Online proceedings - EDA Publishing Association

7-9 October 2009, Leuven, Belgium
of 7 to 10 % has been found [24-26] .
found using the experimental specific heat for nat Si [30]. Due
to the lack of experimental specific heat data for single
It is well known that the fluctuation in the mass silicon isotopes, the quantum corrected temperature is also
distribution throughout a crystal produces thermal resistance obtained using the analytical expression for specific heat
[27-29]. Klemens [29] and Carruthers [28] have suggested (both results are presented in Table I), given by [31]
two similar expressions for the relaxation times associated to cv, a ( T ) =
isotope scattering,
∑ cv,
a ( T,
ωm
)
(7a)
m
−1
4
−1
A 4
τ imp = Aω and τ imp = ω
(4)
⎪⎧
⎪⎫
3
2 exp( hωm
/ kBT
)
v
c ( )
g
v,
a T = ∑ ⎨kB
( h ωm
/ kBT
)
⎬ (7b)
2
m
⎪⎩
[ exp( hωm
/ kBT
) −1]
⎪ ⎭
where, v g is the phonon group velocity and A is a fitted
constant.
where cv, a
( T,
ωm
) is the mode contribution to the specific
Thermal conductivity (W/m-K)
400
350
300
250
200
150
Δk e
= 16.8 %
100
200 210 220 230 240 250 260 270 280 290 300
Temperature (K)
nat Si - Ho et al. 1974
nat Si - Capinski et al. 1997
nat Si - Kremer et al. 2004
28 Si - Capinski et al. 1997
28 Si - Ruf et al. 2000
28 Si - Gusev et al. 2002
28 Si - Kremer et al. 2004 (Sample NN)
Fig. 1 - Experimental values of thermal conductivity of natural and isotopeenriched
silicon. Δ ke
represents the reduction of the thermal conductivity
due to isotope scattering measured in [25].
III. METHODOLOGY
To obtain the quantum corrected temperature, we modify
the approach described in [14] to include the quantization of
the energy per mode basis and the change of the dispersion
relations. In [14], the thermal conductivity correction factor
is written as,
dT c T
MD v,
e(
)
= (5)
dT 3( N −1)
kB
/ V
where, dT MD / dT is now defined as the ratio between the
experimental ( c v , e ) and the classical specific heats, N is the
number of atoms in the simulation, k B is the Boltzmann
constant. The relation between the molecular dynamics and
corrected temperatures can be found integrating Eq. 5 with
respect to temperature,
V T
*
TMD ( T ) = cv, e(
T ) dT + T
3( N −1)
k ∫
(6)
0
B
In this expression the integration constant * T is of the
order of the Debye temperature [14]. The zero-point energy
term is implicitly included since we are using the
experimental specific heat. The corrected temperature (T ) is
heat. In Eq. 7b, the summation is taken over all phonon
modes in [100] and is computed using the dispersion relation
previously determined using MD [11] at different
temperatures, which include the effects of anharmonicity (i.e.
thermal expansion and change of the dispersion relations).
In addition, due to the difference between the experimental
[12] and analytical dispersion relations at room temperature,
it is expected that the experimental Debye temperature ( θ D )
will be different from the one calculated using pair
potentials, such as, Stillinger-Weber, Tersoff, EDIP, etc.
Therefore, the integration constant in Eq. (6) is adjusted to
include this difference, such that
⎛
max
⎞
* θ ⎜
ω
D
a,
k
T =
⎟
∑
(8)
⎜ max
6 ⎟
k = 1,6 ⎝ ω e , k ⎠
max max
where, ω a,k and ω e,k are the maximum frequencies of the
analytical and experimental dispersion relations for each
branch. Based on our previous results obtained using the
Stillinger-Weber potential [11], we found that the integration
constant is T * =790.94 K. Eq. 8 would converge to the
experimental Debye temperature if both dispersion relations
(experimental and analytical) are equal.
To obtain the correction for the thermal conductivity and
specific heat, Eq. 3 is adjusted to include the quantization of
the energy of each mode at T . In addition, the analytic
specific heat is obtained using the dispersion relations
determined using MD at the studied temperatures. The
thermal conductivity is corrected considering that the
specific heat depends on: i) the temperature, ii) the frequency
(ω ) and iii) the ratio between the total experimental and
analytical specific heats ( c T ) / c ( ) ), as
k(
T)
c
⎧
( T)
⎪4π
⎨
( T)
⎪ 3
⎩
v , e ( v,
a T
= v , e
∑ ∫
c
v,
a
In the equation, v g and p
⎡ωm,max
⎤⎫
1 ⎛
⎞
⎢ ⎜
vg
⎟ ⎥⎪ a
r ⎬
( ) ⎢ ⎜
v
2
c , ( T,
ω)
τ ω dω
3
2
2π
v ⎟ ⎥
⎪ ⎭
b,
p ⎢⎣
ωm,min
⎝
p ⎠ ⎥⎦
… (9)
v are the phonon group and phase
velocities and τ r is the phonon relaxation time. The
expression between curly brackets corresponds to the
Tiwari’s thermal conductivity expression [32]. For
simplicity, we assume that the thermal properties of the
©EDA Publishing/THERMINIC 2009 199
ISBN: 978-2-35500-010-2