Molecular Dynamics Prediction of the Thermal Conductivity of Si/ge Superlattices

Abstract

Molecular dynamics simulations and the non-equilibrium direct method are used to predict the thermal conductivity of a Si/Ge superlattice modeled by the Stillinger-Weber potential at a temperature of 300 K. We focus on the methodology of making the thermal conductivity prediction (limited effort has been made to model Si/Ge nanocomposites in the literature) and find that proper selection of the size and composition of the thermal reservoirs is important. NOMENCLATURE Ac cross-sectional area a, b constants used in method to impose the heat flux A, B, p, q constants in the Stillinger-Weber potential c nondimensional cutoff radius CP cross-plane direction Cv specific heat (J/m3-K) Ek,R total kinetic energy of reservoir before scaling ∆Ek kinetic energy added to the hot reservoir or subtracted from the cold reservoir IP in-plane direction k thermal conductivity L, Ltot period length (monolayers), total length (nm) LSi, LGe layer thickness of silicon and germanium (monolayers) ∗Address all correspondence to this author. m mass NR total number of atoms in reservoir PR total momentum of reservoir before scaling q heat flux r, r particle separation, particle position vector, particle separation vector R scaling factor in imposed heat flux scheme, reservoir ∆t time step T temperature v2, v3 twoand three-body terms in the Stillinger-Weber potential v, v′, vsub velocity, velocity after scaling, subtracted velocity VR total velocity of reservoir before scaling y normalized particle separation Greek εi j energy scale for atomic pair (i, j) λ, γ constants in the Stillinger-Weber potential Λ phonon mean free path Φ potential energy ν phonon group velocity σi j length scale for atomic pair (i, j) Subscripts ∞ infinite system size e f f effective i, j, k summation indices, particle labels 1 Copyright c © 2007 by ASME INTRODUCTION Superlattices (i.e., composite, periodic structures containing alternating material layers with thicknesses as small as one atomic monolayer) can now be built with high precision due to advancements in fabrication techniques such as molecular beam epitaxy [1–5]. Superlattices have potential for application in thermoelectric energy conversion devices because the crossplane thermal conductivity can be reduced while maintaining good electron transport properties, resulting in high values of the thermoelectric figure-of-merit [6]. The succesful design and development of superlattices for specific applications is dependent on understanding the thermal transport behavior in these structures. Experimental techniques have been used to characterize the thermal properties of typical semiconductor superlattices (e.g., Si/Si1−xGex, GaAs/AlAs, and Bi2Te3/Sb2Te3) [1–4, 7, 8]. In some cases, the superlattice thermal conductivity is observed to be less than an alloy of similar composition [1, 3, 4, 7, 8]. The reduction below the alloy thermal conductivity for Si/Si1−xGex superlattices [4] has been suggested [9] to be caused by strain induced defects and dislocations resulting from the lattice mismatch. However, experimental studies on different material systems without significant defects [1,3,8] also show reductions below the alloy value. The effect of the period length on the experimental thermal conductivities is also conflicting. Some studies [7, 8] have found that the cross-plane thermal conductivity decreases with decreasing period length until a minimum is reached, beyond which the thermal conductivity increases, while others [1,2] have observed that the cross-plane thermal conductivity decreases monotonically with decreasing period length. Modeling efforts have used lattice dynamics calculations [10–12], the Boltzmann transport equation (BTE) [13], and molecular dynamics (MD) simulations [14–20] to investigate the observed experimental trends. Because assumptions about the nature of phonon transport (e.g., a constant phonon relaxation time) are required in lattice dynamics and BTE approaches, these analysis techniques are not ideal. Molecular dynamics simulations, which require no prior assumptions about the nature of phonon transport, are a good method for studying the thermal transport behavior, provided that a suitable interatomic potential for the given material system is available. Molecular dynamics simulations were performed by this group on Lennard-Jones superlattices containing component species that differed only in their mass [20] (and therefore having identical lattice constants). The superlattice thermal conductivity was found to be greater than the MD predicted alloy thermal conductivity for a variety of unit cell designs. This finding supports the hypothesis