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2
2 e ω/ ( k B T )
( e ω/ ( k B T )
ω
σ ( T ) =
d ω
1) 2 ( ω ),
(3.8)
2 π k B T 2
0
where k B is the Boltzmann constant.
Thenweshowthatmanyimportantfeaturesofmesoscopicther-
mal transport can be revealed by analyzing the Landauer formula
for thermal conductance (see Eq. 3.8). For ballistic transport,
ω
(
)
ω
isequaltothenumberofphononmodesoffrequency
withpositive
group velocity. ( ω ) is an integer number and will not change
when the transport length L is increasing. Thus, σ ( T ) is constant
(independentof L )atagiventemperature.Thethermalconductivity
κ σ L isproportional to L and diverges for infinitelarge L .
Thermalconductanceisaweightedintegralofphonontransmis-
sion function, as shown by the Landauer formula. The weighting
factor is strongly dependent on the phonon frequency. Defining x =
ω/ ( k B T ),Eq.3.8 changes into
k B T
x .
k B T
h
x 2 e x
( e x
σ ( T ) =
1) 2
dx
(3.9)
0
1) 2 )asa
function of x . It clearly shows that the weighting factor deceases
very fast with an increasing x . When x > 10, the weighting factor is
nearlyzero.Thisindicatesthatlower-frequencyphononshavemuch
larger contribution to thermal conductance than higher-frequency
phonons. The phonons of very high frequencies, ω< 10 k B T / ,
have negligible contribution to thermal conductance. At very low
temperatures ( T close to zero), only those acoustic phonons of
nearly zero ω are excited by the temperature. They can give
quantizedthermal conductance aswewill discussnext.
Figure 3.2 presents the weighting factor ( x 2 e x
/
( e x
3.1.5 Quantized Thermal Conductance
Whenthetransmissionfunction ( ω ) 1,Eq.3.9givesthequantum
of thermal conductance as
σ 0 = π
2 k B T
3 h
.
(3.10)
10 13 W/K 2 ) T represents the maximum possible
value of energy transported per phonon mode [5, 6]. Different from
σ 0
=
(9.456
×
 
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