Limits to Fourier theory in high thermal conductivity single crystals

At a Glance

Metadata	Details
Publication Date	2015-11-16
Journal	Applied Physics Letters
Authors	Richard B. Wilson, David G. Cahill
Institutions	University of Illinois Urbana-Champaign
Citations	46
Analysis	Full AI Review Included

6CCVD Technical Analysis & Material Solutions Briefing:

Limits to Fourier Theory in High Thermal Conductivity Single Crystals

This technical brief analyzes the critical findings of Wilson and Cahill regarding thermal transport in high-conductivity single crystals, specifically focusing on the performance of SCD diamond when microscale heat sources are utilized. This research validates diamond as the premier material for thermal management in next-generation nanoelectronics where heat source dimensions are comparable to phonon mean-free-paths (MFP).

Executive Summary

Validation of TDTR: Time-Domain Thermoreflectance (TDTR) was confirmed as a robust metrology tool for accurately measuring the bulk thermal conductivity ($\Lambda$) of nonmetallic single crystals across a wide range of laser spot sizes.
Failure of Fourier Theory: The predictive power of standard Fourier heat diffusion theory breaks down in high thermal conductivity materials (SiC, Ge, GaAs) when heater dimensions approach submicron scales ($w_0 < 2 \mu m$), indicating ballistic phonon transport effects.
Diamond Superiority: Single Crystal Diamond (SCD) exhibited the smallest deviation from Fourier theory predictions compared to all other high-K materials studied, confirming its robust performance under conditions relevant to modern nanoelectronics.
High Thermal Conductivity Confirmed: Measured SCD diamond apparent thermal conductivity ($\Lambda_A$) was confirmed at $\approx 2100$ W m^-1 K^-1 (at $w_0 \approx 5 \mu m$).
Optimal Phonon Scattering: SCD’s thermal response suggests that its heat-carrying phonon population is weighted toward higher frequencies (modeled by an $n=1$ scattering rate), resulting in shorter effective mean-free-paths (0.3 µm to 3 µm) and minimizing long-range ballistic transport effects.
Application Relevance: The study concludes that the breakdown of Fourier theory is not a major impediment to utilizing SCD diamond as a high-performance heat spreader (e.g., in GaN HEMT devices) where active region dimensions are submicron.
Required Thin Films: Experimentation necessitated the precise deposition of ultra-thin Aluminum (Al) transducer layers (45 nm to 90 nm) to facilitate accurate TDTR measurements.

Technical Specifications

Parameter	Value	Unit	Context
Laser Spot Size Radius Range ($w_0$)	0.7 to 12	µm	Range used for TDTR experiments
Pump Modulation Frequency ($f$)	9.8	MHz	Standard TDTR operational frequency
SCD Diamond $\Lambda_A$ (In-Plane, Bulk Limit)	$\approx 2100$	W m^-1 K^-1	Apparent thermal conductivity measured at $w_0 \approx 5 \mu m$
6H-SiC $\Lambda_A$ (In-Plane, $1 \perp c$)	$\approx 340$	W m^-1 K^-1	Apparent $\Lambda_A$ measured at $w_0 \approx 1.2 \mu m$
Fourier Theory Deviation (SiC/GaAs/Ge)	15 to 25	%	Difference between $\Lambda_A(w_0 \approx 1 \mu m)$ and $\Lambda_A(w_0 \approx 12 \mu m)$
Thermal Transducer Material	Al	-	Deposited thin film for TDTR measurement
Transducer Film Thickness	45 to 90	nm	Measured via picosecond acoustics
Diamond Phonon Mean-Free-Path (RTA n=1)	0.3 to 3	µm	Range carrying 80% of thermal energy
Measured $\Lambda$ Agreement (Large $w_0$)	$\pm 7$	%	Experimental uncertainty relative to literature values

Key Methodologies

The TDTR methodology was utilized to probe the thermal response of various single crystal substrates coated with a metallic transducer.

Sample Preparation and Transducer Coating: High thermal conductivity single crystals (including SCD diamond, 6H-SiC, Ge, GaAs) were coated with thin metallic transducer layers (Aluminum) ranging from 45 nm to 90 nm thick.
TDTR System Setup: The samples were excited by a periodically modulated train of pump laser pulses at a frequency of 9.8 MHz. A time-delayed probe beam monitored the temperature-induced changes in the intensity of the reflected light.
Spot Size Control: The laser spot size radius ($w_0$) was systematically varied across a broad range, specifically focusing on the critical submicron dimensions (0.7 µm to < 2 µm) where ballistic heat transfer effects are pronounced.
Data Extraction via Heat Diffusion Model: An isotropic heat diffusion model (based on Fourier theory) was used to fit the in-phase and out-of-phase TDTR voltage signals. The fitting parameters extracted were the apparent thermal conductivity ($\Lambda_A$) and the interfacial thermal conductance ($G$).
Assessment of Fourier Limit Failure: The dependence of $\Lambda_A$ on the laser spot size ($w_0$) was cataloged. A decrease in $\Lambda_A$ at small $w_0$ relative to bulk values (measured at large $w_0$) signified the failure of the continuum Fourier model due to unscattered phonon transport.
Ballistic/Diffusive Modeling: Theoretical predictions were generated using a Relaxation Time Approximation (RTA) model coupled with a ballistic/diffusive transport model. Different phonon scattering exponents ($n=1$ for diamond, $n=2$ for others) were applied to match the observed experimental deviations.

