A Modified Theoretical Model to Predict the Thermal Interface Conductance Considering Interface Roughness

The acoustic mismatch model and the diffuse mismatch model have been widely used to predict the thermal interface conductance. However, the acoustic mismatch model (diffuse mismatch model) is based on the hypothesis of a perfectly smooth (completely disordered) interface. Here, we present a new modified model, named as the mixed mismatch model, which considers the roughness/bonding at the interface. By taking partially specular and partially diffuse transmissions into account, the mixed mismatch model can predict the thermal interface conductance with arbitrary roughness. The proportions of specular and diffuse transmission are determined by the interface roughness which is described by the interfacial density of states. It shows that the predicted results of the mixed mismatch model match well with the values of molecular dynamics simulation and experimental data.

Thermal transport across interface is an important issue for microelectronics, photonics, and thermoelectric devices, and has been studied both experimentally and theoretically recently. [1,2] Generally, the thermal interface conductance (TIC) is used to evaluate the physical properties of thermal transport in devices and materials [3], such as composites [4], superlattices [5], thin-film multilayers [6], nanoscale devices [7], and nanocrystalline materials [8]. Therefore, a deep understanding and an accurate prediction of TIC are crucial to improve the performance of a diverse of devices and materials.
So far, two theoretical models, the acoustic mismatch model (AMM) and the diffuse mismatch model (DMM), have been widely used in predicting the TIC [1]. The AMM makes an essential simplified assumption that phonons incident at an interface undergo specular transmission and are governed by continuum mechanics [1]. Contrarily, the DMM assumes that the interface is completely disordered and phonons lose their memory after reaching the interface. [1,9] Although these two models have advanced the understanding of thermal transport across interface, both the AMM and DMM purely consider the bulk properties, and ignore the interfacial roughness, phonon states [10] and inelastic scattering, which leads to the inaccuracy of the two models. [1,11] In the past decades, some works have improved the accuracy of the AMM and DMM, and sophisticated modifications to the original models that account for the inelastic scattering have been proposed. For example, the maximum transmission model [12], the higher harmonic inelastic model [13], the joint frequency diffuse mismatch model [14], the scattering-media ted acoustic mismatch model [15] and the anharmonic inelastic model [16]. And the electronphonon couplings are included when studying the metal/non-metal interface. [17] Chen proposed the partially diffuse and partially specular interface scattering model to predict the thermal conductivity of superlattice structures [18]. The idea is worth to be applied in calculating the TIC. Besides, it is found that the phonon interface states which are localized close to the interfacial region and different from bulk states have a large effect on TIC. [19][20][21][22][23] However, all of the existing theoretical models do not consider the interfacial states. Therefore, there is a great demand for a theoretical model that accounts for interface states in predicting the TIC. At the interface between material A and B, an incident phonon with frequency ω and mode j can either scatter back or transmit [1]. According to the Landauer formalism [24], we can predict the TIC, G, as where ω is the frequency,  v is the cutoff frequency, D is the phonon density of states, n(ω,T) is the Bose-Einstein distribution function, v is the phonon group velocity, and α is the phonon transmission coefficient. The subscript "j" refers to the phonon polarization.
So, in the MMM, we define the transmission coefficient by mixing AMM and DMM as According to Ziman [25], the p is related to the root mean square roughness (RMSR), η, and phonon wavelength, λ, as The interfacial density of states (IDOS) depends on the interfacial structure and RMSR. For the specular interface, the IDOS is as the same as the DOS of crystal structure, where the RMSR is zero. On the other hand, for the diffuse interface, the IDOS is similar to the DOS of an amorphous structure, where the RMSR goes to infinity. We define a relationship between the value of RMSR and the DOS.
where C is a constant, the Dint, Damo are the DOS at the interface in crystal and in amorphous structure, respectively.
