Inelastic phonon transport across atomically sharp metal/semiconductor interfaces | Panda Anku

We built high-quality metal/semiconductor interfaces by epitaxial growth of Al(111) on Si(111) and GaN(0001) using molecular beam epitaxy (MBE), see Methods and Supplementary Information Note I. Before the growth of Al/Si interface, a Si wafer was cleaned by hydrofluoric acid and then heated in a vacuum at 900 °C to ensure the surface was free of oxide layer and adsorbates. To fabricate an Al/Si interface with controlled interface quality, Al growth was proceeded at different temperatures, 100 °C (denoted as Sample 1) and 300 °C (denoted as Sample 2), knowing that 100 °C is the optimum temperature for Al deposition in our prior work24. For Al/GaN interface, GaN thin film was grown on a sapphire substrate at 800 °C in the MBE chamber first, and then the temperature was ramped down to 150 °C to grow the Al layer. For comparison, an Al/SiO2/Si sample was also prepared by e-beam evaporation of Al on Si substrate in presence of native oxide.

We measured the thermal conductance of Al/Si and Al/GaN interfaces over a wide range of temperatures (80–700 K) by time-domain thermoreflectance (TDTR)25,26. The raw TDTR data, data analysis, and uncertainty estimation can be found in Methods and Supplementary Information Note V-VIII. The measured thermal conductance G of Al/Si and Al/GaN interfaces as a function of temperature are plotted in Fig. 1a and Fig. 1b. At room temperature, Al/Si Sample 1 and Al/GaN show a record high thermal conductance of 379 and 423 MW m−2 K−1, respectively. For both Al/Si Sample 1 and Al/GaN, our results can be clearly divided into two regimes. At temperatures lower than the Debye temperature of Al (TD = 428 K), the thermal conductance of Al/Si Sample 1 and Al/GaN gradually saturate with the increase of temperature, which has the same trend as the previous measured thermal conductance of Al/Si and Al/GaN interfaces. However, when the temperature approaches 400 K (close to the Debye temperature of Al) and beyond, both Al/Si Sample 1 and Al/GaN show a linear increase in thermal conductance with temperature instead of reaching a plateau. For Al/Si Sample 2, it has a thermal conductance of 309 MW m−2 K−1 at room temperature. Throughout the low temperatures (T < TD), the thermal conductance of Al/Si Sample 2 is ~10% lower than that of Al/Si Sample 1. However, unlike Al/Si Sample 1 and Al/GaN, Al/Si Sample 2 shows a saturated thermal conductance when T > TD.

Fig. 1: Thermal conductance of Al/Si and Al/GaN interfaces.
figure 1

a Thermal conductance of Al/Si Sample 1 (red spheres) and Sample 2 (blue spheres). Black dashed line is interface thermal conductance calculated by DMM. For comparison, we show previously measured Al/Si thermal conductance in open squares by Minnich39,  triangle by Wilson40, and diamond by Jiang41. Yellow solid spheres are measured thermal conductance of Al/Si with a native oxide layer, compared with the results by Hopkins, shown in open circles14. b Thermal conductance of Al/GaN interface (red spheres). For comparison, the calculated thermal conductance using DMM (black dashed line) is plotted. Previous measurement results by Donovan42 are shown in open circles, the Al film of which was deposited by e-beam evaporation. Phonon dispersion relations of Al/Si (c) and Al/GaN (d) are calculated from first-principles. The calculation of error bars is detailed in Supplementary Information Note VIII.

