Alumina α-Al2O3 (ī10):
An ab-initio Examination of the Surface Electronic
Structure
Michael Vargas
Dr. Nick Kioussis
Abstract:
First principles calculations were run on bulk and the (ī10) surface of α-Al2O3 in order to
examine the electronic structures and structural relaxations. The Materials Studio package was
used, specifically the program CASTEP, utilizing Perdew Burke Ernzerhof exchange-correlation
pseudo-potentials. The partial and full density of states were calculated, and used to examine
differences between the surface and bulk. An Aluminum vacancy was also created on the
surface, and calculations were run on this. The energy of the surface was found to be higher(less
negative) than bulk, as expected, and changed from a strong insulator of band gap 5.9488 eV to a
conductor.
The calculations in this report will show that Al2O3 is highly ionic, an insulator in bulk, a
conductor at the surface.
Introduction:
Alumina is highly important in many fields of electronics and engineering. It is an
insulator, chemically non-reactive, and has a high melting point. Ruby, which is Alumina with a
Cr3+ impurity, is the earliest and most well known laser crystal. Alumina is used in circuit and
computer components, as well as micro-abrasives.
The α-Al2O3 unit cell has a hexagonal cell structure, with a rhombehedral primitive cell
consisting of 10 atoms.
Four of these atoms are Aluminum, six are Oxygen.
In this
examination, a cell of 60 atoms was created, cleaved at the (ī10), using the Cleave Surface tool
in Materials Studio. A 60 atom cell was chosen so that there would be little interaction between
the surfaces. The cell used has a u-vector, which spans the distance from the origin to the point
A, with length 5.13 Ǻ. The cell is also defined by a v-vector, which is from the origin to the
point B, its length is 6.99 Ǻ. The vectors u and v are separated by an angle γ = 84.16˚. Both u
and v are perpendicular to the vector w.
The last vector necessary to describe the cell
dimensions is the w-vector, which is from the origin to the point C.
Fig. XXXX: Rhombohedral primitive cell. Lattice dimensions of a1=a2=a3= 5.128 Ǻ
α=γ=β=55.28˚
After cleaving the plane, a vacuum slab of thickness 10 Angstroms was added. The cut
and vacuum slab are made precisely such that there is consistent pattern, i.e. ABCDABCDABC
(cleavage) DABCD. The surface is Aluminum terminated, the last layer having four Al atoms.
K-point and Energy Cut-off convergence tests were done to find optimum settings. When
relaxing a surface, atoms nearer to the surfaces are expected to relax with greater displacements
than those atoms located in the inside of the cell. This is due to the fact that there are no longer
atoms providing electrons to the outer atoms, in order for these outer atoms to feel that there
shells are being completed, they sink down into the still "complete" cell. Of course, since there
are only discontinuities along the w-direction, relaxations should be chiefly in this direction. U
and V direction displacements are very slight.
For the (000) surface of Alumina, such behavior was indeed observed. The Aluminum
atoms have the greatest relaxations, as they are able to sink downwards into triangle formations
of Oxygen atoms below them. Near to the surface, large displacements occur, while in the center
of the cell, displacements are so small that the atomic positions those of bulk Alumina. For the
(ī10) plane, relaxations similar to this are not expected, as there are no "convenient" resting spots
for the Aluminum atoms to take. The structures have all been tilted; there are no triangle
formations of Oxygen for the Aluminum atoms to sink into along the w-direction. Results
actually show that there are significant relaxations all along the cell length, in not only the wdirection, but in the u and v as well.
5
4
3
2
1
0
0
10
20
30
40
50
60
-1
-2
-3
-4
-5
Fig XXXX: Percent relaxations for Alumina in the u-direction. Notice that the greatest
displacements are near to the surfaces. Inside the cell, near to the middle, there is very
little relaxation, with the structure being almost bulk.
Before Relaxation
After Relaxation
Fig XXXX: Atomic Positions before and after relaxation, notice the showcased atoms
have moved the greatest.
15
10
5
0
0
10
20
30
40
50
60
-5
-10
-15
Fig. XXXX: Percent relaxation in Alumina in v-direction. Notice that relaxations near to
the surface are greatest. The small relaxations in the center are more apparent here.
Before Relaxation
After Relaxation
Fig. XXXX: Atomic positions after relaxation. Notice the two atoms showcased have
moved the greatest.
1
0.5
0
0
10
20
30
40
50
60
-0.5
-1
-1.5
Fig. XXXX: A graph of the percent relaxation in Alumina along the w-direction. These
relaxations are mostly small, with only slightly larger relaxations near to the surface.
Before Relaxation
After Relaxation
Fig XXXX: Atomic Positions before and after relaxation, notice that there are very little
displacements along the w-direction.
