Environmental Engineering Reference
In-Depth Information
4.2.2
Continuum Elastic Model
In the low temperature region, the thermal conductance is mainly
attributed to the contributions of several lowest phonon modes
with low frequency. The wavelength of these phonons is generally
much larger than any microscopic length, such as the atomic bond
length. Thus, the continuum elastic model is quite adaptive to
describe the ballistic phonon transport in quantum structures. This
is also confirmed by microscopic calculations [76] and Raman
experiments [77] for dielectric continuum model and by phonon
transmission spectroscopy [78] for the continuum elastic model.
According to continuum elastic theory, the phonon displacement
field
U
(
u
x
,
u
y
,
u
z
)intheCartesiancoordinatescanbegiveninterms
of the scalar potential
φ
andthe vector potential
H
[79-81]
U
=∇
φ
+∇×
H
(4.7)
with
H
=
(
H
x
,
H
y
,
H
z
)and
∇=
∂/∂
xi
+
∂/∂
y
j
+
∂/∂
zk
.Eachpotential
function satisfies the following waveequations:
v
L
∇
2
2
φ
+
ω
φ
=
0,
(4.8)
v
T
∇
2
H
x
+
ω
2
H
x
=
0,
(4.9)
v
T
∇
2
H
y
+
ω
2
H
y
=
0,
(4.10)
v
T
∇
2
H
z
+
ω
2
H
z
=
0,
(4.11)
/∂
z
2
.Here,
ω
isthefrequencyofthe
wave and
v
L
,
v
T
denote the sound velocities of the bulk longitudinal
and transverse acousticwaves, respectively, and given as
2
2
/∂
x
2
2
/∂
y
2
2
with
∇
=
∂
+
∂
+
∂
λ
+
μ
2
v
L
=
(4.12)
ρ
μ
ρ
=
v
T
,
(4.13)
where
ρ
and
λ
(
μ
) represent the mass density and the Lame
coe
cients,respectively.AccordingtoEq.4.7,thedisplacementfield
U
can be divided in to three componentforms as
u
x
=
∂φ
∂
x
+
∂
y
−
∂
H
z
∂
H
y
∂
z
,
(4.14)
