Online proceedings - EDA Publishing Association

7-9 October 2009, Leuven, Belgium
silicon crystal are isotropic and equal to those for the [100] 1.2
direction. The thermal conductivity obtained with Eq. 9 is
1.1
further corrected ( k'
→ k /( π / 3)
) to account for the real
1
volume of the first Brillouin zone.
To include the isotope scattering, we find the value of the
coefficient A of Eq. 4 that produces a reduction in the
thermal conductivity equal to the one observed
experimentally due to the presence of isotopes. As shown in
Fig. 1, we compute the decrease of thermal conductivity at
the quantum corrected temperature from [25], avoiding in
this way mixing thermal conductivity values from different
measuring techniques and equipments. The final expression
of the phonon relaxation time, including phonon-phonon and
impurity scattering, is written using the Matthiessen’s rule,
−1 −1
−1
as: τ τ + τ .
r
= ph−
ph imp
c v
/ k B
0.9
0.8
0.7
0.6
0.5
0.4
TA
LA
Before QCs
After QCs
0.3
0 2 4 6
Frequency (rad/s)
8 10 12
x 10 13
Fig. 2 - Mode contribution to the specific heat before and after quantum
corrections at 220 K. Arrows indicate the frequency range of each mode.
LO
TO
IV.
RESULTS
Quantum corrections. Table I shows the value of the
quantum corrected temperature using the experimental
specific heat for nat Si [30] and using the analytical expression
for the specific heat (Eq. 7b), with T * = 790.94 K. Both
estimates provide similar results. At T MD = 300 K, the
corrected temperature is roughly 220 K, while at 1000 K the
temperature quantum corrections are negligible.
TABLE I
Quantum corrected temperature (Eq. 6)
T MD (K) T (K)
Experimental
Analytical
300 218.5 220.6
1000 1011.3 1006.5
Quantum corrections also affect the behavior of the
contribution of each mode to the specific heat with
frequency. Fig. 2 shows the contribution of each mode as a
function frequency at T MD = 300 K. As expected, before
QCs the mode contribution obtained MD is independent of
the frequency, however, when the quantization of the energy
is considered and QCs are applied, the contribution of each
mode decreases as their frequency increases. The quantum
corrected mode contribution starts at almost the same value
of the corresponding for the classical anharmonic system
( c v / k B = 1), however, the difference becomes larger at
higher frequencies. This is also expected for real systems
where c v / k B →1
as ω → 0 and c v / k B < 1 as ω > 0 . In the
figure, the small shift in the specific heat value observed at
zero-frequency is produced due to the difference between the
experimental and analytic specific heats ( cv , e( T ) / cv,
a(
T ) ).
Fig. 3 shows the contribution for each mode to the thermal
conductivity as a function of the frequency in the [100]
direction. Eq. 9 has been arranged such that,
ωm,max
k
k ω dω
(10)
= ∑∫
m
ω
m,min
m
( )
It is observed that the contribution of each mode is
substantially affected by the correction of the specific heat
and decreases as the frequency of the modes increases. This
is evident for the LO mode, whose contribution near the
Brillouin zone before QCs is comparable to that of the
acoustic modes, but reduces significantly after QCs are
applied. Before quantum corrections, the contribution of the
TA and LA modes to the total thermal conductivity is
approximately 33.9% and 55.9%, respectively [11]. As
shown in Table II, when the proposed QCs are applied, the
quantization of the energy changes both the relative
contribution of each mode and overall value of the thermal
conductivity. This is not the case when the standard
procedure is applied. The standard procedure only modifies
the overall value of thermal conductivity by a factor equal to
c v, a(
T ) / 3( N − 1)
kB
. However, the relative contribution of
the different modes remains the same.
k m
(ω) (W/m-K*s/rad)
Before QCs
4
After QCs
3.5
3
LA
2.5
2
1.5
TA
LO
1
0.5
TO
0
0 2 4 6 8 10 12
4.5 x 10-12 Frequency (rad/s)
x 10 13
Fig. 3 - Mode contribution to thermal conductivity before and after QCs.
As noted in Fig. 3, the specific heat of high-frequency
modes is substantially affected. This translates in a reduction
of the thermal conductivity values for the LA, LO and TO
modes, while the value for TA mode remains almost the
same. The contribution of the acoustic modes increases to
©EDA Publishing/THERMINIC 2009 200
ISBN: 978-2-35500-010-2