eLearning

our knowledge, this is the first time such data were calcu-
lated for
-Al2O3 based on whatever structures were as-
sumed. In general,
-Al2O3 has smaller elastic constants than
-Al2O3 except for C14 where
-Al2O3 is significantly larger
by almost 80%
34.9 GPa vs 19.4 GPa.
-Al2O3 also has a
slightly larger Poisson’s ratio which indicates that it is less
compressible. The calculated bulk modulus for
-Al2O3 is
204 GPa, larger than the value of 175 GPa reported in Ref.
35 and smaller than the value of 219 GPa of Ref. 26. Obvi-
ously, the difference comes from the different structural
models used for
-Al2O3. We are not aware of any experi-
mental bulk modulus for
-Al2O3. Even if it is available, it
could be somewhat smaller than the theoretical values be-
cause of the thin-film or porous nature of the samples.
From the elastic tensor, we can compute the elastic wave
velocity along specific directions by solving the Christoffel
wave equationm,
61
v2
ui
= Cijkl
lj
lkul
.
9
In Eq.
9,  is the density, v is the elastic wave velocity, C is
the elastic tensor, l is the direction of the k vector, and ui
is
the polarization of the elastic wave. Figure 6 displays the
inverse sound velocity of the three acoustic modes along all
directions in the x-y, y-z, and z-x planes. The longitudinal
high-frequency mode is nearly spherical, indicating that the
longitudinal wave is isotropic in the x-y plane. The two
transverse modes, however, show a sixfold symmetry which
is a feature of a crystal lattice with a threefold symmetry, i.e.,
the trigonal lattice. We note that the anisotropy is significant
only for the transverse modes. For longitudinal waves, the

-Al2O3 behaves like a homogenous media. This may indi-
cate that some crystalline features remain in the present

-Al2O3 model.
IV. ELECTRONIC STRUCTURE AND BONDING
The electronic structure and bonding in
-Al2O3 is stud-
ied using the first-principles OLCAO method.
47
Over the
years, we have been using this DFT-based method in its local
approximation
LDA for electronic structure and optical
properties calculations of many crystals and complex micro-
structures with great success.
47,62–73
The method has been
amply described in published papers and should not be re-
peated here. In the present study, we used the atomic orbitals
of Al
1s ,2s ,3s ,4s ,5s ,2p,3p,4p,5p,3d,4d and O

1s ,2s ,3s ,4s ,2p,3p,4p for a full basis expansion. A large
number of k points
864 in the irreducible portion of the BZ
are employed in the BZ integration.
The calculated band structure and total DOS are show in
Figs. 7 and 8, respectively. For convenience, we used the
notation of a hexagonal lattice to label the symmetry points
and axes. The calculated band structure for
-Al2O3 is al-
most identical to that of Ref. 30 using a different method.

-Al2O3 is an insulator with a sizable LDA gap of 4.22 eV.
The real gap could be 30%–35% larger to account for the
deficiency of the LDA theory. The top of the valence band

VB is rather flat from A to
and the bottom of the con-
duction band
CB is at
and consists of a single band. As
indicated in Fig. 1, we classify the atoms in
-Al2O3 as Altet
,
Aloct
,O3-fold, and O4-fold on the basis of their local coordina-