6CCVD Solutions & Capabilities

The research highlights the critical necessity of high-purity, high-thermal-conductivity Single Crystal Diamond (SCD) that performs reliably in submicron environments, coupled with precise metalization for accurate thermal characterization and device integration. 6CCVD is uniquely positioned to supply the materials and engineering services required to replicate and advance this research.

Applicable Materials

To achieve and surpass the $2100$ W m^-1 K^-1 thermal conductivity demonstrated in this study for heat spreader applications, 6CCVD recommends:

Optical Grade Single Crystal Diamond (SCD): High purity (low nitrogen/defects) maximizing thermal conductivity and minimizing phonon scattering unrelated to inherent crystal structure, ensuring minimal deviation from Fourier theory at critical submicron length scales.
- Specification: SCD plates/wafers with thickness from 0.1 µm up to 500 µm, ideal for heterogenous integration with materials like GaN.
High-Quality Polycrystalline Diamond (PCD): While the paper focused on SCD, high-quality PCD can be manufactured by 6CCVD with high grain boundary quality to limit ballistic effects, potentially offering a cost-effective alternative for certain applications, with sizes up to 125 mm (inch-size).

Customization Potential

Accurate TDTR requires precise dimensions and specific metal stacks to function as the heat transducer. 6CCVD capabilities directly address the needs documented in this study:

Research Requirement	6CCVD Capability	Benefit to Customer
Ultra-thin Al Transducer (45-90 nm)	Custom Metalization Services	Capability for depositing single or multi-layer stacks (Au, Pt, Pd, Ti, W, Cu) at nanometer precision to optimize transducer function and device integration.
Large Area Substrates (SiC/Diamond)	Large Diameter PCD Wafers	Availability of PCD substrates up to 125 mm diameter, suitable for scaling up thermal management research and manufacturing processes.
Sample Geometry (Small Spots $< 2 \mu m$)	Precision Polishing & Finishing	SCD polishing to Ra < 1 nm, essential for minimizing interface scattering and ensuring reliable thin film deposition and TDTR measurement fidelity.
Non-standard Dimensions	Laser Cutting & Shaping	In-house capability for custom shaping and laser cutting of SCD/PCD to match specific device footprints or experimental configurations.
Global Logistics	Global Shipping DDU/DDP	Reliable, rapid global shipping ensures researchers and engineers worldwide receive materials in specification and on time.

Engineering Support

The robust thermal performance of diamond in the submicron regime is contingent on controlling phonon mean-free-path distribution. 6CCVD’s in-house PhD material science team specializes in manipulating diamond growth parameters to optimize thermal properties for specific applications.

Heat Spreader Design: Our team can provide consultative support for projects focused on near-junction thermal management (e.g., GaN high electron mobility transistors, HEMTs) where minimizing ballistic transport effects is critical.
Material Selection: We assist engineers in selecting the optimal SCD or PCD grade, thickness, and crystallographic orientation (relevant for anisotropic materials like SiC but minimized in isotropic SCD) to meet stringent thermal load requirements.

For custom specifications or material consultation, visit 6ccvd.com or contact our engineering team directly.

View Original Abstract

We report the results of time-domain thermoreflectance (TDTR) experiments that examine the ability of Fourier theory to predict the thermal response in single crystals when heater dimensions are small. We performed TDTR measurements on Al-coated diamond, 6H-SiC, GaP, Ge, MgO, GaAs, and GaSb single crystals with a wide range of laser spot size radii, 0.7 μm &lt; w0 &lt; 12 μm. When the laser spot-size is large, w0 ≈ 12 μm, TDTR data for all crystals are in agreement with predictions of Fourier theory with bulk thermal conductivity values. When the laser spot-size is small, w0 &lt; 2 μm, there are significant differences between the predictions of Fourier theory and TDTR data for all crystals except MgO.

Tech Support

Original Source

DOI: https://doi.org/10.1063/1.4935987

References

2014 - Near-junction thermal management: Thermal conduction in gallium nitride composite substrates

Limits to Fourier theory in high thermal conductivity single crystals

At a Glance

6CCVD Technical Analysis & Material Solutions Briefing: