors or institutions in these references)

  • The 100th anniversary of the four-point probe technique: the role of probe geometries in isotropic and anisotropic systems, I. Miccoli et al., J. Phys.: Condens. Matter, 27, 223201 (2015)
  • Resistivity Measurements on Germanium for Transistors, L. B. Valdes, Proceedings of the I.R.E, February, 420 (1954)
  • Geometric Factors in Four Point Resistivity Measurement, H. Topsøe, (1966)
  • Measurement of Sheet Resistivities with the Four-Point Probe, F. M. Smits, Bell Syst. Tech. J., 711 (1958)