🔬 mesoscale physics

Experimental assessment of the Wiedemann-Franz law in thin metal films using thermoreflectance and electrical measurements

This study combines thermoreflectance and electrical measurements to demonstrate that the Wiedemann-Franz law, when applied with the bulk Sommerfeld-Lorenz number, significantly overestimates the thermal conductivity of sputtered tantalum and moderately deviates for aluminum and titanium thin films due to deposition-induced disorder and grain-boundary scattering.

Original authors: Zhiwei Deng, Jinlong Ma, Puqing Jiang

Published 2026-09-17

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CC BY 4.0

Original authors: Zhiwei Deng, Jinlong Ma, Puqing Jiang

Original paper licensed under CC BY 4.0 (http://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 microscopic world of modern electronics, tiny layers of metal are the unsung heroes that keep our devices running. These films, often thinner than a strand of hair, act as the wires, contacts, and barriers that guide electricity and manage heat within a computer chip. As these devices shrink and pack more power into smaller spaces, the ability of these metal layers to conduct heat becomes critical. If heat cannot escape efficiently, the device overheats and fails. For decades, engineers have relied on a simple rule of thumb to predict how well a metal film conducts heat: they measure how easily it conducts electricity and apply a standard conversion factor. This rule assumes that the relationship between electricity and heat is fixed and universal, regardless of how the metal was made. However, in the real world, these films are not perfect crystals; they are messy, with grain boundaries, defects, and irregular structures that can disrupt the flow of both electrons and heat. The question remains: does the old rule still hold when the metal is a thin, imperfect film, or does the relationship break down?

A team of researchers at Huazhong University of Science and Technology set out to test this assumption directly. Instead of relying on the old rule to guess the thermal properties of metal films, they measured the heat and electricity flow in real samples. They prepared five different metal films—aluminum, titanium, and tantalum—using three distinct manufacturing methods: thermal evaporation, electron-beam evaporation, and magnetron sputtering. Each method creates a slightly different internal structure, with varying levels of disorder and grain size. To find the truth, the team used a technique called square-pulsed source thermoreflectance. In this setup, a laser flashes in a square pattern to heat the metal film, while a second laser watches how the surface reflects light as it warms and cools. By analyzing the timing and shape of these temperature changes across a wide range of speeds, the researchers could calculate exactly how much heat the film could carry and how much energy it stored. They paired this with direct electrical measurements to see how well the same films conducted electricity.

The results revealed a clear picture of how manufacturing changes the physics of these films. The researchers found that the ability of a metal film to store heat, known as its volumetric heat capacity, remained remarkably stable. No matter which metal or which manufacturing method was used, the heat storage values stayed within eight percent of the standard values for the bulk metal. This consistency suggests that the basic energy-holding nature of the atoms remains unchanged even when the film is thin and imperfect. However, the ability to conduct heat told a very different story. The thermal conductivity varied wildly depending on how the film was made. Films made by thermal or electron-beam evaporation, which tend to be denser and more ordered, conducted heat nearly as well as the bulk metal. In contrast, films made by magnetron sputtering, which introduced more disorder and defects, conducted heat significantly worse. The aluminum film made by sputtering conducted only about half as much heat as the bulk material, while the sputtered tantalum film conducted nearly ten times less heat than its bulk counterpart.

When the team compared their direct measurements against the predictions made by the old rule of thumb, the limitations of that rule became apparent. For the aluminum and titanium films, the standard prediction was off by five to twenty percent. While this might seem close, it represents a meaningful error in precision engineering. For the sputtered tantalum film, the error was far more severe, with the standard rule underestimating the thermal resistance by about forty percent. This large gap proves that the simple conversion factor fails for highly disordered, resistive films. The researchers calculated an "apparent Lorenz number," a value that describes the link between heat and electricity flow. In a perfect world, this number would be constant. In their films, however, it varied widely with temperature and the type of disorder present. For the sputtered tantalum, the value was so different from the standard that it indicated a fundamental shift in how heat and electricity move through the material, likely due to the chaotic internal structure and the presence of different atomic phases.

These findings offer a necessary correction for the design of future microelectronic devices. The study confirms that while the old rule of thumb might provide a rough guess for well-made, low-resistance films, it is unreliable for the highly disordered films often found in advanced manufacturing. Relying on the standard conversion for these materials could lead to significant miscalculations in how much a device heats up, potentially causing failure. The researchers conclude that for critical applications, especially those involving highly resistive or disordered metal films, engineers must measure the thermal properties directly rather than guessing from electrical data. By providing these precise measurements, the study gives designers the real data they need to model heat flow accurately, ensuring that the next generation of electronics remains cool and reliable.

Technical Summary: Experimental Assessment of the Wiedemann–Franz Law in Thin Metal Films

Problem Statement
Accurate thermal property data for metal thin films are critical for microelectronic thermal modeling and for interpreting thermoreflectance measurements where signals are sensitive to film thermal transport. In practice, film thermal conductivity is often estimated from electrical resistivity using the Wiedemann–Franz (WF) law ( $k_{WF} = L_0 T / \rho$ ) with the Sommerfeld Lorenz number ( $L_0 = 2.44 \times 10^{-8} \text{ W}\Omega/\text{K}^2$ ). This approach typically neglects phonon contributions and microstructure-dependent scattering. However, in nanostructured films, grain boundaries, deposition-induced defects, and metastable phases can significantly alter the relationship between charge and heat transport compared to bulk metals. While deviations in the Lorenz number have been observed in suspended nanofilms and ultrathin structures, the magnitude and temperature dependence of this deviation for common deposited metal films (specifically Al, Ti, and Ta) prepared by standard industrial methods remain insufficiently quantified.

