Session: 11-09-01: Thermal Transport Across Interfaces I

Paper Number: 77267

Start Time: Wednesday, 10:45 AM

77267 - A Scattering Matrix Scheme to Model the Periodic Heating Problem in Layered Structures 

The transfer matrix approach has been widely applied to model heat transfer problems in layered structures, since Feldman first introduced it to heat transfer community [1]. For example, Borca-Tasciuc et. al. applied this scheme to the 3Omega method [2]. Cahill extended it to the time domain thermoreflectance method [3].

However, in the heat transfer community, it is not yet recognized that this transfer matrix scheme has an intrinsic numerical instability problem, originating from the fact that one diagonal element of the matrix exponentially decays while the other diagonal element exponentially blows up as the film thickness and the heating frequency increases.  This issue has not yet drawn enough attentions since it only manifests itself when the heating frequency or the total thickness of the multilayer stack increases to some critical values.  For example, for a Si- SiO₂ (50nm/50nm) alternating multilayer stack, when the total thickness is larger than 5.5 microns and the heating frequency is higher than 0.7 GHz , the transfer matrix scheme is likely to fail. This 0.7 GHz heating frequency was thought to be unpractical before, but is becoming reasonable in the context of the ever-growing clock frequency of CPUs [4].

On the other hand, in the electromagnetism and optics community, this numerical instability issue has been solved by replacing the transfer matrix approach to a scattering matrix scheme[5]. The core idea of the scattering matrix is to regroup the inputs and outputs bridged by the matrix such that both of the diagonal elements exponentially decay together, which avoid the numerical problem above.  However, as far as we know, heat transfer community has not yet employed this scattering matrix to analyze multilayer structures.

To address the shortcoming of the transfer matrix approach, here we borrow the scattering matrix scheme from the electromagnetism and optics community, and develop a general framework to model periodic heating problems in multilayer structures. In particular, our framework does not require that the heater has to be on top of the sample, but instead it can be buried in any location of the multilayer stack, which significantly releases the experimental design in real-world applications, e.g. the in-situ monitoring of thermal properties of micro-chips using existing metal lines pre-deposited inside the structure. As sanity checks, our framework successfully recovers several well-known solutions, including the simplest heater-on-substrate scenario [6], the heater-on-film-substrate scenario [7], the heater-on-multilayer-substrate scenario [2], and the substrate-heater-substrate scenario [8]. We construct regime maps and give thumb-rules to indicate in which circumstances the transfer matrix approach is anticipated to fail while the scattering matrix is still valid. We further discuss the constraints on the scattering matrix resulting from the energy conservation, reciprocity, and time reversal symmetry.

 

 

References:

1. Feldman, A. Algorithm for Solutions of the Thermal Diffusion Equation in a Stratified Medium with a Modulated Heating Source. High Temp.-High Press. 31, 293-298 (1999).

2. Borca-Tasciuc, T., Kumar, A. R. & Chen, G. Data Reduction in 3Ω Method for Thin-Film Thermal Conductivity Determination. Rev. Sci. Instrum. 72, 2139-2147 (2001).

3. Cahill, D. G. Analysis of Heat Flow in Layered Structures for Time-Domain Thermoreflectance. Rev. Sci. Instrum. 75, 5119-5122 (2004).

4. https://www.ibm.com/cloud/dedicated.

5. Nayfeh, A. H. Wave Propagation in Layered Anisotropic Media. (Elsevier, 1995).

6. Cahill, D. G. Thermal Conductivity Measurement From 30 to 750 K: The 3Omega Method. Rev. Sci. Instrum. 61(2) (1990).

7. Lee, S. M. & Cahill, D. G. Heat Transport in Thin Dielectric Films. J. Appl. Phys. 81, 2590-2595 (1997).

8. Sean D. Lubner et al. Reusable Bi-Directional 3 Ω Sensor to Measure Thermal Conductivity of 100-Μm Thick Biological Tissues. Rev. Sci. Instrum. 86, 14905 (2015).

 

Presenting Author: Tao Li Southeast University

Authors:

Tao Li Southeast University
Zhen Chen Southeast University