PSI - Issue 23

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

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G. P. Das / Structural Integrity Procedia 00 (2019) 000–000

DB germanene

DB silicene

DB stanene

10

(a)

DB-Stanene

DB-Germanene

500

300

0

DB-Silicene

200

400

γ

-10

-20

200

)

300

-1

10

ω (cm

100

(b)

DB-Germanene

200

DB-Stanene

100

5

DB-Silicene

100

v (Km/s)

0

0

0

0

0

100

200

300

400

500

K

Γ

Γ Γ

M

K

M

Γ

Γ

M

K

Γ

-1

ω (cm

)

(a)

(b)

(c)

Fig. 3. (Left) Phonon dispersion of double buckled (a) silicene, (b) germanene, (c) stanene along the high symmetry path in the Brillouin zone. (Right) Comparison of (a) acoustic phonon group velocity and (b) Gru¨neisen parameter γ of DB silicene, germanene, stanene as function of their frequency of vibration.

of the sheets are plotted as a function of frequency in Fig. 3. The average v of DB germanene and stanene is found to be lower than DB silicene. γ TA / LA and v TA / LA of the low buckled sheets are enlisted in Table. 2. Interestingly, γ TA / LA of the DB sheets is found to be lower than those of the LB sheets, even though the DB sheets have higher cohesive energy per atom than the LB sheet (Table. 2). We have plotted the average velocity of the acoustic phonons v s (Km / s) and the average Debye temperature θ D (K) of the sheets as a function of their buckling height in Fig. 4. We find the presence of buckling height leads to lowering of v s and θ D by orders of magnitude from graphene to LB sheets. However, the increase in buckling height from LB to DB sheets, does not lead to any severe change in any of these parameters. This implies that maximum of the phonon scatterings take place as soon as the out of plane vibration is channelized through their buckled geometry; increase in the buckling height does not really lead to change / tuning of these parameters any further. The Debye Callaway formalism has been used along with the parameters enlisted in Table. 2., to calculate the lattice thermal conductivity κ L of the graphene, LB and DB sheets Fig. 4 by including the contributions stemming from their TA and LA modes. For planar graphene, this formalism yields a diverging tendency in γ ZA near the Γ -point. The phonon transport in pure 2D needs to include all the way to zero phonon frequency ω = 0. In order to avoid this, a buckling height of 3.3 Å has been used from experiment (Nika et al., 2009) to estimate κ L . The Umklapp phonon scattering lifetime is essential to capture the in-plane vibrational transport accurately the cut-o ff frequency for Umklapp process cannot be introduced by analogy with bulk graphite. Thus, the Debye Callaway formalism is found to underestimate the κ L of graphene (Fig. 4) by at least a factor of two (Bhattacharya et al., 2018). Unlike the case of graphene, the Debye Callaway formalism yields good agreement for the κ L of the sheets with the low buckled and double buckled geometry (Fig. 4(b), (c), and (d)) (Bhattacharya et al., 2018). The results have been compared (Table. 3) with those obtained by Peng et al. (2016). The notable features are (i) LB stanene has higher κ L than LB germanene (ii) DB sheets have consistently lower κ L as compared to LB sheets (ii) DB stanene has lowest κ L . The thermal conductivity decreases with increasing temperature for all the 2D sheets, as expected for phonon dominated materials. The reason that Debye Callaway formalism works for the LB and DB sheets is the presence of buckling in their geometry, which others the possibility to treat these sheets as a crystal (with a definite third dimension). * Debye Callaway formalism works in presence of buckling i.e. for LB and DB sheets, but not for graphene (see text for details). 3.6. Lattice thermal conductivity