Environmental Engineering Reference
In-Depth Information
total phonon transmission function isexpressed as
ζ
(
ω
)
=
Tr
R
(
ω
)
G
(
ω
)
L
(
ω
)
G
†
(
ω
)
(2.11)
ω
in terms of the retarded Green's function
G
(
) of the central
conductor connected to the left and right leads. Here, the retarded
Green's function is givenby
G
(
ω
)
=
ω
+
i
0
+
−
D
−
L
(
ω
)
−
R
(
ω
)
−
1
,
2
(2.12)
where the underlined quantities represent matrices with a basis in
the central region,
D
is the dynamical matrix of the central region,
which is derived from the second derivative of the total energy with
respecttotheatomcoordinatesinthecentralregion,and
L/R
(
ω
)is
the self energy due to the left/right lead [13-15]. Its imaginary part
relates to thelevel width function
L/R
asseen in Appendix A.
For the estimation of the phonon transmission function through
Eq.2.11,wehavetoknowtheretardedGreen'sfunctionmatrix,
G
(
ω
),
in Eq. 2.12. The direct numerical procedure to calculate Eq. 2.12
requires a higher computational cost as the length of the conductor
becomeslonger.However,Eq.2.11showsthatnotallelementsofthe
matrixareneededinrealspaceandonlypartoftheelements,which
relate phonon propagation between atoms in the left side and those
in the right side in the conductor, are necessary. In the following, we
shortlyreviewtherecursiveGreen'sfunctiontechniquethatenables
ustocalculatethesematrixelementse
ciently[16].Thistechnique
is based on the Dyson's equation that the retarded Green's function
within the harmonic approximation satisfies. In the absence of
many-body interactions such as the phonon-phonon interaction,
we can obtain the perturbed Green's function exactly, as far as the
solution of the equation converges. In this technique, the Green's
function in Eq. 2.12 is obtained by a recursive procedure. At each
step of the procedure, a perturbation is added and some blocks of
the Green's function are exactly calculated based on the Dyson's
equation.Asaresult,wecanreducethecomputationalmemoryand
get a highercomputational speed.
First, the conductor is divided into layers appropriately, de-
pending on its geometry (crystal structure and interatomic bonds).
For simplicity, we consider the conductor having a square lattice
structure, and only the adjacent atoms are assumed to interact with
