Online proceedings - EDA Publishing Association

7-9 October 2009, Leuven, Belgium
crystals form a subset of the face-centered cubic (fcc)
group with a basis 2, four molecular planes with the
coordinates z = 0, z = a/4, z = a/2 and z = 3a/4 (before
binding energy relaxation) have to be utilized to define
the supercell slab as sketched in Fig. 2(a). The slab
section at z = 0 is displayed in Fig. 2(b). Since the space
domain is quantized with an interatomic lattice constant
a, the lengths d x = n x a, h = n y a, B 0 = μ a, B 1 = m x a, and
H = m y a are obtained from the integer size parameters n x ,
n y , μ, m x , and m y , respectively In the four planes with the
same Miller indices (001), the two QD facets are
assumed to follow close-packed directions of the
fcc crystals. As a result, they show a 45° deflection angle
with respect to the height axis y in Figs. 2(a) and 2(b).
From the precedent, the reduced bottom basis m of a QD
has to respect the equality μ = 2(m y -1) + m x . In this
theoretical study, we analyze an example hybrid
nanomaterial with the integer size parameters n x = 20, n y
= 4, m x = 5 and m y = 4 so that μ = 11. Since phonon
scattering is mainly incoherent for T > 5 K, the basis B 0
of the QDs can be modified and their centers offset (with
respect to the supercell median plane) by a fraction of the
quantity d x /2. Consequently, the overall thermal
conductivity of the altered system in x will remain (in
average) close to that of the periodic system. We also
note that the individual QD lengths in z can be quite
different if the inequality L > Λ 0 holds.
The throughput λ has to be minimized or maximized
for the hybrid thermal nanomaterial to operate in either
insulation or dissipation regimes. Getting one or the
other of these contra effects depends on the in-plane
propagation direction, with a deflection angle β with
respect to the membrane longitudinal axis x, as shown in
Fig. 1(b). For a material with a sufficient number of Ge
QDs, an anisotropic thermal conductivity λ β can be
physically defined as a function of the angle β. This is
the case when (i) the membrane is cleaved to form two
sides in a Miller plane parallel to the direction given by β
and (ii) hot and cold reservoirs are connected to the two
other membrane sides. For instance, the device in Fig.
1(a) is for a membrane with β = 90° since hot and cold
reservoirs are connected through the direction z while the
two lateral sides have to be cleaved with respect to the
(100) Miller plane (orthogonal to the x direction). When
β = 90°, the throughput λ is maximal since the QDs are
stretched in the z direction with L > Λ 0 to form phonon
waveguides. This case is different from that with β = 0°
in the QD-constriction direction x where the throughput λ
is minimal.
The key point to understand the directional effects in
our hybrid nanomaterial is to study the curve of the
anisotropic thermal conductivity λ β vs. β taken from 0° (x
direction) to 90° (z direction).
Si chip @ T h
(a)
(b)
z [001]
y [010]
SA stretched Ge QDs
L av Si thin membrane
y
z [001]
B 1
B 0
dc-Si
Si
dc-Ge
β
d x
x [100]
Si chip @ T c
Continuous supercell
Fig. 1 (colors). Hybrid insulating/dissipative nanomaterial:
(a) Cross section in (y, z) if the membrane is connected between hot
and cold reservoirs in the [001] direction (z) where heat dissipation
is maximal (W av being the average length of the QDs in z); (b)
Continuous-medium scheme of a nanomaterial supercell where β is
the average angle of the heat flux in a membrane plane with respect
to the [100] direction (x) showing constriction of the QDs. The
inequalities B 1 ≤ B 0 < Λ 0 have to hold in (b).
If we use polar coordinates (k, φ) taken from the origin
point of the 2D reciprocal space related to a direct plane
(x, z) of the 3D nanomaterial space, this curve is obtained
from:
λ =
β
π / 2
∫
0
+
[cos( β −φ)]
R(
φ)
dφ
+
π / 2
∫
0
2
sin[2( β −φ)]
P(
φ)
dφ.
π / 2
∫
0
[sin( β −φ)]
A(
φ)
dφ
In Eq. (1), the one-dimensional (1D) integration kernels
R(φ), A(φ) and P(φ), with the same metric unit [W/m/K
per radian], depends on the only independent variable φ.
They are obtained in a 2D reciprocal plane by prior
integrations over the radial coordinate k from 0 to the
location K(φ) on the boundary of the first 2D rectangular
Brillouin zone (BZ) for the azimuthal angle φ, as depicted
in Fig. 3(a). R(φ) and A(φ) are related to the square
amplitudes of the radial [ u m ( k,
φ)
] and azimuthal
[ v m ( k,
φ)
] phonon group velocities in the first 2D BZ,
respectively. The cross-term P(φ) in Eq. (1) is a
functional of the product u m ( k,
φ)
× v m ( k,
φ)
.
2
(1)
©EDA Publishing/THERMINIC 2009 205
ISBN: 978-2-35500-010-2