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Development of a voltage-divider based probe method for sheet resistance measurement of copper foils and thin films

This paper presents a voltage-divider-based probe method for measuring the sheet resistance of copper foils and thin films, demonstrating its effectiveness in providing reproducible results for micrometer-thick samples and enabling measurements of nanometer-thick films where conventional four-point probe techniques fail.

Original authors: Mehrangiz Kazemi, Hossein Miar-Naimi, Majid Abbasi, Marzieh Abbasi-Firouzjah

Published 2026-10-21

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CC BY 4.0

Original authors: Mehrangiz Kazemi, Hossein Miar-Naimi, Majid Abbasi, Marzieh Abbasi-Firouzjah

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of modern electronics, the tiny metal layers that carry electricity inside our devices are becoming thinner and thinner. To understand how well these layers work, scientists need to measure a property called sheet resistance. Think of this not as the resistance of a single wire, but as a measure of how difficult it is for electricity to flow across the entire surface of a thin sheet of metal. If a sheet is very thin, or if it has tiny imperfections, the electricity struggles more, and the sheet resistance goes up. This measurement is critical because if the resistance is too high, the device might fail or run too hot. Traditionally, scientists have used a tool with four small metal tips to measure this. They press two outer tips to send a current through the material and use the two inner tips to listen to the voltage. This method is reliable for large, flat samples, but it becomes tricky when the sample is very small, oddly shaped, or so thin that the metal tips might damage it or fail to make a good connection.

A team of researchers at Babol Noshirvani University of Technology and Hakim Sabzevari University in Iran set out to create a simpler way to measure these difficult samples. They developed a new method based on a basic electrical principle called a voltage divider. Instead of using four separate probes, their system uses a single probe setup connected to a circuit that compares the voltage across the unknown metal sample against a known reference resistor. By carefully measuring the voltage that drops across the sample, they can calculate its resistance without needing the complex geometry of the traditional four-tip tool. The researchers built a custom device using a small computer board to control the measurements and a high-precision sensor to read the tiny voltage changes, ensuring the process was repeatable and accurate.

To test their new approach, the team prepared two types of copper samples. First, they took thick copper foils, ranging from twenty to one hundred micrometers, and rolled them thinner and thinner. Second, they created extremely thin copper films on glass, with thicknesses ranging from about thirty-two nanometers to five hundred twenty-nine nanometers. For the thicker foils, they compared their new single-probe method against the standard four-point probe. The results were remarkably close. Both methods showed that as the copper foil got thinner, its electrical resistance increased slightly, a behavior consistent with how metal behaves when squeezed into a smaller space. The new method proved to be just as reliable as the traditional one for these micrometer-thick samples, with measurements that repeated consistently every time.

The real test came with the nanometer-thin films, where the traditional tool began to struggle. For the thinnest copper layers, those around thirty-two and fifty-three nanometers thick, the standard four-point probe could not produce a readable measurement under the conditions used in the lab. The contact between the metal tips and the fragile, ultra-thin film was likely too unreliable. However, the new voltage-divider method worked perfectly. It successfully measured the resistance of these ultrathin films, revealing that as the copper layers became thinner, their electrical resistivity increased significantly. This increase happens because electrons, the particles carrying the electricity, begin to bounce off the surfaces and grain boundaries of the metal more frequently when the film is so thin that it approaches the size of the electrons' natural path.

The researchers found that their new method could measure films as thin as thirty-two nanometers, a thickness where the conventional tool failed. While the numbers from the new method differed slightly from the traditional tool for the thicker nanometer films, the difference was small enough to be expected given the extreme sensitivity of these tiny samples to surface roughness and contact quality. The study also showed that by calibrating the system with known samples, the new device could even estimate the thickness of an unknown copper foil just by measuring its electrical resistance, with a high degree of accuracy.

This work demonstrates that a simple, single-point measurement system can serve as a powerful alternative to complex multi-probe setups, especially when dealing with delicate or very small samples. It does not replace the established four-point probe for all situations, but it fills a specific gap where the old tools struggle. By proving that a voltage-divider circuit can provide reliable data for copper films ranging from thick foils down to nanometer-scale layers, the researchers have offered a practical, reproducible tool for characterizing the electrical properties of materials that are essential for the next generation of electronic devices. The method is straightforward, requires less complex equipment, and opens the door to measuring samples that were previously difficult to test.

Technical Summary: Development of a Voltage-Divider Based Probe Method for Sheet Resistance Measurement

Problem Statement
Accurate measurement of sheet resistance ( $R_S$ ) and electrical resistivity ( $\rho$ ) is critical for characterizing thin metallic films and foils, particularly as sample dimensions and thicknesses approach the limits of conventional probe-based techniques. In thin-film systems, electrical properties often deviate from bulk behavior due to surface, interface, and grain boundary scattering. While the linear four-point probe (4PP) is the standard for such measurements, it faces limitations when sample areas are small, geometries are non-standard, or thicknesses are extremely thin (nanoscale). In these regimes, contact resistance, probe spacing, sample boundaries, and thickness non-uniformity can significantly skew results. There is a practical need for a measurement approach that combines a simple electrical configuration with controlled data acquisition to reduce sensitivity to probe arrangement and sample geometry.

