**5.1 Modelling Si/Ge superlattices with rough interfaces**

90 Molecular Dynamics – Theoretical Developments and Applications in Nanotechnology and Energy

superlattices may be dramatically smaller than the corresponding values of the constituent materials in their bulk form. This decrease has been related to the folding of the Brillouin zone and the related mini-umklapp three-phonon scattering process. Tamura & al 1999 analysed the effect on the phonon spectra in superlattices by three major reasons: a. folding of the phonon branches caused by the periodicity of the superlattices, b. the formation of the mini-band and c. the confinement of the acoustic phonons in the different layers due to the mismatch of the spectra. These three reasons impose reduction of the group velocity in the cross-plane direction, leading to the decrease of the cross-plane thermal conductivity. Chen & Yang 2005a claimed that the group velocity reduction is not sufficient to explain the dramatic decrease of the thermal conductivity, and they argued that one should add the diffusive scattering at the interfaces and treat phonons as incoherent particles. Chen & Neagu 1997 solving the Boltzmann Transport Equation for specular and diffuse interfaces showed that depending on the superlattice period, the thermal conductivity might be influenced either by the diffuse interface scattering or by the scattering induced by the dislocations. The literature is rich in this subject and a series of articles appeared with a lot of experimental (Capinski et al 1999, Huxtable et al 2002, Lee et al 1997) or theoretical results using lattice dynamics method or Equilibrium (Volz 2000, Landry 2008, Termentzidis 2011b, Termentzidis 2011c) and Non-Equilibrium Molecular Dynamics method (Liang & Shi 2000,

One very interesting property of superlattices is their thermal anisotropy. In the paragraphs below the distinction between the in-plane (parallel to the interfaces) and cross-plane (perpendicular to the interfaces) thermal conductivity has been underlined. In general one expects a value of the in-plane thermal conductivity close to the bulk conductivity especially for superlattices with smooth interfaces, where phonons are expected to specularly reflect from the interfaces, and in which each layer behaves like a phonon waveguide. For the cross-plane direction the picture is totally different, with thermal conductivity even smaller than a random alloy of the same material (Mahan 2004). A key point for the physical explanation of the phenomena related to the superlattice thermal conductivity is the thermal

Molecular dynamics simulations have been performed recently to understand the physical mechanisms ruling the transport properties of superlattices (Landry et al, 2008, 2009, Termentzidis et al, 2009,2010,2011a,2011b,2011c). In this contribution, we will discuss the influence of the interface roughness and of the superlattice period on the in-plane and the cross-plane thermal conductivities of Si/Ge superlattices using both the EMD and NEMD methods. This study proves that heat transport in superlattices is controlled by the interfaces. An atomic knowledge (or description) of the interfaces is necessary for the correct prediction of the thermal conductivity. In turn, understanding the link between the interfacial structure and the thermal conductivity will certainly help in tailoring and controlling the phonon behaviour in nanostructures. This can lead to the augmentation of the lifetime and to optimize the working of several nano-devices. The state of the interfaces is crucial for the determination of the behavior of phonons within the nanostructures. When the layer thickness of the superlattice is comparable with the MFP, the thermal conductivity is controlled by the transmission of phonons across the interfaces of the superlattice. In particular, the thermal boundary resistance or the Kapitza resistance, which has been discussed before, will play a key role in the thermal transport in

boundary resistance or the Kapitza resistance, which has been discussed before.

Chen 2004, Termentzidis 2009, Termentzidis 2010).

superlattices with thin layers.

In this subsection, we study the effect of the superlattice period and the structure of the interface on both the in-plane and the cross-plane superlattice thermal conductivities. Simulations have been held for two periods of superlattices 20 and 400, which are comparable to the phonon meat free path at the temperature we are working at. To understand the role of the interfacial structure, we have considered two types of interfaces: smooth interfaces on the one hand, and rough interfaces with a right-isoscele-triangles shape, as shown in fig. 4. In the case of rough interfaces, the height of the interfaces has been varied between 1 monolayer (ML) which is half of a lattice constant for a fcc crystal, up to 120, which is more than the half of the superlattice period for superlattices with period of 200 and from 1ML up to 240 for superlattices with period of 400 (see fig. 4). The maximal interface height is thus more than half the superlattice's period.

