Thermal Conductance Across Harmonic-matched Epitaxial Al-sapphire Heterointerfaces: A Benchmark for Metal-nonmetal Interfaces

A unified understanding of interfacial thermal transport is missing due to the complicated nature of interfaces which involves complex factors such as interfacial bonding, interfacial mixing, surface chemistry, crystal orientation, roughness, contamination, and interfacial disorder. This is especially true for metal nonmetal interfaces which incorporate multiple fundamental heat transport mechanisms such as elastic and inelastic phonon scattering as well as electron phonon coupling in the metal and across the interface. All these factors jointly affect thermal boundary conductance (TBC). As a result, the experimentally measured interfaces may not be the same as the ideally modelled interfaces, thus obfuscating any conclusions drawn from experimental and modeling comparisons. This work provides a systematic study of interfacial thermal conductance across well controlled and ultraclean epitaxial (111) Al parallel (0001) sapphire interfaces, known as harmonic matched interface. A comparison with thermal models such as atomistic Green s function (AGF) and a nonequilibrium Landauer approach shows that elastic phonon scattering dominates the interfacial thermal transport of Al sapphire interface. By scaling the TBC with the Al heat capacity, a nearly constant transmission coefficient is observed, indicating that the phonons on the Al side limits the Al sapphire TBC. This nearly constant transmission coefficient validates the assumptions in AGF and nonequilibrium Landauer calculations. Our work not only provides a benchmark for interfacial thermal conductance across metal nonmetal interfaces and enables a quantitative study of TBC to validate theoretical thermal carrier transport mechanisms, but also acts as a reference when studying how other factors impact TBC.

To explain experimental results, theories of interfacial thermal conductance have been developed since 1950s, such as the acoustic mismatch model (AMM) and the diffuse mismatch model (DMM). 5,6,35 More recently, other theoretical tools were used to calculate TBC, for instance, equilibrium/non-equilibrium molecular dynamics (MD) [36][37][38] , interface conductance modal analysis (ICMA), 39,40 wave packet method 41,42 , atomistic Green's function (AGF) 43,44 , and non-equilibrium Landauer approach 45 . Due to the complicated nature of interfaces, most of the theoretical calculations cannot capture the detailed features of the interface. It still remains an open question whether the modelled interfaces are the same as the measured ones in most of previous works due to the lack of detailed simultaneous material and thermal characterization of the interfaces. 46 This is especially true for metal-nonmetal interfaces because of the complicated interfacial structures and multiple carrier transport mechanisms near the interfaces. Metal growth on nonmetal substrates, usually by sputtering or evaporation, suffers from problems like chemical reaction or interfacial mixing with substrates during deposition processes, the inclusion of an oxide or contaminating layers at the interface, polycrystalline metal films with a mixture of different orientations, or poor adhesion with substrates. 16,25,32,[47][48][49] These result in the growth of metalnonmetal interfaces to deviate from the ideal interface often assumed in theoretical modelling, so direct comparison between theoretical and experimental results limits our ability to draw accurate conclusions. 34,50 Additionally, interfacial thermal transport involves multiple fundamental transport mechanisms, including elastic and inelastic phonon transport across the interface, and electron-phonon coupling in the metal and across the interface. [51][52][53][54][55] The relative contributions of these mechanisms to TBC remain unclear and are still an open question. 51,56 For instance, theoretical calculations show electrons on the metal side can pass some energy directly to phonons on the nonmetal side while experiments show that a 400-fold change in electronic density for otherwise similar metals on the metal side does not impact TBC significantly. 7,13,55,56 A unified understanding of interfacial thermal transport across metal-nonmetal interfaces does not exist possibly due to the lack of systematic benchmark studies of well-controlled interface growth, simultaneous structural and thermal characterizations, and corresponding comparison with thermal modellings because very few computational methods can take interface non-idealities into consideration. 44,57,58 In this work, we fill the gap by epitaxially growing (111) Al on (0001) ultraclean sapphire substrates by Molecular Beam Epitaxy (MBE). The Al-sapphire system is particularly suitable for benchmarking because of the atomically smooth surfaces, easy cleaning by baking at high temperatures in ultrahigh vacuum (UHV) conditions, no surface oxidation during baking, and no reaction with Al during growth. 56 These orientations are selected because of the relatively small lattice mismatch (4%) and similar crystalline structure, as shown in Figure S1. The TBC of Alsapphire interfaces are measured by three TDTR systems in Georgia Institute of Technology, University of Virginia, and University of Notre Dame. Simultaneous structural characterizations are performed with a transmission electron microscope (TEM) and x-ray diffraction (XRD). The calculated TBC by AGF and non-equilibrium Landauer approaches are compared with experimental data over a temperature range of 80-480 K. The phonon and electron transport mechanisms across interfaces and their contributions to TBC are discussed. Our work provides a benchmark for interfacial thermal conductance across metal-nonmetal interfaces and enable quantitative study of TBC to validate theoretical mechanisms.

