Environmental Engineering Reference
In-Depth Information
confinement. In very low temperatures, due to the fact that thermal
wavelengthbecomescomparabletoorlargerthanthedimensionsof
the thermal pathway, the phonons transport ballistically. Following
the Landauer transport theory originally used for the electrical
conductance [28, 29], several groups have derived the expressions
of thermal conductance by ballistic phonons in low temperature
regions [20-23], and predicted that the thermal conductance at low
temperature is quantized in a universal unit
2
k
B
T
/
π
3
h
[20, 24],
similartotheroleof2
e
2
/
h
asaquantumofelectricalconductancein
one-dimensional wires. Afterwards, these predictions were verified
in a beautiful experiment by Schwab
et al.
[14]. Their experiment
shows a result very consistent with the quantum value at temper-
atures below about 0.08 K. To understand the quantum thermal
conductancebehaviors,manyeffortshavebeendevotedtoelucidate
the thermal-transport mechanisms in various nanostructures, such
as thin plates [25], nanowires [30-32], nanoribbons [33-38],
superlattices [39-41], nanotubes [42-46], and so on. Now, it has
become well known that unlike the diffusive phonon transport at
high temperatures in bulk materials, the thermal current in the
ballistic region is mainly carried by a set of the discrete vibrational
phonon modes due to the phonon boundary confinement in the
quantum systems. Therefore, in the low temperature limit, each
mode will contribute
K
0
(
K
0
2
k
B
T
/
3
h
) to the total thermal
conductance; a perfect sample has
N
A
K
0
with
N
A
being the number
of acoustic modeswith zero cutoff frequency at long wavelengths.
However, the smooth quantum plateau of the low temperature
thermal conductance is only presented if the transmission or
thermal phonons in a nanowire is perfect. For most cases, it is
unexpected that the quantum platform is prematurely destroyed
and instead the thermal conductance is decreased below the
quantum value as the temperature goes up in the low temperature
range. It is suggested that the decreases of the ballistic thermal
conductance in low temperature range may stem from the interface
scatteringduetoinhomogeneitiessuchasabruptjunctionsbetween
the wire and the reservoirs [47], rough surfaces [48-50], structural
defects [51-53] in the quantum structure, and so on. Based on
the full elasticity theory, Santamore and Cross investigated the
effect of surface roughness on the ballistic thermal conductance
=
π
