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an increasing function of its TE figure of merit ZT.
Indeed, the non-dimensional ZT is given by ZT = S 2 σ T /
λ, where S, σ and T denote the Seebeck coefficient,
electrical conductivity and absolute temperature,
respectively [11,12]. Therefore, ZT is inversely
proportional to λ but directly proportional to the power
factor S 2 σ. Discovery of a material with ZT higher than 3
would result in a TE yield that is higher than 42 % of the
Carnot efficiency for hot and cold junctions at 800 K and
300 K, respectively [9,12]. Achievement of such a dream
would have an enormous impact for energy conversion
and renewable energies.
In this theoretical study, we investigate a novel type
of thermal membranous nanomaterials that is made up of
a thin dc Si membrane covered by stretched SA Ge QDs
with facets. The QDs are stretched in the [001] direction
with respect to the dc symmetry to form phonon
waveguides. The length L of the QD waveguides in the
stretching direction [001] is defined as larger than the
average phonon mean free path (MFP) in Si given by Λ 0
∼ 100 nm. In contrast, the QDs are constricted in the
orthogonal in-plane direction [100] with a bottom basis
B 0 that is several folds smaller than Λ 0 .
The anisotropic thermal conductivity λ of the QD
device is obtained in a membrane plane as a function of
the deflection angle β taken with respect to the axis x
parallel to the QD constriction direction [100]. The
throughput λ, when hot and cold junction are connected
to both sides of the membrane, is computed with 3D
lattice dynamics for β = 0° to β = 90°. The latter value is
related to the axis z parallel to the QD stretching direction
[001]. The discrete supercell to be encoded to obtain the
phonon dispersion curves is taken as a molecular slab of
the nanomaterial. Since L > Λ 0 , the slab has an out-ofequilibrium
width given by the Si lattice parameter a =
0.5431 nm in the direction z. To respect the dc
symmetry, four molecular planes with the same Miller
indices (001) have to be used to define the supercell slab.
The dispersion curves, computed by lattice dynamics
[13-15], show flat behaviors in the direction x owing to
QD constriction. Consequently, low phonon group
velocities are obtained in this direction leading to a small
throughput λ. In contrast, the slopes of the dispersion
curves are usually higher in the direction z so that the
stretched QDs form nanoscale phonon waveguides with
axes parallel to z. Indeed, the QD average length L is
large in z with L > Λ 0 . As a result, the throughput λ is
much larger in the direction z with respect to that x.
When hot and cold junctions are connected to the
membrane following z, the QD-waveguide nanomaterial
can be used for the design of efficient unidirectional
thermal interface devices for heat sinking. Indeed, the
leakage heat currents, which might affect thermal budget
7-9 October 2009, Leuven, Belgium
of other parts of a device to cool, are small owing to the
much smaller throughput λ in the in-plane orthogonal
direction x. The proposed membranous material presents
a hybrid behavior between thermal dissipation and
insulation. The operation regime depends on the heatflux
direction determined by the hot and cold junctions
with a connection angle β with respect to x.
Consequently, the same nanomaterial can be applied to
the design of heat sinkers and dissipaters as well as TE
generators and coolers.
In the following sections, using an example
molecular-scale QD-waveguide nanomaterial, we show
exaltation of the throughput λ by a significant factor of 4
to 5 folds (i.e. from 0.7 to 2.9 W/m/K) when the
connection angle β is increased from 0° to 90°,
respectively. The thermal transition between the
insulating and dissipative regimes is obtained for β = 45°
that is related to the in-plane close-packed direction
of the Si membrane.
II.
MODEL
To keep a membrane-like geometry, which is
responsible of directional effects on the thermal
conductivity, we set the length L of the stretched Ge QDs
in the in-plane direction z, or [001], as being of the order
or larger than the average MFP Λ 0 ∼ 100 nm of the
phonons in bulk dc Si, as depicted in Fig. 1(a). The QD
bottom basis B 0 in the in-plane x direction, or [100], is
defined as being several folds lower than Λ 0 . Therefore,
the QDs are stretched in the z direction with respect to the
orthogonal direction x so that they form phonon
waveguides. In the continuous-medium representation of
Fig. 1(b), the top basis of the QDs is denoted by B 1 while
d x is the length of a nanomaterial supercell in the QDconstriction
direction x. In the non-repetitive direction y,
or [010], the height of the dc Si membrane is h while that
of the Ge QDs is H.
The throughput thermal conductivity λ of the
suspended nanomaterial is computed from the dispersioncurve
diagram in a range from 0 to ∼ 20 THz using 3D
lattice dynamics and an incoherent approach of the
phonon scattering relaxation times. A discrete model is
derived from a supercell slab of the continuous-medium
representation sketched in Fig. 1(b). As shown in Fig. 2
where the red and blue dots denote the locations of Si and
Ge atoms, respectively, we encode a discrete supercellslab
model at the molecular scale for lattice-dynamics
calculations. Since L ≥ Λ 0 ≥ B 0 , the width in z of the
supercell slab, which is parallel to the crystallographic
plane with the Miller indices (001), can be taken as equal
to the Si lattice parameter a = 0.5431 nm. Since the dc
©EDA Publishing/THERMINIC 2009 204
ISBN: 978-2-35500-010-2