Environmental Engineering Reference
In-Depth Information
with increasing the temperature. This is due to the fact that the
wide frequency integration range in Eq. 4.3 can smear out all the
oscillationsinthetransmissioncoe cient,whichisconfirmedhere.
Clearly in the low temperature region, the thermal conductance is
contributedbytheseverallowestphononmodes,especiallythezero
acoustic mode. As discussed above, all the thermal conductance
approaches the universal quantum value in the limit T
0due
to the perfect transmission of zero mode at
ω
0. Note that
/
the reduced thermal conductance K
T of the zero mode seems to
be insensitive to the variation of temperature, while that for higher
modes rapidly increases as the temperature goes up. In addition,
with the increase of the width or length of T-shaped junction, the
overall reduced thermal conductance K / T decreases noticeably,
which indicates that the thermal conductance may be effectively
tuned by adjusting the geometric parameters of T-shaped junction.
Forcase(ii),Chen et al. [51] found that the dependence of the
thermal conductance on the temperature is qualitatively different
for different types of defects. When the defect is void (see Fig.
4.3), namely the stress-boundary condition is applied between the
defect and surroundings, it is shown that only the zero mode
contributes to the thermal conductance and K approaches the
universal quantum value π
2 k B T / 3 h as the temperature T 0.
Zero mode with m = 0 propagating through the structure is
the peculiarity of the elastic wave, which stems from the stress-
free boundary condition employed for the void defect allowing the
propagationofthemodewith ω = 0.Moreover,thereducedthermal
conductance K
0 is independent of the size
of the defect (Fig. 4.3(a)), which indicates the scattering effect of
the void defect for the long wavelength phonons with
/
T in the limit T
0is
very small. By increasing the temperature, because many higher
modesbeingexcitedbegintocontributetothethermalconductance,
theirreducedthermalconductance K
ω
/
T ismonotonouslyincreased
indicating K T x ( x > 1) for higher modes (Fig. 4.3b and c).
However, for the zero mode the reduced thermal conductance K / T
unexpectedlydecreaseswithincreasingtemperatureindicating K
T x (0 < x < 1) for the zero mode. Due to the main contribution
of the zero mode to the thermal conductance, the decrease of the
overall reduced thermal conductance K / T can be also found in low
 
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