Inversion for Thermal Properties with Frequency Domain Thermoreflectance

At a Glance

Metadata	Details
Publication Date	2024-01-09
Journal	ACS Applied Materials & Interfaces
Authors	Benjamin Treweek, Volkan Akçelik, Wyatt Hodges, Amun Jarzembski, Matthew Bahr
Institutions	Sandia National Laboratories
Citations	10
Analysis	Full AI Review Included

Executive Summary

This research introduces a sophisticated methodology for non-destructive evaluation (NDE) of bond quality in 3D integrated microelectronic systems, specifically focusing on GaN-diamond interfaces.

Core Challenge Addressed: Traditional Frequency Domain Thermoreflectance (FDTR) analysis is limited to highly symmetric geometries. This work overcomes this by integrating FDTR data with complex 3D Finite Element Method (FEM) simulations.
Methodological Breakthrough: A gradient-based optimization technique (adjoint method) is coupled with High-Performance Computing (HPC) FEM simulations, allowing the inversion of FDTR phase-lag data to determine spatially varying thermal properties in arbitrary, complex geometries.
Spatial Mapping Capability: The technique successfully inverts hyperspectral FDTR data to create a spatial map of thermal conductivity (and thus Thermal Boundary Conductance, TBC) in the buried, unknown bonding layer.
Bond Quality Quantification: Inversion results clearly differentiate between unbonded (poor), partially bonded (intermediate), and well-bonded (good) regions along a measurement line.
Quantitative Results: Inferred TBC values ranged from less than 10 MW/m²K (unbonded) to greater than 100 MW/m²K (well-bonded), providing a critical metric for manufacturing quality control.
Engineering Significance: This framework enables advanced characterization and failure analysis for heterogeneously integrated devices, where thermomechanical stress and reliability are major concerns.

Technical Specifications

Parameter	Value	Unit	Context
Pump Beam Radius (w_pump)	3.46	µm	FDTR Experiment Setup
Probe Beam Radius (w_probe)	2.75	µm	FDTR Experiment Setup
Pump Frequency Range (f_pump)	1 kHz to 60	MHz	Range for hyperspectral data cube
Thermal Penetration Depth (δ_th) Range	3.3 µm to 0.1 mm	µm	Corresponds to 1.5 MHz to 1 kHz f_pump
GaN Layer Thickness (d₂)	5	µm	Thinned layer thickness
Au Transducer Layer Thickness (d₁)	134	nm	Fitted value (Au/GaN interface)
Unknown Layer Thickness (L)	100	nm	FEM model input (Bond layer)
Unbonded TBC (G₂)	3.84 ± 0.23	MW/m²K	Fitted value for poor thermal interface
Well-Bonded TBC (G₂)	> 100	MW/m²K	Conservative estimate for good thermal bond
Inferred Thermal Conductivity (Poor Bond)	0.472	W/mK	Inversion result (Location 3)
Inferred Thermal Conductivity (Intermediate Bond)	1.784	W/mK	Inversion result (Location 2)
FEM Mesh Size	231,537	nodes	Total nodes in the 3D quadratic hexahedral mesh
GaN Thermal Conductivity (k₂)	124 ± 9.3	W/mK	Fitted value for bonded region

Key Methodologies

The inversion process relies on a combination of advanced experimental data acquisition and HPC-enabled computational modeling.

Sample Fabrication (GaN-Diamond Stack):
- GaN and Diamond substrates were coated with a TiAu film (5 nm Ti / 120 nm Au).
- Surfaces were cleaned using Ar plasma etching (100 W).
- A cold weld was created via thermocompression bonding (2 kN force, 15 seconds) to promote bonding of the gold surfaces.
- The GaN layer was subsequently thinned and polished to approximately 5 µm.
FDTR Data Acquisition:
- A wide-bandwidth FDTR system was used to measure the phase lag (θ) between the applied heat flux (pump beam) and the surface temperature response (probe beam).
- Hyperspectral mapping was performed by raster-scanning the sample across a wide frequency range (1 kHz to 60 MHz) to obtain depth-dependent thermal information.
Forward Model (FEM Simulation):
- The time-harmonic heat diffusion equation was solved using the Finite Element Method (FEM) within the Sierra/SD structural dynamics code.
- The 3D geometry (Au/GaN/Unknown/Diamond) was discretized using a highly refined mesh (231,537 nodes) to handle the high frequencies (up to 1 MHz used for inversion) and small feature sizes.
- The computed temperature integral (H) was calculated via a surface integral over the measurement area, mimicking the probe beam measurement.
Inverse Problem Formulation:
- The determination of unknown thermal properties (k and ρc_p) in the bonding layer was formulated as a PDE-constrained optimization problem.
- The objective function (J) was defined as the L2-norm difference between the measured and computed phase lags across multiple frequencies and measurement points.
Gradient-Based Optimization:
- The inverse problem was solved using the adjoint method, implemented via the Rapid Optimization Library (ROL).
- This method efficiently calculates the gradient of the objective function with respect to the unknown material properties, allowing for iterative adjustment of thermal conductivity in the discretized domain (element-by-element).
Spatial Inversion:
- For spatial mapping, a heterogeneous model was used, allowing thermal properties to vary along a 1D line segment (40 µm) in the unknown layer, resulting in a spatial map of bond quality (TBC).

Commercial Applications

The ability to nondestructively map subsurface thermal properties in complex 3D stacks is critical for several high-reliability and high-performance engineering sectors.

Heterogeneous Integration (HI): Essential for quality control and reliability assessment in advanced 3D packaging, where combining dissimilar materials (e.g., Si, GaN, Diamond) leads to significant thermomechanical stresses.
High-Power Electronics: Directly applicable to GaN-on-Diamond devices, which are used in high-frequency and high-power applications (e.g., 5G/6G infrastructure, radar). Accurate TBC mapping ensures efficient heat dissipation, preventing thermal runaway and failure.
Quality Assurance and Manufacturing Screening: Provides a non-destructive method to screen for manufacturing defects (voids, poor adhesion) in metallic interconnects and thermocompression bonds, replacing destructive techniques like electron microscopy.
Reliability and Lifetime Prediction: Enables characterization of bond degradation over time due to fatigue from power cycling, providing data crucial for predicting device lifetime and improving material selection.
Thermal Management Solutions: Offers detailed, spatially resolved data necessary for validating and refining thermal models used in the design of advanced microelectronic cooling systems.

View Original Abstract

3D integration of multiple microelectronic devices improves size, weight, and power while increasing the number of interconnections between components. One integration method involves the use of metal bump bonds to connect devices and components on a common interposer platform. Significant variations in the coefficient of thermal expansion in such systems lead to stresses that can cause thermomechanical and electrical failures. More advanced characterization and failure analysis techniques are necessary to assess the bond quality between components. Frequency domain thermoreflectance (FDTR) is a nondestructive, noncontact testing method used to determine thermal properties in a sample by fitting the phase lag between an applied heat flux and the surface temperature response. The typical use of FDTR data involves fitting for thermal properties in geometries with a high degree of symmetry. In this work, finite element method simulations are performed using high performance computing codes to facilitate the modeling of samples with arbitrary geometric complexity. A gradient-based optimization technique is also presented to determine unknown thermal properties in a discretized domain. Using experimental FDTR data from a GaN-diamond sample, thermal conductivity is then determined in an unknown layer to provide a spatial map of bond quality at various points in the sample.

Tech Support

Original Source

DOI: https://doi.org/10.1021/acsami.3c13658