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Fresnel transmission coefficients for thermal phonons at solid interfaces

Interfaces play an essential role in phonon-mediated heat conduction in solids, impacting applications ranging from thermoelectric waste heat recovery to heat dissipation in electronics. From a microscopic perspective, interfacial phonon transport is described by transmission and reflection coefficients, analogous to the well-known Fresnel coefficients for light. However, these coefficients have never been directly measured, and thermal transport processes at interfaces remain poorly understood despite considerable effort. Here, we report the first measurements of the Fresnel transmission coefficients for thermal phonons at a metal-semiconductor interface using ab-initio phonon transport modeling and a thermal characterization technique, time-domain thermoreflectance. Our measurements show that interfaces act as thermal phonon filters that transmit primarily low frequency phonons, leading to these phonons being the dominant energy carriers across the interface despite the larger density of states of high frequency phonons. Our work realizes the long-standing goal of directly measuring thermal phonon transmission coefficients and demonstrates a general route to study microscopic processes governing interfacial heat conduction.

Transport across interfaces in heterogeneous media is a fundamental physical process that forms the basis for numerous widely used technologies. For example, the reflection and transmission of light at interfaces, as described by the Fresnel equations, enables waveguiding with fiber-optics and anti-reflection coatings, among many other functionalities.
Interfaces also play an essential role in phonon-mediated heat conduction in solids. 1 Material discontinuities lead to thermal phonon reflections that are manifested on a macroscopic scale as a thermal boundary resistance (TBR), also called Kapitza resistance, R k , that relates the temperature drop at the interface to the heat flux flowing across it. TBR exists at the interface between any dissimilar materials due to differences in phonon states on each side of the interface. 2 Typical interfaces often possess defects or roughness which can lead to additional phonon reflections and hence higher TBR.
TBR plays an increasingly important role in devices, particularly as device sizes decrease below the intrinsic mean free paths (MFPs) of thermal phonons. 1 At sufficiently small length scales, TBR can dominate the total thermal resistance. For instance, the effective thermal conductivity of a superlattice can be orders of magnitude smaller than that of the constituent materials due to high TBR. [3][4][5][6] This physical effect has been used to realize thermoelectrics with high efficiency 7,8 and dense solids with exceptionally low thermal conductivity. 9 On the other hand, TBR can lead to significant thermal management problems [10][11][12] in applications such as LEDs 13,14 and high power electronics. 12,15 Thus both scientifically and for applications, a fundamental understanding of thermal transport across solid interfaces is essential. In principle, Fresnel transmission coefficients can also be used to provide a microscopic description of thermal phonon transport at interfaces owing to the similarities between photons and phonons. However, despite decades of work, the microscopic perspective of heat transport across interfaces remains poorly developed compared to that available for photons. Today, interfaces are most often studied using macroscopic measurements of TBR or thermal conductivity. For example, numerous works have studied interfacial thermal transport by observing the temperature dependence of the thermal conductivity 16 or interface conductance, G = 1/R k 17-21 or by correlating changes in bonding strength and interface conductance. 22,23 However, these experimental approaches provide limited information about the transmission coefficients because the observable quantities are averaged over all phonons and thus obscure the microscopic processes of interest.  [50][51][52][53][54][55] In this work, we interpret these effects as fundamentally originating from the non-equilibrium phonon distribution emerging from the interface. As While the required data is straightforward to obtain, the key to extracting the transmission coefficients is to rigorously interpret the data with the ab-inito phonon transport modeling based on the BTE. It is this step that has long impeded efforts to study interfaces due to the extreme cost of solving the BTE for the TDTR experiment. A number of simplified models [55][56][57][58][59][60][61] have been proposed to explain these observations. However, all of these models make various approximations that limit the information that can be extracted from the measurements.
In this work, we overcome this challenge using two recent advances we reported for rigorously solving the spectral BTE under the relaxation time approximation (RTA), with no additional simplifications, that yield a factor of 10 4 speedup compared to existing methods and allows the first ab-initio phonon transport modeling of TDTR free of artificial parameters or simplifications of the phonon dispersion. First, we have obtained an analytical solution of the spectral BTE in a semi-infinite substrate subject to an arbitrary heating profile. 62 Second, we have employed a series expansion method to efficiently solve the spectral, one-dimensional (1D) BTE in a finite layer, suitable for the transducer film. 63 In this work, these two solutions are combined using a spectral interface condition 57 that expresses the conservation of heat flux at each phonon frequency. Following the conclusions of prior computational works 17,64 that the energy transmission at the interfaces considered here is elastic, we enforce that phonons maintain their frequencies as they transmit through the interface; direct electronphonon coupling and inelastic scattering is neglected (see Supplementary Information). thermal measurements on variable thickness silicon membranes. 69 This fact implies that quasiballistic transport should be readily observable in a typical TDTR experiment on Si, despite the seemingly correct thermal properties measured.
This apparent contradiction can be resolved by considering the spectral profile of the transmission coefficients. Our BTE calculations reveal that the measurements from TDTR on this sample strongly depend on the spectral profile of the transmitted phonon spectrum, a dependence that does not occur in the heat diffusion regime. This dependence allows the transmission coefficients to be directly obtained from the TDTR data by adjusting them to fit the surface temperature data from the lock-in amplifier in a procedure exactly analogous to that used to measure bulk thermal properties.
We performed the fitting of transmission coefficients by adjusting them until the simulated surface temperature curve and experimental data from the lock-in amplifier matched each other. This comparison for all samples can be found in Supplementary Information. To more compactly report the data, we further process the BTE results by fitting the simulated surface temperature decay curves to the same heat diffusion model used in the experiments to extract the apparent thermal conductivity and interface conductance at different modulation frequencies. If these two parameters at each modulation frequency match, then the fitting curves will also match, enabling a compact comparison of the simulation and experimental data sets. However, we emphasize that the use of the heat diffusion model is for ease of comparison only and was not used in the transmission coefficient measurement. So far, the measurements indicate that low frequency phonons are nearly completely transmitted through a clean interface. We next seek to determine what types of interfacial disorder can reflect these modes. We conducted additional measurements for Al on Si with a native oxide layer (thickness ∼ 1 nm as shown in a TEM image in Fig. 4(a)) and Si with thermally grown oxide layer (thickness ∼ 3.5 nm as shown in a TEM image in Fig. 4(b)).
Since the oxide layers are sufficiently thin to neglect their thermal capacitance, we can treat them as part of the interface that modifies the transmission coefficient profile in our current BTE model. and more phonons, especially those with wavelengths between 1 and 2 nm, are reflected at the interface. Therefore, our measurements show that phonons with wavelength shorter than the interface roughness are more likely to be reflected by the interface than long-wavelength phonons, and as the interface gets rougher, a larger fraction of the phonon spectrum is affected by the interface.

