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Calculate Sheet Resistance Using the Four-Probe Method


This guide explains the theory behind sheet resistance, an electrical property of thin films of materials, and demonstrates how the four-probe method can be used to measure it.

What is Sheet Resistance?


Sheet resistance (also known as surface resistance or surface resistivity) is a common electrical property 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 a square. The key advantage of sheet resistance over other resistance measurements is that it is independent of the size of the square - enabling an easy comparison between different samples.

This property can easily be measured using a four-point probe and is critical in the creation of high-efficiency perovskite photovoltaic devices, where low sheet resistance materials are needed to extract charge.

Examples of Applications


Sheet resistance is a critical property for any thin film of material in which electrical charges are intended to travel along (rather than pass through). For example, thin-film devices (such as perovskite solar cells or organic LEDs) require conducting electrodes which generally have thicknesses in the nanometre to micrometre range. The figure below shows how charges move within an LED device. The electrodes must transport electrical charge laterally and need low sheet resistances to reduce losses during this process. This becomes even more important when attempting to scale up the size of these devices, as the electrical charges will have to travel further along the electrodes before they can be extracted.

Current flowing laterally through the electrodes to the active material on a thin film LED
A schematic diagram of a thin film LED, showing current flowing laterally through the electrodes to the active material. The sheet resistance of the electrodes will effect amount of current that reaches the LED, impacting its performance.

Furthermore, the resistivity and conductivity can be calculated if the sheet resistance and material thickness are known. This allows for the materials to be electrically characterised, purely by taking a sheet resistance measurement.

The Four-Probe Method for Measuring Sheet Resistance


General Theory

The primary technique for measuring sheet resistance is the four-probe method (also known as the Kelvin technique), which is performed using a four-point probe. A four-point probe consists of four equally spaced, co-linear electrical probes, as shown in the schematic below.

Four point probe schematic depicting the four-probe method
A schematic diagram of a four-point probe. The four probes have equal spacing (s) and are shown in contact with a surface. A current (I) is injected through probe 1 and collected through probe 4, whilst the voltage is measured between probes 2 and 3.

It operates by applying a DC current (I) between the outer two probes and measuring the resultant voltage drop between the inner two probes.

The Sheet Resistance Equation

The sheet resistance can then be calculated using the following equation:

Four-probe sheet resistance equation

Rs is the sheet resistance, ΔV is the change in voltage measured between the inner probes, and I is the current applied between the outer probes. The sheet resistance is generally measured using the units Ω/□ (ohms per square), to differentiate it from bulk resistance.

It should be noted that this equation is only valid if:

  • The material being tested is no thicker than 40% of the spacing between the probes
  • 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).

If the thickness of the measured material is known, then the sheet resistance can be used to calculate its resistivity:

Sheet resistance equation to calculate resistivity

Here, ρ is the resistivity, and t is the thickness of the material.

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Eliminating Contact Resistance

One of the primary advantages of using a four-point probe to perform electrical characterisation is the elimination of contact and wire resistances from the measurement. The diagram below shows the circuit resistances of a four-point probe measurement..

Four-point probe equivalent circuit diagram to show circuit resistances
Equivalent circuit diagram of a four-point probe, showing the wire resistances (RW), contact resistances (RC), and sample resistances (RS). The green arrows represent current flow.

The applied current I enters and leaves the sample via the outer probes, and flows through the sample. Voltmeters typically have high electrical impedance to prevent them affecting the circuit being measured, so no current will flow through the inner two probes. Only the voltage is measured between the inner probes, meaning that the wire resistances (RW2 and RW3) and the contact resistances (RC2 and RC3) do not contribute to the measurement. Any measured decrease in voltage (ΔV) will therefore arise entirely from the sample resistance (RS2). This simplifies the sheet resistance equation, so that only ΔV and the applied current are required to find the value of RS2 (i.e. the sheet resistance).

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:

Four-Point Probe Circular Sample 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 below.

Four point probe positioning on the centre of the sample
Illustration of probe positioning and dimensions for a rectangular sample with l ≥ w.

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, l / w = 2 and w / s = 5, so the table is searched for the correction factor which satisfies these two values, looking along the columns for l / w = 2 and the rows for w / s = 5, which is C = 0.7887. The measured sheet resistance is multiplied by this value to get the correct value for the sample.

w / s

l / w = 1

l / w = 2

l / w = 3

l / 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

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 the sample. 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 these can be found in 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.

Four-Point Probe Equation Derivation


In order to determine 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
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 Formula Derivation Step 1 - calculate current density

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 Formula Derivation Step 2 - Apply Ohm's Law

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

Sheet Resistance Formula Derivation Step 3 - integrate between r and r'

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

Sheet Resistance Formula Derivation Step 4 - Apply a boundary condition

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). They are set up so that current is injected through probe 1 and collected by probe 4 as discussed earlier in the General Theory section. 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 Formula Derivation Step 5 - calculate the voltage at any point

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 Formula Derivation Step 6 - calculate voltage at probe 2
Sheet Resistance Formula Derivation Step 7 - calculate voltage at probe 2

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

Sheet Resistance Formula Derivation Step 8 - calculate the change in voltage between probe 2 and 3

Therefore, the resistivity between the probes is:

Four-Point Probe Resistivity Equation - find the resistivity between probes 2 and 3

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
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 Formula Derivation Step 9 - find current density involving thickness t

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 Formula Derivation Step 10 - apply Ohm's Law

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 Formula Derivation Step 11 - Subsitute sheet resistance into the equation

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

Sheet Resistance Formula Derivation Step 12 - integrate between r and r'

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 Formula Derivation Step 13 - find voltage between probes 1 and 4

The voltages measured at probes 2 and 3 are therefore:

Sheet Resistance Formula Derivation Step 14 - find the voltage at probe 2
Sheet Resistance Formula Derivation Step 15 - find the voltage at probe 3

Hence, the change in voltage is:

Sheet Resistance Formula Derivation Step Step 16 - find the difference in voltage between probes 2 and 3

Which can be rearranged to give:

Four-Point Probe (sheet resistance) Equation

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.

Four-Point Probe

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References and Further Reading


  • 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)
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