Molecular dynamics study of the lattice thermal conductivity of Kr/Ar superlattice nanowires

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Abstract

The nonequilibrium molecular dynamics (NEMD) method has been used to calculate the lattice thermal conductivities of Ar and Kr/Ar nanostructures in order to study the effects of interface scattering, boundary scattering, and elastic strain on lattice thermal conductivity. Results show that interface scattering poses significant resistance to phonon transport in superlattices and superlattice nanowires. The thermal conductivity of the Kr/Ar superlattice nanowire is only about 13 of that for pure Ar nanowires with the same cross-sectional area and total length due to the additional interfacial thermal resistance. It is found that nanowire boundary scattering provides significant resistance to phonon transport. As the cross-sectional area increases, the nanowire boundary scattering decreases, which leads to increased nanowire thermal conductivity. The ratio of the interfacial thermal resistance to the total effective thermal resistance increases from 30% for the superlattice nanowire to 42% for the superlattice film. Period length is another important factor affecting the effective thermal conductivity of the nanostructures. Increasing the period length will lead to increased acoustic mismatch between the adjacent layers, and hence increased interfacial thermal resistance. However, if the total length of the superlattice nanowire is fixed, reducing the period length will lead to decreased effective thermal conductivity due to the increased number of interfaces. Finally, it is found that the interfacial thermal resistance decreases as the reference temperature increases, which might be due to the inelastic interface scattering.

Introduction

Semiconductor superlattices (SL) are of great interest due to their potential applications in thermoelectric and optoelectronic devices [1], [2]. Superlattice structures provide the possibility of decreasing materials’ thermal conductivity while retaining their electrical conductivity, thus achieving a high thermoelectric figure-of-merit and improving the performance of thermoelectric devices. A large amount of experimental and theoretical work [3], [4], [5], [6], [7], [8] has been carried out to study the effects of lattice period and interface on the thermal conductivity of various kinds of superlattice films. Results showed that the superlattice may have a much lower thermal conductivity than the value for each of the two materials composing the superlattice structure along both the in-plane and the cross-plane directions. Different mechanisms have been proposed to explain the reduction of the thermal conductivity, including interface phonon scattering due to acoustic impedance mismatch, phonon scattering by crystal imperfections at the interface, phonon spectrum mismatch, and mini-band formation. However, quantitative analysis of the relative importance of these different mechanisms has not been completed yet and it is not clear which mechanism contributes most to the reduction of the thermal conductivity under different conditions. Experimental results [6], [7] indicate that as the period of the superlattice decreases, the thermal conductivity along the cross-plane direction also decreases, and for very short period superlattice, its thermal conductivity could even fall below the alloy limit, i.e. if the two materials mixed homogeneously into an alloy. The in-plane thermal conductivity follows a similar pattern but it jumps to higher values for some very short period superlattice [8]. This phenomenon cannot be explained solely by diffuse mismatch theory about phonon scattering at the interface. Other mechanisms such as acoustic impedance mismatch, mini-band formation or phonon spectrum mismatch have to be taken into account.

Theoretically, lattice dynamics and particle transport models are usually used to study the thermal conductivity of superlattice structures. Some recent lattice dynamics work was carried out by Hyldgaad and Mahan [9] and Tamura and Tanaka [10]. In their work, a simple cubic lattice model [9] and a face-centered cubic model [10] were used to calculate the group velocity of acoustic phonons in the cross-plane direction. Their results suggested that reduction of the phonon group velocity in superlattices could lead to a reduced thermal conductivity. Considering their lattice models were too simple to provide details of phonon dispersion spectra, Kiselev et al. [11] used a diatomic unit cell model to simulate the dispersion spectra of the Si/Ge superlattice. Due to the large mass of Ge atoms in comparison to Si, the most probable acoustic phonons in Si layers at room temperature have no counterpart in the phonon spectra of the Ge layers. In other words, a phonon at a given frequency in the Si layer may not be able to proceed into the Ge layer without scattering with, or into, one or more phonons of different frequency. This leads to highly efficient trapping of high-energy phonons in the Si layer and a drastic reduction of the superlattice thermal conductivity. Although qualitative agreement can be obtained through lattice dynamics, it is difficult to compare those results quantitatively with experimental results for different materials under different temperatures due to the fact that lattice dynamics can only model very simple systems. The particle transport model treats phonons as individual particles and solves the Boltzmann transport equation (BTE) to study the phonon transport in microstructures. Simkin and Mahan [12] showed that for layers thinner than the mean free path (mfp) of phonons, the wave aspect of phonons must be taken into account and wave theory must be applied, while for layers thicker than the mfp of phonons, the particle treatment of phonons was acceptable and BTE could be used to study the phonon transport. The first study on the thermal conductivity of superlattices was carried out by Ren and Dow [13]. They modeled the thermal conductivity of ideal superlattice by combining the BTE with a quantum mechanical treatment of the additional scattering process caused by the mini-bands. However, their predicted results could not match the experimental data. Based on the BTE and the assumption of partially specular and partially diffuse boundary scattering, some interesting results were obtained by Chen et al. [14], [15], [16], [17] for superlattice thermal conductivity along both in-plane and cross-plane directions. In his model the reduction of the superlattice thermal conductivity was mainly attributed to diffuse scattering at the interface.

