PSI - Issue 23

Gour P. Das et al. / Procedia Structural Integrity 23 (2019) 334–341 G. P. Das / Structural Integrity Procedia 00 (2019) 000–000

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mixture of sp 2 and sp 3 hybridization due to presence of inherent buckling (Zhang et al., 2015), in sharp contrast to sp 2 bonded graphene that has zero buckling. The interest in silicene stems from the fact that it could provide an easier compatibility with silicon based microelectronics technology. The extent of buckling facilitates tuning of the band gap by external means, such as by application of an electric field or some in-plane mechanical strain (Bhattacharya et al., 2018). Both these routes pose non-trivial challenge for opening a much-needed band-gap in graphene. Interestingly it has been observed that the Dirac cone characteristics are retained not only in pure silicene and germanene, but also in Si 1 − x Ge x 2D alloys with honeycomb lattice (Jamdagni et al., 2015). Although there is resemblance in the lattice structure as well as electronic structure of graphene and its Group-IV 2D analogues (Bhattacharya et al., 2015), their phonon dynamics are found to be quite di ff erent. The planar geometry, low mass and strong in-plane sp 2 bonding results in a large lattice thermal conductivity ( κ L ) of graphene ( ∼ 2000-4000 Wm − 1 K − 1 at room temperature). Silicene and the subsequent members in this family have much lower κ L ( ∼ 2 orders of magnitude lower) posing a potential challenge on the thermal management of such materials in microelectronics. There are possible routes to increase κ L of silicene viz (a) by increasing the size (length) of the flakes (Gu and Yang, 2015) (b) by hydrogenation (Lin et al., 2017) (c) by making 2D alloys of silicene and graphene viz. 2D-SiC. It may be noted that mechanical exfoliation that works for graphene, does not necessarily work for silicene, ger manene, stanene, and hence the latter ones cannot be realized as free standing. Rather these Gr-IV sheets (monolayer, bilayer etc.) have been grown epitaxially on various substrates, viz. metallic Ag (111) (Vogt et al., 2012), III-V semi conductors (Bhattacharya et al., 2013) and insulating ZrB 2 (Fleurence et al., 2012). More recently, in order to explain the structural phase transition of silicene grown on Ag (111) surface, a double buckled (DB) structure was proposed, which was calculated to have higher cohesive energy per atom than the low buckled silicene (Zhao et al., 2016). Similarly, synthesis of buckled layer of germanene on Al (111) substrate (Derivaz et al., 2015) and stanene (Zhu et al., 2015) have been reported. Several groups have reported thermal conductivity of buckled and double buckled sheets as for instance Nissimagoudar et al. (2016) and Peng et al. (2016), apart from other interesting properties like quantum spin hall e ff ect (Tang et al., 2014). In this article, we explore the lattice dynamics of the members of the grapheme family, and establish connection between the intrinsic parameters (such as group velocity, Gru¨neisen parameter, and Debye temperature of the acous tic phonon modes) and the lattice thermal conductivity. Our calculations (Bhattacharya et al., 2018) show that the presence of buckling reduces the group velocity and the Debye temperature of the sheets down the group, and hence, reduces their lattice thermal conductivity. Density functional theory (DFT) calculations as formulated by Hohenberg and Kohn (1964) and Kohn and Sham (1965) have been performed using FHI-aims (Blum et al., 2009), which is an all electron, full potential electronic structure code that uses numeric, atom-centered basis set. LDA-PW91 (Perdew et al., 1992) has been used for the treatment of electronic exchange and correlation. All numerical settings are chosen so as to ensure convergence in energy di ff erences to better than 10 − 3 eV. The atomic positions and lattice vectors are fully relaxed for all structures and the Hellman-Feynman forces are converged to less than 10 − 3 eV Å − 1 . A vacuum spacing of 15 (20) Å has been used for the LB (DB) sheet in order to prevent interactions between the adjacent layers. Converged reciprocal space grid of 24 × 24 × 1 (16 × 16 × 1) per unit cell is used for the lattice relaxation of the LB (DB) sheets. Converged supercell of 16 × 16 × 1 (4 × 4 × 1) is used to calculate the phonon band structure of the low buckled (double buckled) sheets with two (ten) atoms per unit cell. The phonon band structure is calculated using the finite displacement method as implemented in the phonopy (Togo et al., 2008) code. The phonon group velocity, total ( v ) and mode resolved ( v i ), Gru¨neisen parameter total ( γ ) and mode resolved ( γ i ), have been extracted from the harmonic phonon band dispersion using indigenously developed python based extensions to phonopy-FHI-aims. For calculating γ and γ i , the lattice is subjected to biaxial strains of 2 %. We use the Asen-Palmer modified (Asen-Palmer et al.,1997) version of Debye Callaway theory (Callaway, 1959) as applied for solid in Zhang et al. (2012) and Chauhan et al. (2018) to model the phonon scattering lifetimes and calculate the lattice thermal conductivity of the sheets. Please note that the normal phonon scattering lifetime 2. Methodology