DISCUSSION
Our work has considerable implications for thermal metrology and technological applications. First, TDTR is a widely used characterization tool to measure the thermal properties of thin films and bulk materials. However, this work demonstrates that in the quasiballistic regime, the apparent thermal conductivity and interface conductance measured by TDTR may not reflect bulk properties because both are strongly affected by the transmission coefficient profile. The apparently correct measurement of silicon thermal conductivity appears to be a fortuitous cancellation of two factors: the high transmission coefficient of low frequency phonons leads to an increased contribution to heat flux that offsets the deviation from Fourier's law that occurs due to a lack of scattering. If these two factors were not balanced, the apparent thermal conductivity of Si would not coincide with the bulk value.
In fact, these factors are not balanced in SiGe, resulting in an apparent thermal conductivity that is substantially smaller than the actual value. Additionally, the apparent thermal interface conductance measured in SiGe deviates from the value measured in Si despite the fact that the transmission coefficients are exactly the same. These observations are further evidence that the apparent thermal properties extracted in a conventional TDTR interpretation approach do not necessarily reflect the actual physical properties of the materials.
Although this situation is undesirable from a thermal metrology perspective, our work shows that TDTR is capable of providing considerable microscopic detail about thermal phonons provided the measurements can be properly interpreted.
Second, our measurements show that the spectral profile of transmission coefficients is essential to understanding thermal transport across interfaces. Due to a lack of knowledge about interfaces, the phonon transmission coefficients are often taken to be a constant or calculated using the DMM. However, this work shows that both of these simple models are incapable of explaining the experimental measurements. Although the gray model and measured transmission coefficient profiles yield the same apparent interface conductance, the spectrum of heat transmitted across the interface is completely different, resulting in opposite conclusions regarding the primary heat carriers across the interface. Therefore, considering the spectral transmission coefficient profile across an interface is essential to accurately describing thermal phonon transport.  Ab-initio modeling Thermal transport in TDTR experiments, assuming only cross-plane heat conduction, is described by the spectral, one-dimensional (1D) BTE under RTA. 72 Efficiently solving this equation for the conditions of the TDTR experiment required two of our recent computational advances. First, we have obtained an analytical solution of the spectral BTE in a semi-infinite substrate subject to an arbitrary heating profile. 62 Second, we have employed a series expansion method to efficiently solve the spectral 1D BTE in a finite layer, suitable for the transducer film. 63 In this work, these two solutions are combined using a spectral interface condition 57