The energies of the bulk, un-relaxed, and relaxed cells are:
Energy
(eV)
E/Atom
Bulk
Un-relaxed
Relaxed
-2876.0202
-287.6020
-17223.4408
-287.0573
-17229.7369
-287.1622
In these calculations, 2 k-points were used. Cut off energy was 300 eV. After finding the
energies of the bulk and surface, the surface formation energy can be calculated. The surface
formation energy is the energy required to create the surface. The surface energy is the change
in energy from bulk to the surface divided by the area of each surface. The energy at the surface
(SE) can be calculated thus:
SE = (Esurface - N*(Ebulk/n))/(2*A)
where N is the number of atoms in the Surface cell,
n is the number of atoms in the bulk cell,
and A is the area of the surface.
The surface area for this system can be found in this manner:
(uXv) = |u|*|v| * sin(θ)
A = (5.13 Ǻ)*(6.99 Ǻ)*sin(84.16˚)
A = 35.67 Ǻ2
The calculated surface formation energy for Alumina is:
SE = ((-17223.4408 eV) – 60*(-2876.02028 eV/10 atoms))/(2*35.67 Ǻ2)
SE = 0.458 eV/ Ǻ2 =
7.328 J/m2
In comparison with other metal oxide surfaces, this surface energy is quite large.
A published surface energy for the (000) surface is 3.5678 eV/ Ǻ2.
We can now calculate the surface energy for the relaxed surface.
SE = ((-17229.7369 eV) – 60*(-2876.02028 eV/10 atoms))/(2*35.67 Ǻ2)
SE = .369 eV/Ǻ2 = 5.904 J/m2
5.904 J/m2 is closer to previous results for a similar surface.
The density of states for bulk Alumina shows that it is plainly a very strong insulator,
with a very wide band gap. Most of the valence electrons near the Fermi energy are in the P
states, while most of the states in lower energies in S states. The states past the Fermi energy
consist of a more equal mix of s and p states, with d states not present. After cleaving the
surface, adding the vacuum slab, and relaxing the surface, the density of states can be run for the
surface. In comparison with the bulk density of states, it can be seen that the surface is actually a
conductor.
10
9
8
7
6
5
4
3
2
1
0
-25
-20
-15
-10
-5
0
s
5
p
10
15
20
d
Fig. XXXX: Density of states for bulk Alumina. Notice that states from -20 to -15 are
chiefly s states, while states from -10 to 5 are chiefly p states. States outside the Fermi
energy are more balanced between s and p, although not filled at zero degrees.
35
30
25
20
15
10
5
0
-30
-20
-10
s
p
0
10
d
Fig. XXXX: Density of states for Alumina relaxed surface. Notice that there is now no
Fermi gap, this system has become a conductor. The lower energies are still mostly s
states, and the higher energies are still mostly p states.
Now that the examination of the bulk and surface energies and density of states is
completed, we can look at the charge density for our systems. The charge density analysis
option shows how the charge is distributed inside the cell. For a highly ionic system, charge will
be localized at certain elements. This means that these particular elements have more electron
negativity, and will attract the free electrons from the other elements. In an ionic system, these
free electrons are drawn completely to the other attractive elements.
Fig XXXX: Charge density along two different planes in the Alumina bulk primitive cell. Notice
that all charge is localized about the red Oxygen atoms. Oxygen is a highly electro-negative, and
so makes the bonds of this system ionic.
Aluminum Terminated
Oxygen Terminated
Fig XXXX: Charge density at the two surfaces, notice that the Aluminum terminated surface has
only small amounts of charge, due to the Oxygen atoms immediately below the surface. The
Oxygen terminated surface shows high charge around the O atoms.
These figures show that there is no change in the behavior of charge between the bulk
and surface charge densities. There is still strong charge localization about the highly ionic
Oxygen atoms, and still strong charge de-localizations about the Aluminum atoms.
Such
behavior, again, is the sign of ionic bonds, and an ionic system. The absence of charge at the
upper surface and collection of charge at the lower surface may bring rise to an electric field,
although this effect has not been studied in this report.
Conclusions:
As can be seen from this report, Alumina is indeed an insulator that changes into a
conductor at the (ī10) surface. It has strong ionic bonds, with charge localized at the Oxygen
atoms. The high surface energy suggests that this surface is difficult to create, and may fall apart
quickly.
Future Work:
After having completed an analysis of the (ī10) surface in Alumina, the next step is to
discover whether or not a vacancy at the surface can be made more easily than at the (001) plane.
Such behavior can be seen in some other metal-oxides. Another interesting project would be to
see if adding a magnetic atom in the system such as Cobalt or Iron can change the magnetic
moment of the entire system.