Environmental Engineering Reference
In-Depth Information
to 20% when
W
varies from 0.5 nm to 2 nm, and decreases
monotonously to 14% as
W
increases to 10 nm. Since thermal
conductance is isotropic in graphene sheets [42], we expect the
anisotropy factor of GNRs will eventually decease to zero when
W
is large enough. But what is the critical width at which the
anisotropy of thermal conductance disappears? To determine the
criticalwidth,weconsideredwiderGNRswith
W
upto35nm.Their
scaledthermalconductance,asshownintheinsetofFig.3.8a,varies
very slowly with
W
. Same is for the anisotropy factor. The room
temperature anisotropy factor of 35 nm wide GNR still has a value
of 13%. Obviously, a direct calculation of thermal conductance at
critical width using the NEGF method is beyond our computational
capability. So we apply linear regression to fit the data from 4 to
35 nm for ZGNRs and AGNRs, and find that the room temperature
anisotropy may disappear when
W
∼
140 nm.
The temperature-dependence of the anisotropy factor is also
investigated as shown in Fig. 3.8b. Varying temperature from 100 K
to 500 K only slightly affects the anisotropy factor except for very
narrow GNRs. While the anisotropy factor will decrease rapidly
when the temperature decreases from 100 K (the data is not shown
here). An extreme case is that at nearly zero temperature the
anisotropyfactorwillbezerobecausethermalconductanceofGNRs,
irrespective of edge shapes, are all 4
σ
0
,where
σ
0
is the quantum of
thermal conductance.
We have analyzed the size-dependence and temperature-
dependence of the anisotropy. What is the underlying mechanism
for this anisotropy? Similar anisotropy is found in the thermal con-
ductance of silicon nanowires (SiNWs) [43]. There, the anisotropic
phonon structure of bulk silicon is proposed to be the reason.
However, same mechanism does not apply for GNRs, because their
bulk counterpart, graphene, is isotropic in thermal conductance.
A MD calculation on thermal conductance of GNRs attributes the
anisotropytodifferentphononscatteringatedges.Thisisobviously
not the case here, because there is not boundary scattering for
periodic structure studied here. We showed that this anisotropy of
thermal conductance appears even without phonon scattering. Our
results suggest that it is the different boundary conditions at edges
