Effect of Amorphisation on the Thermal Properties of Nanostructured Membranes

2.1 Modelling the Structures

We modelled nanoporous silicon membranes using the methodology described in France-Lanord et al. [7] and which is fully reproducible. The nanopores in the current study have cylindrical shape with the axis of cylinders vertical to the free surfaces of the membranes. We modelled pores without and with shells of amorphous Si (a-Si) or SiO₂ (Fig. 1). The system is relaxed at T=300 K with an NVT ensemble during 200 ps. At the end of this procedure, the pressure was 0 Pa. Five cases are studied: nanoporous crystalline Silicon (c-Si) without shell around the pores (Fig. 1a), nanoporous c-Si with a-Si shell (Fig. 1b and c), and nanoporous c-Si with amorphous SiO₂ shell (Fig. 1d and e). The axes of the cylinders are along z-axis. Periodic boundary conditions are set in x and y directions, while the conditions on z are fix. The dimensions of the simulation box for all configurations were set to 10a₀×10a₀×10a₀ with a₀=5.43 Å.

Figure 1:

Cross sections of Si membranes with cylindrical pores without and with shells: (a) no shell, (b) amorphous Silicon shell with 0.5 nm thickness, (c) amorphous Silicon shell with 1 nm thickness, (d) amorphous SiO₂ shell with 0.5 nm thickness, and (e) amorphous SiO₂ shell with 1 nm thickness. Grey atoms represent silicon and red the oxygen.

2.2 Molecular Dynamics and Thermal Conductivity

Molecular dynamics simulations are performed with LAMMPS [8]. Equilibrium molecular dynamics (EMD) method is used to determine the TC via the Green-Kubo formula:

where x and y subscripts denote directions, 𝒱 the system’s volume, J the heat flux, t the time, T the system’s temperature, and k_B the Boltzmann constant. With this formula, the TC can be obtained for all directions. In our study, we give the results for only the in-plane thermal conductivity (averaged in x and y directions).

The time step is set to 0.5 fs. At the beginning of the simulation, the system is relaxed at T=300 K with an NVT ensemble during 200 ps. Then the computation of the TC is performed every 40 ps under an NVT ensemble, estimating the flux fluctuations correlation every 10 fs, for a total duration of 10 ns. The TC is averaged over the second half of the simulation, when the steady state is established, in order to reduce the uncertainty on the results [6]. Moreover, each simulation is repeated 10 times with different initial seeds and the TC is averaged. Periodic boundary conditions are applied on surfaces perpendicular to the directions of interest (x and y directions), while nonperiodic and shrink-wrapped boundary conditions are applied on the free surfaces (z direction for membranes).

For systems containing only Silicon atoms, their interactions are described with the Stillinger–Weber potential [9] with modified coefficients by Vink et al. [10] as the latest parameterisation is able to describe thermal transport in silicon, in both crystalline and amorphous phases. When the system contains oxygen (in SiO₂ shells), the interatomic potential is Tersoff SiO [11]. In the bulk crystalline silicon case, the TC obtained with Stillinger–Weber potential is k=164±21 W/m K. That is in good agreement with reported experimental values k=148 W/m K at 300 K (EL-CAT Inc.). The consistence of our study has been checked by computing the TC of one of our systems with both potentials and the results are reasonably in agreement (difference less than 5 %).

2.3 DOS

With molecular dynamics, the phonon’s density of states (DOS) of the system can also be extracted from the trajectory of the atoms. Indeed, the displacement of the atoms can be decomposed by Fourier transform to find the density of frequencies of phonons. In this article, the specden plugin of the VMD Signal Processing Plugin Package was used to calculate the phonon DOS of the porous membranes.

3.1 Impact of Thickness and Amorphous Material Shells on Thermal Conductivity

In this section, the TC of Si membranes with cylindrical nanopores with different kind of shells as showed in the previous section is investigated with molecular dynamics. The periodicity of the structure is set to a=15 a₀ and the diameter of the pore is d=10 a₀ (8.145 and 5.43 nm, respectively). The thickness of membranes is set to h=15 a₀. The shell thickness e is 0.5 or 1 nm. The thermal conductivity is computed in the inplane direction of the membranes (perpendicular to the axis of the cylinder). For comparison, the thermal conductivity of a silica membrane of thickenss of 8 nm measured experimentally is 0.7 W/mK [Goodson et al., J. Heat Transfer 116, 317 (1994)] and for bulk amorphous silicon reported values are in the range 1–1.8 W/mK [Wada, Japanese J.A.P. 35, 5B (1996)].

