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Figure 3.10 Phonon dispersion of graphene computed by density func-
tionaltheory(DFT).Thecalculationusesnormconservingpseudopotential,
planewavebasiswithanenergycutoffof40Hartree,andTroullier-Martins
density functional in local density approximation (LDA) as implemented
in the ABINIT code. Reprinted from Ref. [45], Copyright (2009), with
permission from Elsevier.
longitudinal optical (LO) phonons, which correspond to in-plane
longitudinalvibrations.
At long wavelengths the LA and TA phonon modes have a linear
dispersion. In contrast, due to rotational symmetry the flexural
ZA phonon mode obeys a quadratic dispersion in the vicinity
of the point of the Brillouin zone, which is a characteristic
feature of layered crystals [46]. Due to the existence of this
quadratic dispersion, at low temperatures thermal conductance
of graphene is dominated by the T 1.5 contribution of the ZA
phonon mode, in contrast to the T 2 contribution of the linear LA
and TA phonon modes [41]. Note that the numerical calculations
may slightly break the rotation symmetry, which can cause the
dispersion of the ZA mode deviate from quadratic. Therefore,
numerically reproducing the quadratic dispersion is di cult. In
fact, our calculated ZA branch is linear close to the point.
This inaccuracy in phonon dispersion changes the temperature
dependence of thermal conductance at low-temperature region
(less than 50 K). Nevertheless, the temperature dependence of
 
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