For the metal/semiconductor interface, there are four thermal transport interactions, which are the phonon–phonon transport across interface including both elastic and inelastic phonon transport, as well as the electron–phonon coupling in the metal and across interfaces. The effect of electron–phonon coupling across interfaces is negligible13. The electron–phonon coupling in Al adds an additional thermal resistance in series with the phonon–phonon interactions27, which is driven by the thermal non-equilibrium between electrons and phonons near the interface. To calculate the phonon–phonon transport induced interface thermal conductance alone, we followed Majumdar and Reddy’s treatment of electron–phonon coupling to use ({{{{{rm{G}}}}}}=frac{{{{{{{rm{G}}}}}}}_{{{{{{rm{ep}}}}}}}{{{{{{rm{G}}}}}}}_{{{{{{rm{pp}}}}}}}}{{{{{{{rm{G}}}}}}}_{{{{{{rm{ep}}}}}}}+{{{{{{rm{G}}}}}}}_{{{{{{rm{pp}}}}}}}}), where G is the total thermal conductance, Gep is electron–phonon coupling and Gpp is phonon–phonon transport induced thermal conductance27. The electron–phonon coupling induces the conductance ({{{{{{rm{G}}}}}}}_{{{{{{rm{ep}}}}}}}=sqrt{{{{{{rm{g}}}}}}{Lambda }_{{{{{{rm{p}}}}}}}}), where g is the electron cooling rate and Λp is the lattice thermal conductivity of Al. We used experimentally determined g28 and first-principles calculated Λp29, which have been reported previously and are well accepted, to determine Gep. The calculated Gpp for Al/Si Sample 1, Sample 2, and Al/GaN are shown in Supplementary Fig. 10. It shows that Gpp of Al/Si Sample 1 and Al/GaN interfaces has a stronger temperature dependence than G, while Gpp of Al/Si Sample 2 changes slightly with the temperature at high temperatures. Besides, the electron–phonon coupling reduces the measured G below that prediction from phonon–phonon interactions alone.

To study the relationship between interface quality and thermal conductance and understand the difference between the two Al/Si interfaces, high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) was used to study the cross-sectional interface structure, as shown in Fig. 2. The STEM images clearly shows that Al/Si Sample 1 has a sharp interface, while Sample 2 has a diffuse interface. The interface structure of Al/Si Sample 1 is shown in Fig. 2a, showing the interface between Al and Si is atomically sharp, with only 1–2 distorted layers of interface atoms observed. The cross-sectional HAADF-STEM image of Al/Si Sample 2 is shown in Fig. 2b and Supplementary Fig. 7. Unlike the Al/Si Sample 1, it shows a diffuse interface with an interdiffusion region of 1.38 nm and 1.2 nm for two randomly chosen spots, which roughly equals the thickness of 3–5 atomic layers of Si. Further structure analysis in Supplementary Fig. 9 shows that the diffuse interface of Sample 2 results from the intermixing of Al and Si atoms. Both interfaces of Sample 1 and Sample 2 are quite homogenous, as seen in Supplementary Figs. 3 and 7. We have monitored that there is no trace of oxygen residue at the interface (see Supplementary Fig. 1a). The strain analysis shows that the strain localizes at the nearest layers adjacent to the sharp interface, as in Supplementary Fig. 5. Though the as-grown Al films are of high quality, there are still lattice imperfections. The main imperfections in the Al layer are domains, which are micron sized and much larger than the phonon or electron mean free path in Al, thus have little effect on thermal transport.

Fig. 2: Interface structure of Al/Si Sample 1 and Sample 2.
figure 2

Cross-sectional TEM image of Al(111)/Si(111) Sample 1 (a) and Sample 2 (b). Scale bars are 2.5 nm.

When the samples are heated up, interdiffusion across the interface will inevitably occur. The diffused Al or Si atoms create some disorder in the other side. And according to some previous calculations, such disorder will facile phonon transport and increase interface thermal conductance30,31,32. To test this, we fabricated sharp Al/Si interfaces with both intrinsic Si and doped Si, and we found that the two interfaces have nearly identical thermal conductance (see Supplementary Fig. 11). We infer that a small portion of foreign atoms will not affect interface thermal conductance, thus the increase of thermal conductance with temperature is intrinsic.

