Environmental Engineering Reference
In-Depth Information
For cases without phonon-phonon interactions, phonon trans-
missionfunctionsoftheleftandrightthermalleadsare,respectively,
L
(
ω
)
=
Tr[
G
r
R
G
a
L
]
(3.64)
and
R
(
ω
)
=
Tr[
G
r
L
G
a
R
].
(3.65)
Forcaseswithphonon-phononinteractions,weshouldadoptthe
effectivephonontransmissionfunctionshowninEqs.3.61and3.63.
The two transmission functions could be different, but they give
the same total thermal current. For thermal rectification effect, it is
meaningfulandimportanttoinvestigatethedifferencesbetweenthe
two transmission functions.
3.2.6
First-Principles-Based NEGF
We have systematically introduced the NEGF method, from basic
theory to practical calculation. Starting from the Hamiltonian, the
NEGF method can give all the one-particle observables of quantum
thermal transport. However, the NEGF method itself is not first-
principles, because the Hamiltonian, the most important quantity
of the method, requires external input. Following we discuss the
calculation ofthe Hamiltonian ofthermal-transport systems.
The Hamiltonian of phonons is actually composed of force
constants, that is, the derivatives of energy with respective to
displacement at equilibrium positions. Computing the Hamiltonian
by using empirical potential is very e
cient and feasible for large
systems. Successful empirical potentials are now available for few
elements,suchasC,O,H,andSi.However,theaccuracyoftheresults
strongly relies on the employed empirical potential parameters.
First-principles approaches, based on quantum mechanics, can
predict various materials' properties, without the need of any
empirical parameters. The first-principles approach we mention
is based on the Kohn-Sham density functional theory (DFT). DFT
has been widely and successfully used in the fields of physics,
chemistry, material science, biology and so on. It turns out to
be an important research tool for predicting materials' electronic,
mechanical, thermal, magnetic, andoptical properties.
