# An Analytical Modeling Study on the Thermal Behavior of Copper–Carbon Nanotube Composite Through-Silicon Via (TSV)

## TL;DR

This study models the thermal behavior of copper-carbon nanotube composites in through-silicon vias, finding that thermal conductivity does not significantly increase with higher CNT volume due to interfacial resistance.

## Contribution

A novel analytical model for thermal conductivity of Cu-CNT TSVs using Monte Carlo and Gauss–Hermite quadrature, validated against experimental data.

## Key findings

- Thermal conductivity of TSV does not significantly increase with CNT volume fraction due to interfacial thermal resistance.
- The model maintains less than 2% error for CNT volume fractions below 10%.
- The effective Cu-CNT interfacial thermal resistance is estimated to be on the order of 10−7 m2K/W.

## Abstract

In this study, the Monte Carlo (MC) method is employed to generate the diameter and relative positional distributions of carbon nanotubes (CNTs). Based on this, we develop a three-layer thermal model for a copper-carbon nanotube (Cu-CNT) through-silicon via (TSV). By integrating Gauss–Hermite quadrature with the Law of Large Numbers (LLN), an analytical expression for thermal conductivity is derived, enabling efficient and accurate estimation of the thermal conductivity of Cu-CNT-filled TSV. Contrary to expectations, the thermal conductivity of TSV does not increase significantly with CNT volume fraction, primarily due to the interfacial thermal resistance at Cu-CNT and CNT-CNT junctions. Through calibration against previously reported experimental data, the effective Cu-CNT interfacial thermal resistance is estimated to be on the order of 10−7 m2K/W. Comparison with previously reported effective thermal conductivity data of Cu-CNT composites shows that the model maintains an error below 2% when the CNT volume fraction is below 10%. The model is therefore most suitable for low CNT volume fractions, where the assumed spatial distribution and structural simplifications remain physically valid. Furthermore, this study investigates the influence of TSV length on thermal performance, predicts the variation in thermal conductivity of Cu-CNT composites under different volume fractions, and the extracted thermal conductivity values are further used as material inputs for device-level electro-thermal COMSOL 6.1 simulations.