Environmental Engineering Reference
In-Depth Information
that all optical phonons are not excited. For that case, only
those acoustic phonons with frequencies close to zero are
excited and each of them contributes one quantum of thermal
conductance. For quasi-one-dimensional (quasi-1D) systems
(such as, nanowires, nanotubes, and nanoribbons), there are
four eigenmodes of zero frequency, originating from three
rigid translation invariances and one rigid rotation invariance.
Consequently, their thermal conductance is 4
σ
0
at nearly zero
temperature.
2. Accurately measuring
σ
0
, which is a very small quantity at
low temperatures, requires extremely sensitive calorimeters and
e
cient approaches to exclude influence of environment, and
thus is very challenging in experiments. As one of the most
important breakthroughs in the research of quantum thermal
transport, Schwab
et al
. measured thermal conductance of
suspended insulating nanostructures at very low temperatures
(below 1 K), and for the first time experimentally proved the
existenceof quantizedthermal conductance [6].
3.2 The Non-equilibrium Green's Function Method
In this chapter, we will systematically discuss the NEGF formalism
and try to offer a complete theory for the investigation of quantum
thermaltransport.TheNEGFmethodiswidelyandsuccessfullyused
forthestudyofelectronictransport.Herethemethodisgeneralized
for thermal transport. Since we focus on the thermal transport
contributed by phonons, the NEGF represents the NEGF of phonons
here.
3.2.1
Hamiltonian of Thermal-Transport Systems
Figure 3.3, using graphene junction as an example, shows a
representative thermal-transport system. The center part (C) is
connected to two thermal leads on the left (L) and right (R).
The thermal leads are composed of semi-infinite ideal systems, in
which phonon transport experiences no geometric scattering. The
inclusion of many-body interactions in thermal leads on the one