Methodology
The authors employed a combined experimental approach to directly characterize both thermal and electrical transport in five metal films (Al, Ti, and Ta) deposited via thermal evaporation, e-beam evaporation, and magnetron sputtering on fused silica substrates.

Thermal Characterization (SPS Thermoreflectance): The study utilized Square-Pulsed Source (SPS) thermoreflectance. In this pump-probe configuration, a pump laser modulated by a square waveform generates periodic heating, while a probe laser monitors the time-resolved surface temperature via reflectance changes. By acquiring full periodic waveforms across a broad modulation frequency range (1 Hz to MHz) and varying the laser spot size, the technique decouples sensitivity to thermal diffusivity and thermal mass. This allows for the simultaneous extraction of the film's thermal conductivity ( $k_m$ ) and volumetric heat capacity ( $C_m$ ), provided the film thickness is known.
Electrical Characterization: Electrical resistivity ( $\rho_{meas}$ ) was measured independently using the van der Pauw method over the 150–300 K range.
Comparative Analysis: The measured thermal conductivity ( $k_m$ ) was compared against WF-based estimates ( $k_{WF,model} = L_0 T / \rho_{model}$ ), where $\rho_{model}$ was constructed by adding a residual resistivity offset to bulk temperature-dependent resistivity ( $\rho_{bulk} + \rho_0$ ). Additionally, an "apparent Lorenz number" ( $L_{app} = k_m \rho_{meas} / T$ ) was calculated to quantify the overall heat-charge transport relationship, acknowledging that $k_m$ includes both electronic and phonon contributions.

Key Results

Thermal Conductivity vs. Microstructure: The measured thermal conductivity ( $k_m$ $k_{m}$ ) exhibited strong dependence on the deposition method and microstructure.
- Aluminum: Thermally and e-beam evaporated films approached bulk values. Magnetron-sputtered Al showed a substantial reduction (50–60% of bulk) due to enhanced grain-boundary and defect scattering.
- Titanium: Sputtered Ti showed an intermediate reduction (30–40% below bulk).
- Tantalum: Sputtered Ta exhibited the most significant suppression, with $k_m$ nearly an order of magnitude lower than bulk $\alpha$ -Ta, attributed to strong disorder and possible $\beta$ -Ta phase formation.
Volumetric Heat Capacity: In contrast to thermal conductivity, the volumetric heat capacity ( $C_m$ ) of all films agreed with bulk reference values within $\pm 8\%$ , regardless of deposition method. This validates the SPS extraction protocol and suggests $C_m$ can be approximated by bulk values in modeling.
Deviation from WF Estimates: Comparing measured $k_m$ $k_{m}$ with WF-based estimates revealed significant discrepancies:
- For Al and Ti films, deviations ranged from 5% to 20%.
- For sputtered Ta, the deviation reached approximately 40% at 300 K.
Apparent Lorenz Number ( $L_{app}$ ): $L_{app}$ $L_{a pp}$ varied systematically with deposition route and temperature.
- Al films showed $L_{app}$ near or slightly below $L_0$ at low temperatures, rising above $L_0$ near room temperature.
- Sputtered Ti and Ta films exhibited $L_{app}$ values consistently above $L_0$ (up to $3.7 \times 10^{-8} \text{ W}\Omega/\text{K}^2$ for Ta), indicating that using the fixed Sommerfeld value $L_0$ would underestimate the total thermal conductivity in these disordered films.
Resistivity Behavior: While some films (thermally evaporated Al, sputtered Ti) followed a residual-resistivity model ( $\rho_{bulk} + \rho_0$ ), sputtered Al and Ta showed non-parallel temperature dependence relative to bulk, indicating that simple residual-offset models introduce additional uncertainty in WF estimates for highly disordered films.

Significance and Claims
The paper claims to clarify the applicability of the WF law to deposited metal thin films and offers practical guidance for thermal modeling and thermoreflectance analysis. The authors assert that:

Limitations of Universal $L_0$ : Applying the bulk Sommerfeld Lorenz number to highly resistive or strongly disordered metal films (specifically sputtered Ta) leads to substantial systematic errors (up to 40%), well outside experimental uncertainty.
Deposition-Specific Data: For low-resistivity, weakly disordered films, WF-based estimates may serve as a first approximation with a 10–20% uncertainty margin. However, for highly resistive films, direct thermal characterization is necessary to avoid significant under- or over-prediction of local temperature rise in microelectronic devices.
Thermoreflectance Implications: While transducer thermal conductivity is often a minor input in standard cross-plane thermoreflectance, deposition-specific $k_m$ values are crucial when measurement geometries retain sensitivity to lateral heat spreading or when high-resistivity transducers are used.
Interpretation of $L_{app}$ : The apparent Lorenz number should be viewed as an experimental descriptor of the total heat-charge transport relationship (including phonon contributions and modified electronic transport) rather than a direct measurement of the intrinsic electronic Lorenz number.

The study concludes that while volumetric heat capacity can generally be approximated by bulk values, thermal conductivity requires deposition-specific inputs, and reliance on the WF law with fixed parameters is insufficient for accurate modeling of disordered thin films.

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