Online proceedings - EDA Publishing Association

From
7-9 October 2009, Leuven, Belgium
TABLE II
Overall thermal conductivity and relative mode contribution before and after QCs at T = 220 K
Factor
Factor
value
MD n/a n/a
+Standard QCs
+Proposed QCs
c
v,
a
( T )
3( N −1)
k
c
c
v , e
v,
a
( T )
( T )
B
0.645
1.037
k
TA
(W/m-K)
128.615
(33.87%)
82.932
(33.87%)
125.303
(38.62%)
k
LA
(W/m-K)
212.151
(55.87%)
136.797
(55.87%)
180.660
(55.67%)
k
LO
(W/m-K)
35.800
(9.43%)
23.084
(9.43%)
17.310
(5.33%)
Total 94.29 % 5.71 %
k
k ' = k *(3/ π )
TO
(W/m-K) (W/m-K)
3.092
(0.82%)
362.549
1.994
(0.82%)
1.218
(0.38%)
233.773
309.866
94.29 % (i.e. an increase of 4.54 % from the values before
QCs), while the contribution of the optical modes becomes
5.71 %. The contribution of the LO and TO modes reduces
from 9.43 % and 0.82 % before QCs to 5.33 % and 0.38 %
after QCs, respectively, and the TA mode increases from 33.87
to 38.62 %. These results are in excellent agreement with
recent ab initio predictions [33], in which acoustic modes
provide 95% of the contribution to the thermal conductivity
and that the contribution of LA is higher than that from the TA
mode.
Isotope scattering. Based on the experimental data for nat Si
and 28 Si [25] at the corrected temperature, the reduction in the
thermal conductivity is estimated to be 16.80 % (measured
relatively to the nat Si thermal conductivity). Table III shows the
MD thermal conductivity before and after QCs including the
isotropic scattering term. Before QCs are applied, the deviation
of the MD-predicted thermal conductivity with respect to the
experimental value is 44.48 % for 28 Si. When QCs and the
isotope scattering are applied this difference reduces to 23.48
and 13.94 % respectively.
TABLE III
Thermal conductivity before and after isotopic scattering at T = 220 K
From
k
k
k
−1*100
−1*100
(W/m-K) ke( 28 nat
Si)
ke(
Si)
MD 362.5 44.48 %
+QC 309.9 23.48 %
+ISO 265.3 13.94 %
k ( 28 e Si) = 250.934 W/m-K [25], ke ( Si)
= 232.844 W/m-K [21], k e
:
experimental thermal conductivity.
The inclusion of the isotope scattering term modifies all
properties that depend on the phonon relation times. Although,
both isotope scattering expressions (Eq. 4) lead to the same
value of thermal conductivity, the relative contributions of the
modes change. The contribution of the optical modes
decreases, while the on from the acoustical modes increases.
The acoustical modes contribute 97.6 % when the velocity term
is neglected and 98.5 % when is included. Additionally, the
contribution of optical modes becomes almost negligible (less
that 2.4 %).
Fig. 4 shows the mode thermal conductivity as a function of
frequency. In terms of frequency, the isotope scattering lowers
significantly the contribution of optical modes. For both
scattering terms, the contribution of the LO mode reduces
more than half, while the one from TO becomes negligible
(less than 0.01 %). At the same time, the mode thermal
conductivity of the TA does not experience a significant
change, while the LA changes as the frequency increases.
k m
(ω) (W/m-K*s/rad)
Before isotope scattering
4 Isotope scattering: A*ω 4 3 x 10-13 LO
Isotope scattering: A/v *ω 4 2.5
3.5
g
2
3
1.5
1
TO
2.5
LA
0.5
2
0.95 1 1.05 1.1
x 10 14
1.5
1
TA
LO
0.5
TO
0
0 2 4 6 8 10 12
4.5 x 10-12 Frequency (rad/s)
x 10 13
Fig. 4 - Contribution to the thermal conductivity of the TA, LA, LO and TO
modes before (solid thick line) and after isotope scattering.
V. HIGH TEMPERATURE IMPLICATIONS
T ≥ θD
At high temperatures (
), both the quantization of
the energy and the presence of isotopes are expected to have
a minor effect on the reduction of the thermal conductivity.
Note that the contribution to the thermal conductivity from
the high frequency modes as the temperature of the system
is increased would progressively become similar to the one
estimated with MD (before QCs).
VI. SUMMARY AND CONCLUSIONS
In this work a new quantum correction procedure was
proposed to correct silicon thermal properties obtained from
molecular dynamics. The procedure considers the energy
quantization per mode basis and the anharmonic nature of
the potential energy function, and involves the use of
experimental or analytical specific heat values. In addition,
the effect of isotope scattering was analyzed in terms of the
change of the mode thermal conductivity.
In the standard quantum correction procedure, the specific
©EDA Publishing/THERMINIC 2009 201
ISBN: 978-2-35500-010-2