Results and Discussion
Two samples (Sub100 and Sub200) were studied in the round robin analysis in this work. The substrate temperature for Al growth was kept at 473 K for sample Sub200 and at 373 K for Sub100 while all the other growth conditions remained the same. More details about MBE sample growth can be found in the Methods section. A key step is that the sapphire was cleaned through a high temperature annealing step in UHV prior to the epitaxial deposition of the Al in-situ by MBE.  Al || [1010] Al2O3. We see a sharp, distinct, and well-matched Al-sapphire boundary with evidence of only a sub-nm interfacial re-arrangement. 59 Electron back-scattered diffraction measurements (EBSD) (in SI) confirm the film is completely (111) oriented with the 3 twin boundaries and the non-twinned regions extend as much as several tens of microns. More characterization in the SI also shows that the Al films have no strain. We note that these dimensions are more indicative of a high-quality crystalline film than those had been previously obtained for low temperature Al grown on sapphire. 60 These characteristics support the contention that the Al-sapphire interface is an ideal interface which could be compared with modeling results directly.  Electrons dominate thermal transport in metals while phonons dominate in non-metals. For thermal transport across metal and non-metal interfaces, thermal energy carried by electrons in the metals needs to be transferred to phonons in metals first. Then phonons in the metals transfer energy across the interface to phonons in non-metals through elastic and inelastic processes. Near the Alsapphire interface, we assume that the boundary condition for electron transport is adiabatic while phonons in the Al side can transmit through the interface to the sapphire side. 12,13,54 There is a temperature difference between the near-interface electrons and phonons in the Al side. This local non-equilibrium results in a corresponding electron-phonon coupling thermal resistance in Al.
Phonons in the Al side transfer energy to phonons in the sapphire side through elastic and inelastic channels. Additionally, some theoretical calculations show electrons in the metal side could directly pass energy to phonons in the nonmetal side while some other calculations and experimental data show that this cross-interface electron-phonon coupling does not contribute to TBC significantly. 7,13,51,55,61,62 We add this possible heat transfer channel in Figure 2 For instance, the measured TBC reaches a plateau above room temperature, which has a different trend from the measured values in the literature 63 where the increased TBC with temperature was attributed to inelastic phonon scattering. Many factors, such as crystalline orientation, roughness, and interfacial disorder or contamination, jointly affect TBC of Al-sapphire interfaces. This highlights the importance of a benchmarking study on TBC of ideal interfaces, which can not only be used to validate theoretical thermal models across perfect interfaces, but also act as a reference when studying how other factors impact TBC. non-equilibrium Landauer approach can be found in the Methods section and Ref. 45 The small difference in AGF results between our result and Ref. 64 should be attributed to different force constants. Here we used DFT calculation to generate force constants while Ref. 64  Al. We will discuss the role of each heat transfer mechanism below.
First, electron-phonon coupling in the Al side could result in a thermal resistance in series with the interfacial phonon-phonon thermal resistance. 54 The thermal resistance can be estimated as 1 √ ⁄ . 54 Here, .and are the lattice thermal conductivity and electron-phonon coupling constant of Al. We consider independent of temperature and its value at room temperature is 5.38 ×10 17 W/m 3 -K. 65 of Al at room temperature is 6 W/m-K based on first-principle calculations. 65 The thermal resistance derived from electron-phonon coupling in Al is 0.557 m 2 -K/GW, 19% of the measured overall thermal resistance across Al-sapphire interfaces. As the temperature decreases, the lattice thermal conductivity increases, leading to a reduced electronphonon coupling thermal resistance. The phonon-phonon TBC decreases with temperature, leading to a larger phonon-phonon thermal resistance. As a result, the effect of the electron-phonon coupling thermal resistance on overall TBC would become smaller at low temperatures. However, for temperatures comparable or higher than the Debye temperature of Al (428 K), the lattice thermal conductivity Al decreases with temperature ( ~1⁄ ) so electron-phonon coupling thermal resistance is proportional to √ . From 300 K to 480 K considered in this work, the electron-phonon coupling thermal resistance in the metal side accounts for about 20% of the overall measured thermal resistance across Al-sapphire interfaces.
In terms of cross-interface electron-phonon coupling for Al-sapphire interface, we tend to believe that it does not contribute to TBC significantly because the measured TBC is so close to the elastic phonon-phonon TBC for the whole temperature range, especially for low temperatures where inelastic phonon contribution and electron-phonon coupling thermal resistance in the metal are not important. This cross-interface electron-phonon coupling is an additional thermal channel across the interface which could increase TBC. However, the modeled TBC by AGF and non-equilibrium Landauer approach are slightly larger than the measured TBC at low temperatures. Furthermore, a recent theoretical work shows that this cross-interface coupling effect for Si-Cu interface contribute slightly to the overall TBC. 62 Other previous experimental measurements also show that cross-interface electron-phonon coupling is not important for interfacial thermal conductance. 7,13 AGF TBC and Intrinsic Landauer TBC match quite well with measured TBC which shows that elastic phonon contribution to TBC dominates interfacial thermal transport, and inelastic phonon contribution to TBC is not important for Al-sapphire interfaces. At high temperatures where inelastic phonon contribution to TBC is larger, the measured TBC keeps constant. As shown in Therefore, the inelastic phonon contribution to TBC counteracts some of the effect of electronphonon coupling thermal resistance in Al at high temperatures and both do not contribute much to overall TBC. This contradicts the case of metal-diamond interfaces where inelastic contribution to TBC is so large that the measured TBC is greatly higher than the radiation limit (the maximum TBC only considering elastic contribution). 13,53 It may result from the high phonon DOS mismatch of metal and diamond. But it should be noted that the conclusion about metal-diamond interfaces needs to be revisited because of the unknown interfacial structure of these interfaces. (b) temperature dependence of measured sapphire thermal conductivity. The "Cahill" data is from Ref. 66 and the "Monchamp" data is from Ref. 67 The temperature dependence of the measured sapphire thermal conductivity is shown in Figure   3(b). For Sub100 and Sub200, TDTR measurements are more sensitive to cross-plane direction so the measured thermal conductivity is perpendicular to the c-plane. The measured room temperature value is 32.5 W/m-K, very close to the room temperature value (33 W/m-K) in the Ref. 67 as shown in the inset of Figure 3(b). The thermal conductivity of sapphire is weakly anisotropic for parallel or perpendicular to the c-plane. Sapphire with a crystal orientation parallel to the c-plane has a slightly larger thermal conductivity (35 W/m-K at room temperature). Ref. 66 did not include crystal orientation information in the paper but its measured value is very close to the measured thermal conductivity of sapphire with a crystal orientation parallel to the c-plane in Ref. 67 , as shown in the inset of Figure 3(b). Therefore, we speculate that the crystal orientation is parallel to the c-plane.
Our temperature-dependent measured sapphire thermal conductivity matches well with the values in Ref. 66 and we attribute the small difference to the weak anisotropy.