The thermal conductivity of the modelled structures is shown in Figure 2. The TC of the thin crystalline silicon membrane with pores is only 3.5 W/mK, which is lower than the bulk silicon by a factor of 40. This reduction is first due to the decreasing of dimensions (from 3D to 2D) and also due to the pores that increase considerably the scattering rate of phonons on the free surfaces.

Figure 2:

Thermal conductivity of nanoporous silicon membranes and the influence of amorphous shell around the pores depending on the amorphous material and the thickness of the shell. The black squares is for a-Si and the red cyrcles for the a-SiO₂ around the pores. Two shell thickness are studied: 0.5 and 1.0 nm.

When there is an a-Si shell around the pores, the thermal conductivity reduces rapidly and quite linearly. In increasing the thickness of the amorphous shell around the pores, the thermal conductivity reduces further and can reach subamorphous thermal conduction keeping an important ratio of crystalline material. For a shell thickness of 0.5 nm, the TC is divided by 2 compared to the system without shell. This reduction is more pronounced when the shell thickness is increased. Subamorphous TCs are even reached for e=1 nm. So the presence of an amorphous shell around the pore contributes to lower thermal transport, as has been observed in spherical pores [6]. In fact, the TC is reduced as if the diameter of the pore was the crystalline/amorphous interface. This is an evidence that the a-Si have the same effect as void on the thermal conductivity.

The impact of a-SiO₂ is different than this of a-Si. The TC is again lowered by the presence of an amorphous phase around the pore, but this time the reduction is striking. Comparing shame thicknesses of shell with a-Si or a-SiO₂, we can see that their relative reduction is of the rate of 5–12. Another important difference between the two amorphous materials as shells is the in the case of silica the TC does not depend on the shell thickness anymore (this will be explained later with DOS). The thermal conductivity reached by nanostructuration and oxidation can be lower than the bulk amorphous SiO₂ TC (about 1.4 W/m.K).

3.2 Density of States

In order to understand the behaviour of the TC of the precedent structures, their phonon densities of states (DOS) have been computed. Figures 3 and 4 show the impact of the nanostructuration and thickness of amorphous shells on the phonon DOS. Figure 3 shows the effect of the a-Si shell on the DOS. We note that when the shell thickness increases, the DOS becomes closer to the one of bulk amorphous Silicon. This gives a large decrease in the TO peak, a red shift of all the phonons (offset towards lower frequencies), and a smoothing of the two peaks in the middle, which correspond to LA and LO polarisations. All these modifications can explain the reduction of the thermal transport and the more or less linear dependence of the TC with the thickness of shells for the case of a-Si.

Figure 3:

Comparison of the phonon density of states of several nanoporous systems with and without a-Si shells around the pores.

Figure 4:

Comparison of the phonon density of states of several nanoporous systems with and without a-SiO₂ shells around the pores.

Figure 4 shows the influence of a SiO₂ shell on the DOS of the structure. The DOS of bulk SiO₂ is also depicted to help the comparison. As for the a-Si shell, the presence of a-SiO₂ shell moves the DOS of the system towards the one of bulk SiO₂. Thus, the three lower frequency peaks are smoothed and red shifted, whereas the TO peak is decreased and blue shifted (towards higher frequencies). Because of the presence of oxygen, three high-frequency peaks appear with the SiO₂ shell. This might explain why the TC of systems with SiO₂ shells seems independent from the thickness of the shell: increasing the shell thickness reduces the low-frequency population but increases the number of very high frequency phonons and this two effects might be balanced out. We have to add here that the large DOS mismatch between the crystalline silicon and the amorphous silica should yield large interfacial thermal resistance between the two materials. The DOS mismatch for the case of crystalline/amoprhous silicon is not so pronounced as here and this might explain partially the differences between the effective thermal conductivities of the two systems with amorphous silicon and amorphous silica shells.