that defects and dislocations resulting from the strain associated with lattice mismatch are the cause of the experimentally observed thermal conductivities below the corresponding alloy values. The MD method has also been used by Chen et al. [15] to examine the conditions required to produce a minimum in the cross-plane thermal conductivity of LennardJones superlattices. It was found that a minimum exists when there is no lattice mismatch, but when the species were given a lattice mismatch of 4%, the thermal conductivity decreased monotonically with decreasing period length. Daly et al. [16] and Imamura et al. [17] used MD to predict the effect of interface roughness on the minimum thermal conductivity for model GaAs/AlAs superlattices. In their simulations, the crystal structures were simplified so that the two-atom basis for GaAs was treated as a single atom with a mass equal to the average masses of Ga and As, and the AlAs two-atom basis was treated in a similar manner. Both groups found that a minimum existed in the cross-plane thermal conductivity for perfect interfaces, and that the thermal conductivity decreased monotonically with decreasing period length for rough interfaces. To our knowledge, there has only been one limited study to examine the thermal conductivity of a realistic superlattice system (Si/Ge) that has not simplified the superlattice structure [18]. In this work we use MD simulations and the non-equilibrium direct method to investigate the cross-plane thermal conductivity of a Si/Ge superlattice. Because there has only been limited work using MD to predict the thermal conductivity of Si/Ge nanocomposites, we focus this investigation on the methodology used in making the thermal conductivity prediction. The techniques developed in this study will serve as the starting point for further investigation into the thermal conductivity design space associated with the Si/Ge material system. We begin by describing the superlattice model, the basics of the MD simulations, and the direct method for predicting the thermal conductivity. Results showing the effect of in-plane dimensions, and the size and composition of the thermal reservoirs on the thermal conductivity prediction are then presented. MOLECULAR DYNAMICS SIMULATIONS AND THE STILLINGER-WEBER POTENTIAL The atomic interactions are modeled using the StillingerWeber potential. This potential was originally developed to model silicon [21] and later parameterized for germanium [22]. Mixing rules were introduced to model the silicon-germanium interactions [23]. The total system potential energy, Φ, is the sum of twoand three-body terms, and is given by Φ = ∑ i ∑ j>i v2(i, j)+∑ i ∑ j ∑ k> j v3(i, j,k), (1) where v2 and v3 are functions of the positions and species of atoms i, j, and k [21]. The first term is a summation over all atom pairs, and the second term is a summation over all triplets 2 Copyright c © 2007 by ASME Table 1. STILLINGER-WEBER ENERGY AND LENGTH SCALES FOR SI-SI, GE-GE, AND SI-GE INTERACTIONS [21–23]. Interaction Si-Si Ge-Ge Si-Ge/Ge-Si εi j (eV) 2.170 1.930 2.043 σi j (Å) 2.095 2.181 2.138 with atom i at the vertex. The two-body term is v2(i, j) = { εi jA(By −p i j − y−q i j )exp[(yi j− c)−1], if yi j < c 0 otherwise, (2) where A, B, p, and q are constants and c is the nondimensional cutoff radius. yi j is a dimensionless pair separation defined as ri j/σi j, where ri j = |ri− r j|, and ri and r j are the positions of atoms i and j. εi j and σi j are the energy and length scales for the atomic pair (i, j) and are given in Table 1. The three-body term is v3(i, j,k) =    (εi jεik)(λ jλi λk) × exp [ γ yi j−c + γ yik−c ] × (cosθ jik + 1 3 )2 , (3) for both yi j < c and yik < c, and zero otherwise. λ and γ are constants. The cosine of the angle formed by ri j and rik is cosθ jik = r i j ·r ik ri jrik . (4) The constants in Eqs. (2) and (3) are A = 7.0496 B = 0.6022 p = 4 q = 0 a = 1.80 γ = 1.20 λSi = 21.0 λGe = 31.0. (5) The superlattice examined here consists of silicon and germanium layers with thicknesses LSi and LGe. LSi and LGe are both equal to sixteen monolayers, giving a total period length of L = LSi + LGe = 32 monolayers. One period of this superlattice structure is shown in Fig. 1. The interfaces between the layers are perfect (no species mixing), and the atoms are initially located on diamond lattice sites. Note that the directions perpendicular and LSi Cross-Plane (CP) Direction In-Plane (IP) Directions LGe

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Cite this paper

@inproceedings{Landry2007MolecularDP, title={Molecular Dynamics Prediction of the Thermal Conductivity of Si/ge Superlattices}, author={Eric S. Landry and Alan J. H. McGaughey}, year={2007} }