PHYSICAL REVIEW B 87, 195210 (2013)
Transition energies and direct-indirect band gap crossing in zinc-blende Al
x
Ga
1−x
N
M. Landmann,
*
E. Rauls, and W. G. Schmidt
Lehrstuhl f¨ ur Theoretische Physik, Universit¨ at Paderborn, 33095 Paderborn, Germany
Marcus R¨ oppischer, Christoph Cobet, and Norbert Esser
Leibniz-Institut f ¨ ur Analytische Wissenschaften - ISAS - e.V., Albert-Einstein-Str. 9, 12489 Berlin, Germany
Thorsten Schupp and Donat J. As
Experimentalphysik, University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany
Martin Feneberg and R¨ udiger Goldhahn
Institut f ¨ ur Experimentelle Physik, Otto-von-Guericke-Universit¨ at Magdeburg, Universit¨ atsplatz 2, 39106 Magdeburg, Germany
(Received 2 March 2013; published 29 May 2013)
The electronic and optical properties of zinc-blende (zb) Al
x
Ga
1−x
N over the whole alloy composition range are
presented in a joint theoretical and experimental study. Because zb-GaN is a direct (
v
→
c
) semiconductor and
zb-AlN shows an indirect (
v
→ X
c
) fundamental band gap, the ternary alloy exhibits a concentration-dependent
direct-indirect band gap crossing point the position of which is highly controversial. The dielectric functions
of zb-Al
x
Ga
1−x
N alloys are measured employing synchrotron-based ellipsometry in an energy range between
1 and 20 eV. The experimentally determined fundamental energy transitions originating from the , X, and L
points are identified by comparison to theoretical band-to-band transition energies. In order to determine the
direct-indirect band gap crossing point, the measured transition energies at the X point have to be aligned by
the calculated position of the highest valence state. Thereby density-functional theory (DFT) based approaches
to the electronic structure, ranging from the standard (semi)local generalized gradient approximation (GGA),
self-energy corrected local density approximation (LDA-1/2), and meta-GGA DFT (TB-mBJLDA) to hybrid
functional DFT and many-body perturbation theory in the GW approximation, are applied to random and special
quasirandom structure models of zb-Al
x
Ga
1−x
N. This study provides interesting insights into the accuracy of
the various numerical approaches and contains reliable ab initio data on the electronic structure and fundamental
alloy band gaps of zb-Al
x
Ga
1−x
N. Nonlocal Heyd-Scuseria-Ernzerhof-type hybrid-functional DFT calculations
or, alternatively, GW quasiparticle calculations are required to reproduce prominent features of the electronic
structure. The direct-indirect band gap crossing point of zb-Al
x
Ga
1−x
N is found in the Al rich composition range
at an Al content between x = 0.64 and 0.69 in hybrid functional DFT, which is in good agreement with x = 0.71
determined from the aligned experimental transition energies. Thus our study solves the long-standing debate on
the nature of the fundamental zb-Al
x
Ga
1−x
N alloy band gap.
DOI: 10.1103/PhysRevB.87.195210 PACS number(s): 71.15.Mb, 78.20.−e, 81.05.Ea
I. INTRODUCTION
Throughout the last two decades, scientific breakthroughs
on the field of group-III nitride semiconductor materials have
always stimulated rapid technological progress in the man-
ufacturing of optoelectronic and electronic devices. Group-
III nitrides naturally crystallize in stable hexagonal lattices
with wurtzite (wz) structure (space group: P 6
3
mc – C
4
6v
).
Artificially grown, group-III nitrides may adopt a metastable
cubic phase with a zinc-blende (zb) structure (space group:
F 43m – T
2
d
). Probably the most appealing characteristic of
binary group-III nitrides and their multicomponent alloys is
the extremely large accessible range of band gap energies,
i.e., from ∼0.7 eV (wz-InN)
1
to ∼6.0 eV (wz-AlN)
2,3
for
the hexagonal compounds and from ∼0.6 eV (zb-InN)
4
to
∼5.3 eV (zb-AlN)
5
for the cubic ones.
For a long time almost exclusively grown along the polar
(0001) c direction of the hexagonal crystal, group-III nitrides,
and their multicomponent alloys exhibit strong internal piezo-
electric and spontaneous polarization electric fields, which
are undesirable for optoelectronic applications since they
inherently limit the device performance.
6,7
The occurrence
of such polarization fields is a direct consequence of the lack
of inversion symmetry in the hexagonal crystal that naturally
show both piezoelectric and spontaneous polarization parallel
to the c axis.
8
The growth of nonpolar and semipolar nitrides
has found increasing technological interest over the past years
to avoid these strong internal fields. In these nonpolar or
semipolar nitrides, the c axis is orthogonal or inclined to the
growth direction, thus eliminating or limiting the field effects
in the growth direction.
9
However, the electrical, optical, and
structural properties are affected by strong lateral anisotropies.
A way to fabricate group-III nitride based optoelectronic
devices, fundamentally free of polarization fields,
7
is the
growth
5,10–14
of cubic nitride alloys with (001) orientation and
the engineering of nanostructured material compounds (e.g.,
quantum wells and superlattices).
15–20
Among all group-III nitride semiconductors, zb-AlN is
the only binary semiconductor with an nondirect
v
→ X
c
fundamental bulk band gap.
5
In the ternary (also pseudobinary)
zb-Al
x
Ga
1−x
N alloys, the electronic states, responsible for the
band gaps in the pure bulk phases, intermix with respect to the
relative concentration of each material component. Thus there
195210-1 1098-0121/2013/87(19)/195210(21) ©2013 American Physical Society