Environmental Engineering Reference
In-Depth Information
where
T
C
denotes the temperature of the classical system. The
quantum-mechanicaldescription of lattice energy
E
Q
is defined as
n
λ
+
,
1
2
E
Q
=
ω
λ
(1.29)
λ
where
ω
λ
and
n
λ
aretheeigenfrequencyandoccupationnumberfor
a given eigen-mode
λ
. When the system temperature is higher than
the Debye temperature, the occupation number given by the Bose-
Einstein distribution
n
λ
=
1
/
e
ω
λ
/
k
B
T
Q
−
1
can be approximated
by
k
B
T
Q
/
ω
λ
, and thus the associated energy for each mode is
retarded to the classical case of
k
B
T
Q
, which means all the modes
contribute equally to the total energy (excluding the zero-point
energy).However,atlowtemperature,thereisafreezingoutofhigh-
frequency modes in the quantum system so that energy associated
with different modes deviates from the classical case.
To qualitatively account for this discrepancy, a system-level
mapping between the classical temperature used in MD simulation
(
T
MD
)andthequantumtemperaturefortherealenvironment(
T
real
)
has been proposed by equating the total energy in the classical and
quantumdescriptions
ω
D
3
Nk
B
T
MD
=
DOS
(
ω
)
n
(
ω
,
T
real
)
ω
d
ω
,
(1.30)
0
DOS
(
ω
)
1
2
+
n
(
ω
,
T
real
)
ω
D
3
Nk
B
T
MD
=
ω
d
ω
,
(1.31)
0
where
ω
D
is the Debye frequency,
DOS
(
ω
) is the density of states
for modes with frequency from
ω
to
ω
+
ω
.Thisequalitypermits
a quantitative relation between
T
MD
and
T
real
based on the lattice
dynamics information of the system, and both schemes with and
withoutzero-point energy have been used in literature.
Figure 1.2 shows the typical relation between
T
MD
and
T
real
in
SiNWs. Without zero-point energy (Eq. 1.30), the classical temper-
ature
T
MD
can reach zero temperature and increases monotonically
withtheincreaseofquantumtemperature
T
real
.Duetotheinclusion
of zero-point energy in Eq. 1.31, there exists nonzero minimum
temperature for the classical temperature
T
MD
, which shifts up
T
MD
byaconstantcomparedtothatwithoutzero-pointenergy(Fig.1.2).
