Environmental Engineering Reference
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and the anharmonic term H anh . Here, s i α ( t )isanoperatorin
the Heisenberg picture for atomic displacement from equilibrium
along the α direction of the i th atom with mass M i . p i α ( t )isthe
momentumoperatorconjugatedtothedisplacementoperator s i α ( t ),
and k i α , j β represents the spring constant between the i th atom in
the
direction. The system
Hamiltonian is assumed to be divided into fiveparts:
H sys = H L + H LS + H S + H RS + H R (A.2)
Here, H L/R is the Hamiltonian for the left/right thermal lead, H S
is that for the scattering region, and H LS(RS) is the Hamiltonian for
the coupling between the scattering region and the left (right) lead.
Theanharmonicterm H anh isassumedtoexistonlyinthescattering
term H S . Different temperatures, T L and T R (
α
direction and the j th atom in the
β
<
T L ), are assigned to
the left and right regionsof the system, respectively.
The thermal current flowing through the interface between the
left lead and the scattering region can be calculated from the time
evolution ofthe energy of the left lead:
dH L
dt
H L , H sys
i
=−
=
J
(A.3)
wherethebracketsinEq.A.3denotesthenon-equilibriumstatistical
averageofphysicalobservable.Thethermalcurrent J isrewrittenas
i
G i α , j β ( t , t ) + H . c .
t
i L , j S
αβ =
k i α , j β
2
d
dt
=− lim
t
J
(A.4)
xyz
using the greater and lesser Green's functions associated with the
contact between the left lead and the scattering region:
i G i α , j β ( t , t ) = s i α ( t ), s j β ( t )
(A.5)
and
i G i α , j β ( t , t ) = s j β ( t ), s j α ( t ) (A.6)
SincetheGreen'sfunctionsdependonlyonthetimedifferencein
a steady state, it is convenient to work in Fourier space ( ω space).
Then, in the steady state, the thermal current is expressed as
k i α , j β G i α , j β (
H . c . (A.7)
d
ω
G j β , i α (
π ω
J
=−
ω
)
ω
)
+
2
0
i
S
αβ = xyz
L , j
where the relation G i α , j β (
ω
=
G j β , i α (
ω
)
) is used.
 
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