BAND-GAP ENGINEERING OF FUNCTIONAL . . . PHYSICAL REVIEW B 91, 165309 (2015)
FIG. 5. (Color online) The figure illustrates the weights of the
CBM state in real space. The vertical axes are along the stacking
direction of the material and the areas of the circles indicate the
weights of the CBM state for a particular atomic xy plane. The boxes
show the extent of the supercell in the direction of the stacking (z),
and the dashed lines mark the interfaces between the α and β layers.
Above the figure, the calculated band gaps for the different stacking
sequences are denoted. The CBM state is mainly composed of Ta d
orbitals, as shown in Fig. 4(b).
To understand the variation of the conduction-band minimum
in Fig. 3, we first consider the compounds with the
formula αβnβ
, where the band gap decreases as a function of
the number of β layers. The CBM states for these systems
are located only on the TaON plane, as shown in Fig. 5, and
generated by the Ta5d orbitals, as plotted in Fig. 4(b) for the
αβ case. The variation of the band gap as a function of nβ
is a result of quantum confinement. The empty states in the
single α layer are shifted up out of reach, and the CBM state
in β becomes less confined with the increase of the number
of β layers in the αβstructures, as can be seen in Fig. 5.
nβ
The reduction of the confinement results in a down-shift of the
CBM level and thus a reduced band gap, as also sketched in
Fig. 3.
The situation is radically different for all the combinations
αnα
βnβ
, with nα 2. Now the CBM state is not located in β,
but in α. It is located mainly on the Sn5s orbitals, as shown in
Fig. 4(d). If, for example, we consider the compounds α2βn,
the CBM state is localized in the α2 layers and essentially
looks the same, as seen in Fig. 6. The band gap is therefore
also largely unchanged for n 3. For n = 2, a small reduction
relative to the situation with n 3 is seen and this reduction
becomes even larger for n = 1 (see Fig. 2). We ascribe this
reduction to quantum tunneling through the thin β layers. As
can be seen in Fig. 6, the CBM states decay into the β layers
and the tunneling coupling, for small thicknesses, will result
in a lowering of the CBM level and thus a decrease of the band
gap.
The interplay between quantum confinement and tunneling
is seen most clearly for the nβ = 1 systems (Fig. 7). Ignoring
the αβ structure that has a different nature for the CBM level
with respect to the other systems, the CBM state becomes
FIG. 6. (Color online) Weight of the CBM state in each xy plane
of the nα = 2 structure. The CBM state is now composed of Sn s
states, with some tunneling through the TaON plane Fig. 4(d). The
tunneling progressively reduces with the increase of β layers.
less confined with the increase of the number of α layers,
with the consequence of a decrease in the band gap. And
as we have seen, the band gap is further reduced because
of tunneling through the single β TaON layer. However,
for larger thicknesses of the α layer, the tunneling effect is
reduced because the now less-confined CBM state has lower
amplitude at the interface. This interplay between confinement
and tunneling leads to the increase of the band gap between
nα = 4 and nα = 5.
As we have seen, the variation of the band gaps for the
different periodic αnβm compounds can be understood from
the confinement effects shown in the level diagram in Fig. 3
together with additional tunneling effects if β1 or β2 layers are
present. Does this lesson apply tomore complicated sequences
FIG. 7. (Color online) Weight of the CBM state in each xy plane
of the nβ = 1 structure. The character of the CBM state changes
drastically with nα: TaON is responsible for the CBM state for the
nα = 1 structure, while for nα > 1, it is SnO2 Figs. 4(b) and 4(d).
The tunneling across the TaON plane has an effect until nα = 4.
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