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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-12 -8 -4 γ v (Km/s) 10 15 20 25
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Fig. 2. Phonon dispersion along the high symmetry path, dispersion of Gru¨neisen parameter ( γ ) along the high symmetry path, Gru¨neisen pa rameter ( γ ) plotted as a function of frequency, and group velocity ( v ) plotted as a function of frequency, for planar sheets of Graphene (top left), LB-silicene (top right) LB-germanene (bottom left) and LB-stanene (bottom right).
Debye temperature also follows the same trend, caused by the spread of the acoustic phonon modes and the low mass of Carbon. The lower τ s, lower κ L s lower θ D s for the LB sheets are consequences of their lower group velocities.
Table 2. The Gru¨neisen parameter γ TA / LA , the highest phonon group velocity v TA / LA (Km / s), average velocity of the acoustic phonons v s (Km / s), mode Debye temperature θ TA / LA (K), and average Debye temperature θ D (K) of the transverse acoustic (TA) and longitudinal acoustic (LA) branches of the planar, low, and double buckled sheets. System γ TA γ LA v TA v LA v s θ TA θ LA θ D Graphene 0.80 1.5 14.2 23.1 15.4 829 1177 1946 LB silicene 0.50 0.9 6.0 10.5 6.7 140 195 715 LB germanene 0.90 2.1 3.4 6.0 3.8 80 104 392 LB stanene 0.73 0.6 2.2 4.2 2.4 45 72 222 DB silicene 0.20 1.8 5.4 9.6 6.0 61 100 751 DB germanene 0.70 1.7 3.0 5.5 3.3 35 50 401 DB stanene 0.02 1.8 1.9 3.9 2.1 21 24 221
3.5. Double buckled planar
The phonon dispersion of double buckled silicene, germanene, and stanene is shown in Fig. 3. Similar to the case of graphene and LB sheets, ZA mode has a parabolic dispersion in all the three DB sheets, while the TA and LA modes have linear dispersion at the point. The phonon spectrum of the DB sheets is found to be essentially similar to each other with their spectral width reducing from DB silicene to stanene. The reduction in spectrum width is purely due to increase in phonon mass down the group that manifests in lowering of the group velocity and Debye temperature of the acoustic phonons of the DB sheets down the group (Table. 2). The Gru¨neisen parameter ( γ ) and group velocity ( v )
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