Conclusions
In this work, we provided a systematical benchmark study of interfacial thermal conductance

Methods Section
Sample Preparation: In this study, ~80 nm Al was deposited on high temperature annealed sapphire substrates in a Riber 32 Molecular Beam Epitaxy (MBE) system. A multi-step annealing approach was used in order to achieve highly ordered terrace-and-step structure over the surface Non-equilibrium Landauer Approach: The Landauer approach is a widely used method to predict TBC, and the general form of the Landauer formula is from the particle description of phonons and the TBC is calculated from net heat flux and temperature drop across the interface.
In previous studies of non-equilibrium effect at the interface, it is pointed out that the phonons are in strong non-equilibrium because of the difference in modal transmission coefficients and reservoir temperatures, and this non-equilibrium effect should be considered in Landauer formula.
With the recently developed non-equilibrium Landauer approach which can capture the nonequilibrium effect, the theoretical predictions agree much better with experimental results. We have applied the non-equilibrium Landauer approach to our aluminum-sapphire interface to predict the TBC. In our calculation, the phonon properties of both Al and Al2O3 are obtained from ab initio calculations within the framework of DFT, as implemented in the Vienna Ab initio Simulation Package (VASP), and the second order force constants are obtained from Phonopy 71 .
AGF: AGF is a widely used method to calculate the transmission and related thermal properties of a system. More detailed introduction could be found in literature. 43,72,73 With the harmonic assumption, only the second order force constants are needed for Green's function calculation.
Here, the second-order DFT force constants of the leads and the interface are separately obtained from the frozen-phonon method 74

AGF:
Briefly, the system ( Figure S5) could be separated into three parts: two semi-infinite leads (L & R) and a finite size central region(C). In this case, L is bulk Al, R is bulk sapphire. Figure S6 is plotted via VESTA. 2 Under harmonic assumption, the transmission function of the system is