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III.
RESULTS
A. Anisotropic dispersion curves
The definitions in the 2D reciprocal space of the
independent variables k and φ to obtain the throughput λ
are explained in Fig. 3(a). The 2D BZ is elongated by a
20-fold factor in the reciprocal direction k z with respect to
that k x . Indeed, the reduced length of the supercell slab in
x is given by n x = 20. The dispersion-curve diagrams for
φ = 0° (direction k x with k z = 0) and φ = 90° (direction k z
with k x = 0) are displayed in Figs. 3(b) and 3(c),
respectively.
(a)
(b)
φ = 0 o
K z =π/a
k
k = (0,0)
K(φ)
φ
K x =π/d x =π/(n x a)
7-9 October 2009, Leuven, Belgium
Different colors (blue, red, green, magenta and black
from the lowest to the highest frequencies) represent
different frequency ranges for the dispersion curves.
As shown in Fig. 3(b), the dispersion curves for φ =
0° are very flat leading to low radial group velocities
u m (k, 0) in the direction k x . In contrast, for φ = 90°, the
radial group velocities u m (k, π / 2) are usually much
higher than those obtained for φ = 0°. Indeed, as depicted
in Fig. 3(c), the slopes of the dispersion curves for φ =
90° are usually much larger than those for φ = 0° [Fig.
3(b)]. As a consequence, we can already expect from the
evolution of the dispersion curves a significant exaltation
of the throughput thermal conductivity λ in the direction
z (β = 90°) with respect to that x (β = 0°), as discussed in
the following.
B. Anisotropic thermal conductivity
For the example nanodevice presented in Figs. 1 and
2, a low λ β varying from only 0.7 to 1.0 W/m/K is
computed when β is increased from 0° to 24.4°,
respectively, as shown in Fig. 4. Since 1 W/m/K
corresponds (approximately) to the extreme low limit of
the thermal conductivity of bulk amorphous Si [17,18],
the hybrid nanomaterial can have a insulating behavior
when β ≤ 25°. Second, when β is increased from 26.6 to
90°, the material operation regime becomes more
dissipative since λ β grows, with a sigmoid curve, from
1.1 to 2.9 W/m/K, respectively.
(c)
φ = 90 o
Fig. 3 (colors). Phonon band diagrams with the definition of the polar
reciprocal coordinates k and φ (a) and dispersion curves obtained by
lattice dynamics for φ = 0° (b) and φ = 90° (c).
Fig. 4. Computed λ β vs. β curve with a sigmoid shape:
The hybrid nanomaterial is that in Figs. 1 and 2 at room
temperature T = 300 K. The circles, interpolated by the dotted line,
are values for fcc crystallographic directions with integer Miller
indices.
©EDA Publishing/THERMINIC 2009 207
ISBN: 978-2-35500-010-2