We examined also the effect of the shape of the interface on the cross-plane and the in-plane thermal conductivities. In this case, the superlattice period is kept constant and equal to 20<sup>0</sup> and we considered only one height of interfaces, the 60 or 12MLs. Fig. 5 shows the smooth interfaces, the periodic triangular isosceles interfaces and the additional 4 other shapes. The structure shown in 5.iii is obtained by superposing the periodic isosceles of small lengths with the periodic triangular interfaces shown in 5.ii. Cosine like, random like and square like interfaces are also examined.

#### **5.2 Thermal conductivity of Si/Ge like superlattices with rough interfaces**

The in-plane and cross-plane thermal conductivities have been calculated using the EMD and the NEMD methods. The results are displayed in figure 6 as a function of the interface

Molecular Dynamics Simulations and Thermal Transport at the Nano-Scale 93

increasing the roughness's height can be interpreted if we assume that the phonon scattering at the interface remains diffusive, phenomena of back scattering and specular reflection, playing a secondary role here. Hence, we conclude that the variation in the ratio of interface roughness to the superlattice period can tailor the thermal properties of

Fig. 6. In-plane and cross-plane thermal conductivity as a function of the interface roughness

The cross-plane and in-plane thermal conductivities obtained by NEMD and EMD for superlattices with various shapes of interfaces are plotted in figure 7. Further details about the modelling of these interfaces and the physical explanation of the results are given in Termentzidis et al, 2011b. It is striking to note that for rough interfaces, the anisotropy of the thermal conductivity is drastically reduced. Regarding the anisotropy between the in-plane and cross-plane directions, we can categorize the interfaces in three different groups, first

obtained by EMD and NEMD for superlattice with period 200 (top) and with 40<sup>0</sup>

superlattices.

(bottom).

Fig. 5. Shapes of different superlattice interfaces i) smooth interfaces, ii) periodic isosceles triangles, iii) superposition of small triangles over the interfaces of the type ii, iv) cosine like v) random like, vi) periodic square like interfaces. In all cases, the height of the rough interfaces is kept constant equal to 12ML (taken from Termentzidis et al 2011b).

roughness. Again, two superlattice periods have been considered 200 (left) and 400 (right). The figures show also some points named "intra-plane" thermal conductivity, which is defined as the thermal conductivity in the direction of 45° both of the in-plane or cross-plane directions. For the case of infinite roughness (with isosceles periodic triangles) the thermal conductivity in both the in-plane and the cross-plane directions is expected to be equal to this of the intra-plane conductivity, which is verified in our simulations.

For smooth interfaces, it is expected that the transmission of phonons across the interface is specular and depends only on the acoustic impedance mismatch between the bulk materials of the superlattice (Swartz and Pohl, 1987, 1989). This is actually not the case since it is observed that the thermal conductivity of the layer decreases when the film thickness decreases (at least enough to be of the same order of magnitude as the phonon mean-free path). For rough surfaces with small roughness the transmission of phonons becomes more diffusive and the transmitted phonons are distributed over a wide range of angles, which induces an additional resistance to in-plane transport. This is consistent with the MD results which conclude to a decrease of the in-plane conductivity with the interfacial roughness. For rough surfaces with large roughness there is a combination of specular and diffusive transmission. This last case shows some similarities with the smooth surface case but now specular reflection is accompanied by back scattering.

This back-scattering explains the existence of a minimum in the thermal conductivity observed for free surfaces, as well as for the in-plane conductivity of superlattices. A further increase in the surface roughness leads to higher thermal conductivity. This further increase in the thermal conductivity is related to the fact that on average, phonons experience less reflections on the asperities of the superlattice, and the mean free path length between scattering events increases, leading to an enhanced in plane transport. For the randomlike roughness, the monotonous decrease in the thermal conductivity in

Fig. 5. Shapes of different superlattice interfaces i) smooth interfaces, ii) periodic isosceles triangles, iii) superposition of small triangles over the interfaces of the type ii, iv) cosine like v) random like, vi) periodic square like interfaces. In all cases, the height of the rough interfaces is kept constant equal to 12ML (taken from Termentzidis et al 2011b).

roughness. Again, two superlattice periods have been considered 200 (left) and 400 (right). The figures show also some points named "intra-plane" thermal conductivity, which is defined as the thermal conductivity in the direction of 45° both of the in-plane or cross-plane directions. For the case of infinite roughness (with isosceles periodic triangles) the thermal conductivity in both the in-plane and the cross-plane directions is expected to be equal to

