Figure 4.8 shows the transmission functions for the four surface orientations
of interest along two particular transport orientations for each case, that, as shown
below, provide the lowest and the highest thermal conductance for that particular surface.
The layer thickness in all cases is
. In the case of the thin-layer with
surface orientation, in Fig. 4.8-a we consider the
and the
transport channels. The transmissions of the two channels are almost the same, indicating negligible anisotropy. In the case of the thin-layer with
surface orientation, in Fig. 4.8-c we consider the
and the
transport channels. Again in this case, the transmissions are almost the same.
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The transmission function of the thin-layers with
and
surfaces, on
the other hand, is orientation-dependent. For the
surface thin-layers in Fig. 4.8-b, the
channel (blue line) shows the highest transmission function, and the
channel (red-dotted line) the lowest. An even larger difference is observed in the case of the
surface thin-layers in Fig. 4.8-d. The highest transmission is observed for the
channel (blue line), and the lowest for the
channel (red-dotted line). The difference in the transmission of the channels in different transport orientations is largest for energies between
for both the
and the
thin-layers.
Using the transmission functions extracted from the bandstructures, the ballistic lattice thermal conductance is calculated using the Landauer formula for the thin layers with the four different surface orientations of interest. The thermal conductance as a function of the transport orientation
, varying from 0 to
is shown in Fig. 4.9 for room
temperature. We calculate the conductance of thin layers for thicknesses of
,
,
and
. With symbols the high symmetry orientations are denoted using the Miller index
notation, i.e.
- circle,
- star,
- triangle, and
- square. These orientations are marked on the
thin-layer result in Fig. 4.9. In all cases, the conductance increases linearly as the thickness increases because the thicker layers contain more phonon modes that contribute to the thermal conductance.
With regards to anisotropy, for the thin-layers with
surface in Fig. 4.9-a, the conductance has a maximum along the
direction (square), and a minimum is along the
direction (circle), although the difference is small (only
). Interestingly, this observation is the same for all thicknesses considered. The conductance of the channels with
surface is shown in Fig. 4.9-b. The conductance is biggest in the
transport orientation (
, circle) and smallest for the
channels (
, square). The variation between the maximum and minimum values, however, in this case is
for the
thin layer, and decreases to
for the
layer. The conductance of channels with
surface is shown in Fig.4.9-c. The conductance in this case also peaks along the
direction (circle) and is smallest along the
direction (triangle). The variation of the conductance with transport orientation in this case is negligible for the thinner layers, but increases to
in the
case. The thermal conductance for channels with
surface is shown in Fig. 4.9-d. The maximum and minimum conductance is observed along
(circle) and
(star), respectively. Channels with this surface orientation exhibit the largest variation in thermal conductance compared to other surfaces. The difference varies from
for the
layers to
for the
layers. Overall, considering all surface and transport orientations, the maximum thermal conductance is observed for the
channels, and the minimum for the
channels. Interestingly, however, regardless of surface orientation, the thermal conductance is high in
direction. This agrees well with previous works on silicon nanowires, where it is reported that the
oriented nanowires have the highest thermal conductance [111,107]. A similar conclusion was found for thin layers of larger sizes [112]. As it will be explained in next section, the phonon dispersions along the
orientations are more dispersive compared to other orientations, which yield higher group velocities and, therefore, higher thermal conductance.
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Figure 4.10 shows the thermal conductance of the
layers as a function of temperature. For every surface orientation two transport orientations, the one
with the maximum and the one with the minimum conductance are shown (as in Fig. 4.8). The conductance increases with temperature as expected from a ballistic quantity, and starts
to saturate around
. The reason is that the phononic window function [57]:
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(4.2) |
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