To understand the phonon transport mechanism, we used non-equilibrium molecular dynamics (NEMD) to compute the interface thermal conductance of both sharp and diffuse Al/Si interfaces (see Methods). The Al/Si structure along the (111) orientation was established with sharp and diffuse interfaces, as shown in Supplementary Figs. 1a and 1b. The sharp interface is composed of Si(111) 3 × 3 unit cell and Al(111) 4 × 4 unit cell. Before calculating the thermal interface conductance, the structure was fully relaxed to reduce the stress at the interface. A diffuse interface was attained by locally melting the interface at 3000 K and quenched to 300 K in the MD simulation, and the thickness of the diffuse interface is about 1.3 nm. The simulation results of the sharp interface at 500 K with a heat bath temperature difference of 60 K are shown in Supplementary Information Note IX, and the thermal conductance was calculated as 738 MW m−2 K−1. The thermal conductance predicted by MD at high temperature is shown in Fig. 3a. The interface thermal conductance at the diffuse interface is lower than the value of the sharp interface, which is consistent with the experiment results. As temperature rises, the increasing slope of thermal conductance as a function of temperature at the sharp interface is much higher than that of the diffuse interface, demonstrating that the surface sharpness is crucial for the observation of inelastic phonon scattering, which occurs across the atomically sharp interfaces. Considering that MD simulation does not make assumptions on the phonon scattering mechanisms, the temperature dependence of thermal conductance at high temperatures is possibly universal, indicating that inelastic phonon scattering is always expected at high-quality interfaces.

Fig. 3: Phonon transport behavior across Al/Si interface computed by molecular dynamics.
figure 3

a Calculated thermal conductance of sharp (red dashed line) and diffuse (blue dashed line) Al/Si interfaces. b Phonon transmission coefficient for sharp (red) and diffuse (blue) interfaces. c Schematic of temperature distributions near the sharp and diffuse interfaces. Here Tp,h and Tp,l represent temperatures of high- and low-energy phonons. ΔTs and ΔTr are temperatures drop across sharp and diffuse interfaces.

To further explain the observed distinct temperature dependence of the thermal conductance, we calculated the spectral phonon transmissivity using atomistic Green’s function, as shown in Fig. 3b. For the sharp interface, the transmissivity of low-energy phonons (here we define phonons lower than 4 THz as low-energy phonons) is higher than the high-energy phonons. This will result in a relatively large temperature difference between these phonons, as sketched in Fig. 3c. Such a thermal non-equilibrium between high-energy phonons and low-energy phonons leads to mode conversion and energy communication between them through phonon scatterings. Thus, the high-energy phonons with low transmission probability will convert to high transmissivity low-energy phonons before they undergo the transport process across the interface, leading to inelastic phonon transport.

For diffuse interfaces, the difference of transmissivity between the low- and high-energy phonons becomes smaller compared with that of sharp interfaces, as the transmissivity for all phonons reduces. As a result, the temperature difference of different phonons is expected to be smaller, thus the phonon non-equilibrium is smaller for diffuse interfaces. The reduced phonon non-equilibrium leads to less energy communication between high and low-energy phonons, and the inelastic phonon transport will diminish.

Our results point out that a large Debye temperature difference is not required for the inelastic phonon transport process to occur, as the Debye temperature ratio of both Al/Si and Al/GaN is <1.5. This finding is in stark contrast to what previous experiments suggested, where inelastic transport can be only observed across interfaces formed with a large Debye temperature difference12,13,21.

A recent work by Cheng et al. claimed that no inelastic phonon transport is observed in atomically sharp Al/Al2O3 interface grown by MBE6. We would like to point out that at least five monolayers of Al2O3 near the interface are distorted, as shown in their TEM image, which is similar to our Al/Si Sample 2 and not as sharp as our Al/Si Sample 1. Their work actually echoes with our finding that significant inelastic thermal transport can only be observed at atomically sharp interfaces.

In applications with high power density, high frequency, and small sizes, such as power electronics and RF devices, heat is mostly localized, leading to hotspots. In such cases, interface heat transport becomes more important33. Our work is particularly useful to improve heat dissipation in semiconductor interfaces and metal/semiconductor interfaces at elevated temperatures, especially when the interfaces are formed with low Debye temperature materials such as Au, Ti, and GaAs, as an additional channel across interfaces is open for heat conduction when they are atomically sharp, which is still a technological challenge to date.

In summary, we report the observation of inelastic phonon transport across high-quality Al/Si and Al/GaN interfaces grown by MBE. We observed a continuously increasing thermal conductance at high temperatures, which is attributed to the inelastic phonon transport process across the interface. The inelastic phonon transport is expected to occur at atomically sharp interfaces where the strong phonon non-equilibrium exists, in contrast to diffuse interfaces. This work sheds light on increasing thermal conductance across the interface at high temperatures and improving heat dissipation of electronic devices.

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