For smooth interfaces, it is expected that the transmission of phonons across the interface is specular and depends only on the acoustic impedance mismatch between the bulk materials of the superlattice (Swartz and Pohl, 1987, 1989). This is actually not the case since it is observed that the thermal conductivity of the layer decreases when the film thickness decreases (at least enough to be of the same order of magnitude as the phonon mean-free path). For rough surfaces with small roughness the transmission of phonons becomes more diffusive and the transmitted phonons are distributed over a wide range of angles, which induces an additional resistance to in-plane transport. This is consistent with the MD results which conclude to a decrease of the in-plane conductivity with the interfacial roughness. For rough surfaces with large roughness there is a combination of specular and diffusive transmission. This last case shows some similarities with the smooth surface case but now

This back-scattering explains the existence of a minimum in the thermal conductivity observed for free surfaces, as well as for the in-plane conductivity of superlattices. A further increase in the surface roughness leads to higher thermal conductivity. This further increase in the thermal conductivity is related to the fact that on average, phonons experience less reflections on the asperities of the superlattice, and the mean free path length between scattering events increases, leading to an enhanced in plane transport. For the randomlike roughness, the monotonous decrease in the thermal conductivity in

this of the intra-plane conductivity, which is verified in our simulations.

specular reflection is accompanied by back scattering.

increasing the roughness's height can be interpreted if we assume that the phonon scattering at the interface remains diffusive, phenomena of back scattering and specular reflection, playing a secondary role here. Hence, we conclude that the variation in the ratio of interface roughness to the superlattice period can tailor the thermal properties of superlattices.

Fig. 6. In-plane and cross-plane thermal conductivity as a function of the interface roughness obtained by EMD and NEMD for superlattice with period 200 (top) and with 40<sup>0</sup> (bottom).

The cross-plane and in-plane thermal conductivities obtained by NEMD and EMD for superlattices with various shapes of interfaces are plotted in figure 7. Further details about the modelling of these interfaces and the physical explanation of the results are given in Termentzidis et al, 2011b. It is striking to note that for rough interfaces, the anisotropy of the thermal conductivity is drastically reduced. Regarding the anisotropy between the in-plane and cross-plane directions, we can categorize the interfaces in three different groups, first

Molecular Dynamics Simulations and Thermal Transport at the Nano-Scale 95

A new modeling of realistic interfaces is considered with the present study (Termentzidis et al 2010, 2011a). Interfaces with square formed islands of one monolayers and pyramide like islands of two monolayers are modelled. Furthermore there are two characteristic lengths of square islands depending if GaAs or AlAs is on top. It has been proven that large scale islands are formed when an AlAs layer grows on a GaAs layer while small scale islands are formed when a GaAs layer grows on an AlAs layer (Tanaka & Sakaki 1987, Jusserand et al 1990). Furthermore interfaces with interdiffusion parts are also considered. Figure 8 depicts the two kind of interfaces with interface height of one monolayer and in figure 9 with two

Fig. 8. Schematic representation of the two different scale interfaces between the GaAs/AlAs and the AlAs/GaAs, for height of interfaces of one monolayer – square like interfaces.

Fig. 9. Schematic representation of the two different scale interfaces between the GaAs/AlAs and the AlAs/GaAs, for height of interfaces of two monolayers – pyramoidal like interfaces

Figure 10 shows the predicted cross-plane thermal conductivity as a function of the superlattice period, for a variety of interface configurations. In figure 10 at left the thermal conductivity is plotted for smooth interfaces, for rough interfaces with height of one monolayer and three coverage factors (1%, 10% and 50%) and finally for interfaces with

**6.2 Thermal conductivity of GaAs/AlAs like superlattices with rough interfaces** 

(taken from Termentzidis et al 2010).

**6.1 Modelling rough interfaces for the GaAs/AlAs superlattices** 

monolayers (these figures are taken from Termentzidis et al 2010)

the smooth interfaces which displays the maximum anisotropy, with the in-plane thermal conductivity being more than twice larger than the cross-plane thermal conductivity; A second group contains totally random interfaces for which the two thermal conductivities exhibit their minimum values and a third group with periodic rough interfaces of a specific shape (triangular, square or cosine) where the thermal anisotropy is negligible compared to the uncertainties of the methods.