Methodology
The authors developed a voltage-divider-based single-point probe system to measure the sheet resistance of copper samples across micrometer and nanometer scales.

Sample Preparation:
- Micrometer-scale: Copper foils were produced via cold rolling, with thicknesses ranging from 20 to 100 $\mu$ m. Thickness was verified using a precision thickness gauge, micrometer, and optical microscopy.
- Nanometer-scale: Copper thin films (approx. 32–529 nm) were deposited on glass substrates via thermal evaporation. Thickness was determined via cross-sectional Field-Emission Scanning Electron Microscopy (FE-SEM).
Measurement System:
- Circuit Design: The system utilizes a series voltage-divider configuration where the unknown sample resistance ( $R_s$ ) is connected in series with a calibrated reference resistance ( $R_1$ ). The output voltage ( $V_o$ ) across the sample is measured relative to the input voltage ( $V_i$ ).
- Hardware: The circuit includes a signal amplification stage (differential amplifier) and a relay to control current duration, minimizing Joule heating. Data acquisition is handled by an Arduino UNO microcontroller interfaced with a high-resolution 16-bit ADS1115 ADC, replacing the standard 10-bit internal ADC to improve voltage resolution.
- Calibration: The effective reference resistance was determined experimentally by short-circuiting the contacts to account for lead and contact resistances.
Theoretical Basis:
- The method relies on solving the Laplace equation for electric potential on an infinite conducting plane.
- A dimensional condition (Eq. 20) was derived, stating that the distance between measurement points ( $d$ ) must be less than approximately one-fifth of the sample's characteristic lateral dimension ( $D$ ) ( $D/d \geq 5$ ). Under this condition, the finite sample boundary effects are considered negligible, allowing the use of the infinite-plane approximation.
Validation: The proposed method was compared against a commercial Jandel four-point probe (1–3 mm spacing) under identical experimental conditions. Twenty repeated measurements were taken for each sample to assess repeatability.

Key Contributions

Novel Measurement Configuration: The development of a low-cost, single-point probe system based on a voltage-divider circuit, integrated with high-resolution digital acquisition, specifically designed to reduce sensitivity to probe spacing within defined geometric limits.
Theoretical Derivation: Establishment of a theoretical model linking the measured voltage response to sheet resistance using an infinite-plane approximation, accompanied by a specific dimensional condition ( $D/d \geq 5$ ) to validate the model's application to finite samples.
Cross-Scale Validation: Comprehensive evaluation of the method across two distinct regimes:
- Micrometer range (20–100 $\mu$ m): Comparison with 4PP on rolled copper foils.
- Nanometer range (32–529 nm): Application to thermally evaporated films where conventional 4PP measurements failed for the thinnest samples.
Indirect Thickness Estimation: Demonstration of a calibration relationship between sheet resistance and thickness for copper foils, enabling the estimation of unknown foil thicknesses based on electrical measurements.

Results

Micrometer-Scale Performance: For copper foils (20–100 $\mu$ m), the proposed single-point probe method yielded sheet resistance and resistivity values closely comparable to the conventional four-point probe. The resistivity trends were consistent, showing a slight increase as thickness decreased, attributed to microstructural changes from cold rolling. The method demonstrated high repeatability, with measurement variations as low as $5 \times 10^{-5} \Omega$ .
Thickness Estimation: A regression analysis of the micrometer-scale data yielded a coefficient of determination ( $R^2$ ) of 99.62%. This calibration curve successfully estimated the thickness of unknown samples (approx. 45 and 60 $\mu$ m) with agreement against independent mechanical thickness measurements.
Nanometer-Scale Performance:
- For films thicker than 90 nm, the proposed method produced measurable results comparable to the 4PP, though differences increased as thickness decreased (5% difference at 90 nm, rising to 28% at 529 nm).
- Critical Advantage: For the thinnest films (32 nm and 53 nm), the conventional four-point probe system failed to provide measurable values under the study's experimental conditions. In contrast, the proposed voltage-divider method successfully provided repeatable, measurable sheet resistance values.
- Physical Trends: As film thickness decreased below the electron mean free path (~39 nm for Cu), both sheet resistance and electrical resistivity increased significantly. This aligns with established size-effect models where surface and grain boundary scattering dominate electron transport.

Significance and Claims
The paper claims that the proposed voltage-divider-based method offers a simple, reproducible, and complementary approach for sheet resistance characterization. Its primary significance lies in its ability to:

Provide reliable measurements for copper foils and thin films where conventional four-point probes may be difficult to apply due to sample size or geometry constraints.
Extend electrical characterization to ultrathin films (<100 nm) where standard 4PP techniques failed in this specific experimental setup.
Reduce sensitivity to probe spacing when the sample dimensions satisfy the derived geometric condition ( $D/d \geq 5$ ).

The authors modestly position the method not as a universal replacement for established four-point probe techniques, but as a valuable alternative for specific dimensional regimes. They acknowledge that accuracy is dependent on assumptions regarding sample homogeneity, thickness uniformity, and contact conditions, particularly for non-uniform or discontinuous ultrathin films. The study concludes that while the method is effective, future work should focus on quantitative uncertainty analysis and systematic investigation of contact resistance and surface morphology.

Technical Summary: Development of a Voltage-Divider Based Probe Method for Sheet Resistance Measurement

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