According to Eq. (5) and Eq. (6), the specular parameter, p, is obtained This is the key equation of this letter. By substituting Eq. (7) into Eq. (4), we can get the transmission coefficient α, then the TIC can be figured out by Eq. (1). It is shown that the interfacial structure/IDOS will greatly affect the phonon transport. As shown in Fig. 3, for the perfectly smooth interface, the specular reflection does not change the energy and the momentum components along the direction of the temperature gradient [18], so the IDOS should be the same as the DOS of crystal structures. Contrarily, for the extremely rough interface, the IDOS is similar to the DOS of amorphous structures. For a partially specular and partially diffuse interface, the IDOS is between the two extreme situations.
For a real interface, the roughness of interfaces could bring two problems: 1) the disorder but continuity of interfacial stress and displacement; 2) the discontinuity of interfacial stress and displacement [30,31]. The specular parameter p, however, only considers the continuity part. This will lead to an overvaluation of the MMM prediction to experimental values. So, we introduce a contact coefficient, S, which is less than 1 for a discontinuity interface and equal to 1 for a continuity interface. Thus the measured TIC, G, can be written as where GMMM is the value predicted by our model corresponding to a continuity interface.
The MD simulation is an ideal method in predicting the TIC because there is no assumptio n on the phonon scattering [32]. To demonstrate the accuracy of the MMM, we calculated the TIC of interfaces with different roughness, and then compared the results of MMM with MD values.
In our MD simulations, we create two Al/Si interface structures with different roughness, structure (a) and (b) (shown in Fig. S1). The structures after relaxation are shown in Fig. 1(a) and Fig. 1(b). The interface roughness of structure (b) is larger than that of structure (a). And the IDOS of structure (a) and (b) is calculated as the average phonon modes of atoms at the interface. And the interface thickness is set as 0.815nm. We also simulate an amorphous (completely disordered) structure (shown in Fig. 1(c)) to obtain the Damo,Al and Damo,Si in Eq.
(8). The MD simulation details are shown in Table S1.
The main results are shown in Fig. 2(a) predictions, and the TIC of structure (b) is larger than that of structure (a). In the calculation of MMM, the value of C is unknown, but can be obtained by fitting to the MD data of structure (a), as 1.024×10 -11 . Then we applied Eq. (7) to predict TIC of structure (b) by MMM which matches well with the data of MD (Fig. 2(a)). It declares the validity of MMM.
The specular parameter is the most important parameter in the MMM which presents the proportion of phonons specular transmitted across the interface. As shown in Fig. 2(b), the dependence of specular parameters on frequency for difference interface structures was calculated and compared. There is an obvious decline in the p-curves of both structure (a) and (b). For lower frequency phonons in each p-curve, with longer wavelength, it shows that the values of p for both (a) and (b) are close to one. That is, the phonons with longer wavelength are not sensitive to the nanoscale interface roughness and transit across them similar to a specular interface. For higher frequency phonons, the shorter wavelength phonons transmit across the interface with more diffusive scatterings. Besides, the decline frequency of (a), ~ 6.5 THz, is higher than that of (b), ~ 3.5 THz, because the roughness of structure (a) is smaller than that of structure (b). As shown in Eq. (5) the IDOS dominates the value of p, which makes the MMM consider the interfacial structure. To show the relationship between the IDOS and the interfacial structure, we calculated the IDOS of different interfacial structures. Figure 3 To further validate the MMM, we compare the transmission coefficient calculated by the MMM with the experimental results [33]. In Fig. 4 (a) Fig. 4(b)). To get the parameter S, we fitted the TIC of structure (a) calculated by MMM to the experimental value [33] at 300 K. Then, we use the value of S to obtain the TIC at 350 K and 400 K, respectively, which matches well with the corresponding experimental value.
In summary, we have developed a new theoretical model, named as the mixed mismatc h model, which predicts the thermal interface conductance by considering the interfac ia l structure. The MMM takes partially specular and partially diffuse transmissions into account using the IDOS to determine the proportion of specular parameter. The value of specular parameter has an obvious dependence on the phonon wavelength and interfacial roughness.
The MMM is validated by comparing its prediction with both MD results and measureme nts.
Both the value of TIC and the transmission coefficient calculated by the MMM agrees well with the values of MD and the experimental data.

Parameters
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