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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Fig. 1. (a) The top view of the unit cell of the two dimensional sheets (shown in dotted lines). The lateral view of 2D sheets with (b) planar, (c) low buckled, and (d) double buckled geometry.

Table 1. The calculated lattice parameter a 0 (Å), nearest neighbor distance d and d’ (Å), buckling height ∆ (Å), angle between neighboring bonds θ and θ ’ ( ◦ ) and cohesive energy E c (eV / atom) of graphene, LB and DB silicene, germanene, and stanene. The parameters a 0 , d, d’, θ and θ ’ are symbolically shown in Fig. 1(a). System a 0 d d’ ∆ θ θ E c Graphene 2.44 1.41 1.41 - 120.0 120.0 -10.11 LB silicene 3.82 2.25 2.25 0.43 116.2 116.2 -5.23 LB germanene 4.02 2.38 2.38 0.68 112.8 112.8 -4.58 LB stanene 4.55 2.76 2.76 0.84 111.5 111.6 -3.90 DB silicene 7.34 2.32 2.34 2.69 90.3 114.2 -5.43 DB germanene 7.65 2.41 2.47 2.88 89.4 114.0 -4.77 DB stanene 8.82 2.79 2.84 3.34 89.1 113.9 -4.12 germanene and stanene are separated completely from the optical phonon dispersion. Unlike graphene, the acoustic modes are more localized, suggesting lower group velocities v i s for the buckled systems. From the point of view of thermal conductivity, the crucial component is the flexural out-of-pane acoustic mode (ZA branch) that strictly fol lows quadratic behavior ( ω ZA is proportional to q 2 ) near the Γ -point for graphene. Whereas for silicene (and also for germanene and stanene) the dispersion of the ZA branch always contains a linear component just like in sp 3 bonded material like Si. The phonon scattering lifetimes τ are inversely proportional to γ 2 and directly proportional to powers of group velocities v i s (Eqn. 3). High γ signifies low τ and hence low κ L ; similarly, high v i indicates high τ and hence high κ L . For a compound with harmonic potential γ 1 and any deviation form unity is due to anharmonicity of the compound. In graphene, the ZA mode converges in the M → K region that mainly contributes to the high thermal conductivity of graphene. In contrast, the ZA modes in silicene, germanene, stanene do not show any dispersion corresponding to | γ | < 4 and hence do not contribute to κ L . The general observation is that the ZA branch in Graphene contributes ∼ 75% to the large κ L (remaining 25% from the LA / TA branches and minute contribution from optical branches). While in case of silicene, the ZA branch contribution is < 10% while the predominant contribution is from TA / LA and optical branches (Gu and Yang, 2015). 3.3. Gr u¨ neisen parameter ( γ )

3.4. Group velocity (v)

The group velocities of the acoustic modes in graphene ( v TA 14.2, v LA 23.1 Km / s ) are found to be higher than all other sheets (Table. 2), resulting in the anomalously high τ s and hence κ L of the sheet. The corresponding average