Because some assumptions must be introduced to get closed form solutions, the theoretical results usually deviate significantly from the experimental data and sometimes the thermal conductivity reduction mechanisms cannot be readily explained from those theoretical studies. Classical Molecular Dynamics (MD) simulation provides an alternative approach to investigate heat transport in nanostructures. Given the interaction potential between atoms, the force acting on each atom can be calculated. Based on Newton's second law, the motion of a large number of atoms can be described. Without any further assumptions, statistical physical properties can be derived from the ensemble of atoms. In particular, if the size of the nanostructure is smaller than the phonon mfp, it is questionable to use the BTE to describe phonon transport, while MD can be conveniently used to analyze the effects of size confinement on lattice thermal conductivities. Volz et al. [18] demonstrated by MD simulation that Si nanowire thermal conductivity could be two orders of magnitude smaller than the corresponding bulk value and they further argued that by adjusting the specularity parameter, results from the solution of the BTE could fit their MD simulation. Liang et al. [19] applied MD simulation to investigate the effects of atomic mass ratio in the alternating layers of a superlattice on the lattice thermal conductivity. Their results indicated that the thermal conductivity has a minimum value for some specific atomic mass ratio. Abramson et al. [20] studied the effects of interface number and elastic strain on the lattice thermal conductivity of Kr/Ar superlattices with MD simulation. It was argued that increase of the interface number per unit length does not necessarily result in decreased lattice thermal conductivity from their simulation results. Daly et al. [21], [22] reported MD simulation of a classical face centered cubic (FCC) lattice model to study the effects of interface roughness and isotope scattering on thermal conductivities. Simulation results predicted the similar trends for the lattice thermal conductivities of GaAs/AlAs superlattice along both in-plane and cross-plane directions compared with the experimental data [23]. In their model, it was also demonstrated that the lattice thermal conductivity of GaAs/AlAs superlattice had a minimum value with different layer thickness. This conclusion supported the hypothesis that zone folding was the dominant effect on lattice thermal conductivity in the short period superlattice. However, it should be noted that no lattice mismatch between the alternative materials of the superlattice was considered in their model. Volz [24] introduced the conjugate gradient method to minimize the potential energy of Si/Ge superlattices in order to relax the elastic strain on the alternating layers. Simulation results predicted an increasing trend of the superlattice thermal conductivity with the layer thickness.

Recently, experimental investigation on the thermal conductivity of Si and Si/SiGe superlattice nanowires has been carried out [25], [26]. Their results showed that for Si nanowires, the thermal conductivity could be greatly reduced compared with that of bulk Si because of the strong nanowire boundary scattering. For Si/SiGe nanowires, the thermal conductivity was below that of the two-dimensional Si/SiGe superlattice films, which was ascribed to the additional scattering mechanism provided by the nanowire boundary. In this paper, we apply molecular dynamics to study the lattice thermal conductivity of Kr/Ar superlattice nanowires in order to investigate the effects of interface scattering, nanowire boundary scattering, and period length.

Section snippets

Theoretical model and analysis

Nonequilibrium molecular dynamics (NEMD) was used to calculate the lattice thermal conductivities of solid Ar and Kr/Ar superlattice nanowires in the present study. The Lennard-Jones (L-J) potential was used to represent the interaction between two particlesVij(r)=4eijσijr12σijr6,where subscripts i and j stand for either argon or krypton particles, eij and σij represent the energy and length scale of the potential, and r denotes the distance between the two particles. Table 1 gives the

Effect of the number of interfaces

NEMD was used to predict lattice thermal conductivities of pure Ar and Kr/Ar superlattice nanowires. Fig. 2 shows the thermal conductivities of Ar and Kr/Ar superlattice nanowires versus nanowire length for a fixed period length of 8UC. For the Kr/Ar superlattice nanowire, one period is composed of two layers and the thickness of each layer is 4UC. The total length of the Ar nanowire was chosen to be the same as that of the Kr/Ar superlattice nanowire. Also, the two kinds of nanowires are of

Conclusion

In summary, the lattice thermal conductivity of Ar nanowires and Kr/Ar nanowires were studied by NEMD. The interfacial thermal resistance in the Kr/Ar nanowires contributes significantly to the thermal resistance and results in a lattice thermal conductivity that is only one-third of that of pure Ar nanowires. Interfacial thermal resistance increases with the length of the superlattice period for the Kr/Ar superlattice nanowires, which is attributed to increased acoustic impedance mismatch

Acknowledgements

Y.C. would like to acknowledge the financial support of the Chinese Natural Science Foundation (Project No. 50276011, 50275026) and the 863 High Technology Program (Project No. 2003AA404160).

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