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

336

3

τ i N corresponding to the transverse acoustic ( i = TA / TA ) and the longitudinal acoustic ( i = LA ) modes, τ TA N /τ TA N , and τ LA N respectively, can be written as a function x = ω K B T .

4 γ 2

TA V

K B

1

K B

) xT 5

(1)

(

=

τ TA

M 3 v 5

N ( x )

TA

3 γ 2

LA V

K B

1

K B

) 2 x 2 T 5

(2)

(

=

τ LA

M 2 v 5

N ( x )

LA

where, V is the crystal volume (V = Area × ∆ ). The scattering lifetime for the Umklapp phonon scattering process for i = LA , TA is denoted by τ i U 1 τ i U ( x ) = γ 2 M v 2 i θ i ( K B ) 2 x 2 T 3 e − θ i / 3 T . Using all the three scattering lifetimes (as calculated using Eq. 2, 1, and 3), the lattice thermal conductivity κ L i for i = TA , LA is calculated using; (3)

[ θ i / T 0 θ i / T 0

τ i

4 e x

c x

] 2

3 {

θ i / T

τ i 4 e x ( e x − 1) 2 c x

1 3

x − 1) 2

τ i

N ( e

i =

(4)

C i T

dx +

}

κ L

τ i

4 e x

c x

0

x − 1) 2

τ i

i N ( e

U τ

depends on the group velocity v i of the concerned modes and τ i c = τ i U τ i N τ i U + τ i N

K B 4 2 π 2 3 v i

where, C i =

is the relaxation time.

The thermal conductivity is written as κ L = κ TA L + κ LA L .

3. Results and Discussion

3.1. Structure and stability

Supercells (Fig. 1(a)) have been constructed with 2 atoms per unit cell, for planar graphene (Fig. 1(b)) (space group p6 / mmm). Low buckled (LB) silicene, germanene, and stanene (Fig. 1(c)) (space group p3m1) also have their unit cell comprising of two atoms (Fig. 1(a)). In contrast, for double buckled (DB) silicene, germanene and stanene, two ad atoms are added to the 2 × 2 × 1 supercell of LB structures, resulting in a unit cell with ten atoms. DB sheet is constructed by addition of two ad-atoms to the 2 × 2 × 1 supercell of LB supercell, resulting in a dumble-type arrangement (Fig. 1(d)) with 10 atoms per unit cell. Unlike graphene and LB sheets, the bond distances and angles in the DB sheets are found to be non-degenerate [as shown by d and d’, θ and θ in Fig. 1(a)]. The neighboring bonds meeting at the two para positions of the hexagonal ring to form the dumbells are found to be uniformly stretched uniformly stretched (d’ in Table. 1), while the bonds between the inplane atoms are found to be equal but shorter (d in Table. 1). The lattice parameter a 0 , buckling height ∆ , nearest neighbor distance (d and d’), and angle between neighboring bonds ( θ and θ ) in the planar sheets Table. 1 are in good agreement with literature (Mortazavi et al., 2017). The degree of buckling of the DB sheets is found to be several times higher than those in the LB sheets Table. 1. The cohesive energy, enlisted in Table. 1, is calculated from the DFT total energy of the sheet using standard prescription. The cohesive energy of graphene is found to be highest. The low buckled sheets have lower cohesive energy (by 0.2 eV) than their corresponding double buckled counterpart.

3.2. Vibrational Properties

We start with the phonon dispersion of graphene within harmonic approximation, that is well known to be consist ing of 6 phonon branches, 3 acoustic (LA, TA and ZA) and 3 optical (LO, TO, ZO) modes. The dispersion of acoustic branches merges with that of optical branches, The TA and LA phonon modes have almost linear dispersion near the Γ -point, while ZA mode has quadratic dispersion. In contrast, the acoustic phonon dispersion of LB of silicene,