Unbounded Systematic Error in Thin Film Conductivity Measurements

TL;DR

This work reveals that four-bar conductivity measurements in thin-film thermoelectric materials can suffer from unbounded systematic errors due to finite metal electrode conductivity, even when using patterning. Using coupled thermo-electrical finite-element modeling and experimental validation on rub-aligned PBTTT:TFSI, it shows that electrode and device geometry can bias measured conductivity by orders of magnitude and distort temperature-dependent interpretations. The authors derive a general limit condition $\frac{\sigma_F}{\sigma_E} \frac{W^2 t_F}{L_E L_C t_E} \ll 1$ for ideal operation and propose geometry strategies (long, narrow channels with fat outer electrodes; optimized inner electrode dimensions) to minimize errors while maintaining low measurement resistance. These findings have significant implications for the reproducibility and interpretation of conductivity and Seebeck data in thin-film thermoelectrics, and offer practical guidelines to design low-bias devices and validate measurements.

Abstract

Electrical conductivity is the most fundamental charge transport parameter, and measurements of conductivity are a basic part of materials characterization for nearly all conducting materials. In thin films, conductivity is often measured in four bar architectures in which the current source and voltage measurement are spatially separated to eliminate systematic error due to contact resistance. Despite the apparent simplicity of these measurements, we demonstrate here that the four bar architecture is subject to significant systematic error arising from the finite conductivity of the metal electrodes. Remarkably, these systematic errors can in some cases become unbounded, producing arbitrarily high measured conductivity at modest true film conductivities, within the range relevant to emerging thin film thermoelectric materials such as conducting polymers. These unbounded errors, which can occur even in properly conducted four-point measurements of patterned films, likely explain literature reports of extremely high conductivities in conducting polymers, and can lead to anomalous scaling in temperature dependent studies, potentially leading to incorrect interpretation of the relevant charge transport mechanism. We characterize the device geometric factors that control these errors, which stand partially at odds with those required for accurate Seebeck coefficient measurements. Our analyses allow us to identify device architectures that provide small systematic errors for conductivity and Seebeck coefficient while still providing a low measurement resistance, critical to reducing noise in thermal voltage measurements. These findings provide important guidelines for accurate measurements in the growing field of thin-film thermoelectric materials.

Figures (6)

• Figure 1: (a) Schematic of four‑point probe (FPP) conductivity measurement with thin‑film conductivity of 1000 S/cm. Current (red) is injected at the source and drain electrodes (red hollow circles). Voltage is measured at the middle voltage‑probe electrodes (blue hollow circles). Four configurations correspond to different choices of current and voltage probe positions. Color scale (blue-white-red) indicates the potential generated within the device by a fixed measurement current $J_{meas}$. (b) Ratio of measured conductivity to true conductivity ($\sigma_{M} / \sigma_{F}$) versus film conductivity for the four configurations. (c) Potential maps for configuration 4 at several film conductivities.
• Figure 2: Geometric factors affecting systematic conductivity error $\sigma_M/\sigma_F$. a) Effect of varying channel length $L_C$, b) varying electrode length $L_E$, and c) varying channel width $W$. Non-varying geometric factors are identical to those in Figure \ref{['fig:4PP EE distribution']}. d) Map of systematic error for anistropic film conductivity; $\sigma_{Fx}$ is oriented along the channel length while $\sigma_{Fy}$ is oriented along the width. All simulations use configuration 4. Other parameters are as in Table S1 unless specified.
• Figure 3: a) Experimental conductivity measurements of a highly doped rub-aligned PBTTT:TFSI film in four bar architecture ($L_E = 220 \mu$m, $L_C = 580 \mu$m). The conductivity is measured in different probe configurations as film width is progressively reduced by scratching off the film edges with a toothpick. Symbols are experimental data; solid lines are fit using our FEM model using film conductivities $\sigma_{Fx} = 2774$ S/cm, $\sigma_{Fy} = 440$ S/cm, and electrode conductivity $\sigma_{E} = 1.443 \times 10^5$ S/cm and experimentally measured film widths. b) The same data replotted on an expanded conductivity scale, showing an experimentally measured conductivity of $7\times10^5$ S/m near the divergence point.
• Figure 4: (a) Temperature‑dependent conductivity used as simulation input (3D Mott VRH). (b) Current density maps in the Hall bar geometry at 10 K, 100 K, and 300 K. Other parameters as in Figure \ref{['fig:4PP EE distribution']}. The gold electrode resistance also varies with temperature.
• Figure 5: (a) Device geometry and dimensions. A temperature gradient is applied along $x$ (hot: 301 K; cold: 300 K). Thermal voltage is measured between voltage probes 1 and 2. The colormap shows the thermal‑voltage potential. (b) Voltage and temperature along $y=0$. (c) Enlarged view of (b). $V_{\text{measure}}$ and $T_{\text{measure}}$ denote probe‑measured values at the pad; $V_{\text{true}}$ and $T_{\text{true}}$ are values at the channel edge. $\Delta V$ and $\Delta T$ are the corresponding differences. (d) Absolute Seebeck error versus probe offset for different choices of $V$/$T$ in the calculation. Film $S=50\,\mu\mathrm{V\,K^{-1}}$ and $\sigma=500$ S/cm; other parameters as in Figure \ref{['fig:4PP EE distribution']}.