eLearning

principles approach. The VASP code in conjunction with an
in-house package GPT stands for G
P,T were used. A 2
2    1 supercell was used for the phonon calculation of

-Al2O3. The calculation is very demanding because the
equilibrium structure must be highly accurate and, since the
160 atom supercell has no internal symmetry to reduce the
computational burden, the interatomic forces for the dis-
placement of every atom must be explicitly computed. In
the present calculation, we used:
1 ultrasoft pseudopoten-
tials with Ceperley-Alder approximation for exchange-
correlation;
49

2 a single
k point for structure optimization
and phonon calculation for the supercell which is sufficient
since the volume of the supercell is kept fixed;
3 “accurate”
accuracy for the VASP calculations with an energy cutoff of
400eV;
4 energy and force convergence criteria of 1.0
10−5
eV and 0.001 eV/Å, respectively;
5 a20    20
20 k-space mesh for Brillouin-zone
BZ integration of the
phonon density of state
DOS. The phonon density of states
g
 and the site specific partial density of states
PDOS
g
 are evaluated according to,
g
 =
1
n
 q,i
 − i

q

g
 =
1
n
 q
,i,
e,
i

q

2
 − i

q

1
where i
is the phonon frequency, n is the number of q
points, q and i are wave vector and phonon branch indices, 
is the thermal broadening function and e,
i

q
 is the polar-
ization vector of the th
atom along the th
direction for the
i
th
branch phonon at wave vector q. In the present calcula-
tion, we did not include the correction for the longitudinal
optical
LO and transverse optical
TO splitting which re-
quires calculation of the optical dielectric constant and Born
effective charges of each nonequivalent atom using the Berry
phase method.
50 We do not expect it to have a large effect in
the present study for
-Al2O3.
Figure 2 shows the calculated phonon-dispersion curves
along the high-symmetry directions of the BZ. For a better
comparison with corundum, we approximated the BZ to be
that of a hexagonal lattice since in the present model for

-Al2O3
ab,  and 90°, and
120°, see Table I for
more details. The 40 atom cell gives a total of 120 branches
with the maximum frequency just shy of 900 cm−1
. The cor-responding phonon DOS and partial DOS are shown in Fig.
3. The notable Von Hove peak at the high frequency of
750 cm−1
is due to “breathing”-like Al–O bond stretching of
the tetrahedral AlO4 unit, which is notably higher than cor-
responding modes in corundum.
46
This can be explained by
the shorter Al–O bond distance of about 1.73 Å found in the
present model. The Von Hove peak near 500 cm−1
is due to
the breathing mode of the octahedral AlO6 unit. The overall
DOS of
-Al2O3 resembles that calculated for -Al2O3 using
first-principles methods.
33
There is notable broadening of the
few localized modes in comparison to corundum. The broad-
ening of these localized modes and the reduction of some
more diffusive peaks reflects the disruption of longer range
order while maintaining short-range order in the
-Al2O3
model. This may be viewed as a signature of an amorphou-
slike structure.
The zone-center vibrational modes give information on
the Raman and IR active modes in a solid. Because of the
reduced symmetry, symmetry analysis for these modes in

-Al2O3 will not be informative. Instead, we plot the zone-
center mode frequencies as a histogram in Fig. 4 with a small
window of 5 cm−1
. According to the calculation, the four
lowest frequencies are at 97.6, 103.8, 136.0, and
136.4 cm−1
. The highest frequency mode is at 870.8 cm−1
.
A peak at 750 cm−1
is also present which contributes to the
peak in the DOS of Fig. 3.
From the phonon spectrum of
-Al2O3 at different vol-
umes V
see below, we can obtain the mode-Grüneisen pa-
rameter for each phonon branch defined as: