CASTELLI, PANDEY, THYGESEN, AND JACOBSEN PHYSICAL REVIEW B 91, 165309 (2015)
of layers? Some test calculations seem to indicate so. A
compound with the periodic repetition of αβαβnβ reproduces
exactly the same gaps as the αβnβ . This is to be expected since
from the level diagram in Fig. 3 we should expect the CBM
state to be located in the βnβ layer with little tunneling through
the α layers. Another example is the systems with sequence
αβαnαβ, which exhibit a small increase (up to about 0.2 eV) of
the band gap for nα 2 as compared to the αnαβ compounds.
This can be understood in terms of reduced tunneling because
the αnα layers are now separated by βαβ instead of a single β
layer reducing the tunneling effect.
III. OPTICAL PROPERTIES
A number of technological applications such as photovoltaics
or photocatalysis depend on the availability of efficient
absorbers of light in the visible spectrum. This requires an
appropriate band gap of the material, and band-gap tuning is
therefore a key issue. However, the band gap does not by itself
provide any information about the magnitude of the matrix
elements responsible for light absorption. For symmetry
reasons, the light absorption can be dipole allowed or forbidden
and—in particular for heterostructures—the transitions may
take place between states with different degree of spatial
overlap, giving rise to large variations of the absorption
strengths.
To address this issue, we perform linear response calculations
25, using the adiabatic local density approximation
(ALDA), and determine the optical absorption of the
investigated systems focusing for simplicity on the systems
with nα = 1 or nβ = 1 26. The optical absorption spectrum
is calculated using time-dependent density functional
theory (TDDFT) from the density response function χ.
The response function evaluated at point r to first order
in a time-dependent perturbation of frequency ω applied at
point r is χ(r,r
,ω) = δn(r,ω)/δVext(r
,ω), where δn is the
induced density under the perturbation caused by the external
potential Vext.
The microscopic dielectric matrix is defined as
−1
GG (q,ω) = δGG + 4π
|q + G|2 χGG (q,ω), (1)
where G and G are reciprocal lattice vectors, and q is a
wave vector of the first Brillouin zone. The optical absorption
spectrum is given by Im(q → 0,ω), where (q → 0,ω) =
1
−.
1
00 (q→0,ω) Figure 8 shows the optical absorption for the nα = 1
systems. In the plot, we distinguish between the case where
the light is polarized along the xy and the z directions. The
xy plane, in fact, maintains the cubic symmetry, while the
stacking of the layers takes place in the z direction. For these
compounds, the CBM and VBM states are located in the same
region of space, namely, in the TaON layers, and thus the
absorption starts at the direct band gap and is quite intense,
especially for polarizations in the xy direction. The situation
is different for the nβ = 1 systems (Fig. 9), where the VBM
state is located in the TaON layer, while the CBM state has
most weight on the BaO2 layers. The absorption here starts
at much higher energies than the band gap (except for the αβ
compound, in black in the figure, which has the VBM and
FIG. 8. (Color online) Calculated optical absorption for the nα =
1 systems. The direct band gaps are indicated with vertical
arrows.
CBM states located in the same region). The first transition
with appreciable weight is between two TaON states in the β
layer, and the absorption curves are therefore fairly similar,
independent of the band gap.
Table I reports the efficiencies of the two sequences. The
efficiency is calculated as the percentage of the collected
photons of the global solar spectrum at AM1.5 27. As also
shown in Figs. 8 and 9, the efficiency is higher for light
polarized in the xy direction than along z. αβ2 and α2β are
almost comparable because the higher absorption properties of
the former are balanced by the lower band gap of the latter, and
the two systems collect almost the same amount of photons.
The efficiency of the αβnβ sequence always increases with the
number of layers, while the one of αnαβ decreases even though
the gap closes.
The calculations thus indicate that the absorption cross
section at the α − β interface is quite limited and that the
FIG. 9. (Color online) Calculated optical absorption for the nβ =
1 systems. The direct band gaps are indicated with vertical arrows.
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