Journal of Modern Physics, 2011, 2, 158-161
doi:10.4236/jmp.2011.23024 Published Online March 2011 (http://www.SciRP.org/journal/jmp)
Structural and Spin Polarization Effects of Cr, Fe and Ti
Elements on Electronical Properties of α–Al2O3 by First
Principle Calculations
Hossein Asghar Rahnamaye Alibad, Shaban Reza Ghorbani
Department of physics, Sabzevar Ta rbiat Moallem University, Sabzevar, Iran
Received October 3, 2010; revised December 7, 2010; accepted December 11, 2010
Abstract
Structural and spin polarization effects of Cr, Fe and Ti elements on electronical properties of alumina have
been studied by using of Local spin density approximation within density functional theory. The calculated
results indicated that substituting aluminium atoms by these dopants have a significant influence on the
structural and electronic properties of α–Al2O3 crystals. Band gap of alumina decreases with the substitution
of these impurities. Results show that band gap is different for spin-up and down (spin splitting effect).
Among these impurities the effect of Ti on size of the energy gap is small in comparison with Cr and Fe. It is
suggested that the origin of electrons spin splitting is appeared from exchange energy of d-states. These
results may be useful to obtain a physical beheviour of transition metals for electrons spin polarization in
d-states.
Keywords: LSDA, Spin Polarization, Alumina, Transition Metals
1. Introduction
A unique combination of alumina with transition metals is
very important due to their possible industrial applications.
The corundum or sapphire phase of alumina (α–Al2O3)
has widespread applications in ceramic and semiconduc-
tor industry [1]. In order to improve electrical and optical
properties of alumina, it can be doped by other metals; this
requires the variation of electrical properties by theoreti-
cal calculations. Good substitutions on the Al atom sites
are transition metals (TM), because the d-bands in these
metals are partially filled and extended over the band gap.
The substitutions of these metals change the band gap size
and improve alumina properties.
Nearly all atoms have multiple electrons but most of
them are paired up with another opposite spin electrons in
the orbital. Solid magnetic properties are derived from the
ground state properties of incompletely filled electron
shells. Observed magnetic response, in a particular system
largely depends on how the spin and orbital properties of
these electrons end up in consideration of Pauli’s exclu-
sion principle and minimizing Coulomb repulsion [2].
In this work, the influence of spin polarization Cr, Fe
and Ti elements on electrical properties of alumina have
been studied. Spin polarization is the first rule of Hound
for determining the ground state (lowest energy) of elec-
tronic configuration in an atom. According to these rules
and Pauli’s exclusion principle, electrons have been ar-
ranged in a way to have maximum total spin, S. In fact,
these minimize Coulomb energy so that two parallel spins
can not be in the same state. Energy decreasing due to the
preference of being parallel spins is called exchange en-
ergy. In solids, depending on whether the crystal is insu-
lating or conducting, magnetism has historically been
approached from two different schools of thinking: either
a localized or itinerant point of view. In the localized
concept of magnetism, the electrons and their magnetic
properties remain associated with their respective para-
magnetic ion in an insulating crystal. Conversely, in the
itinerant picture, the conduction electrons are responsible
for magnetism. The magnetic of ordering may arise based
on the specific alignment of the atomic magnetic mo-
ments, favoured by atomic exchange interactions and
itinerant magnetism which is associated with metallic
behaviour.
Normally, in a metal, there is an equal number of
spin-up and spin-down electrons which fill up states to the
Fermi energy. In the absence of an external field, a stable

H. A. R. ALIBAD ET AL.159
ferromagnetic state can only arise if there is a spontaneous
splitting of the spin population in the bands. This property
is called spin polarization [2]. Pauli Hamiltonian in a
magnetic system is

2
2
2
pefBef
HV
m
  
Bl
 (1)
where the third and fourth terms refer to spin polarization
and spin orbit interaction, respectively, so that in this
work we have neglected from spin orbit interaction.
ef and ef are electrostatic potential and effective
magnetic field, respectively, and they are defined by the
following Equations:
V B
efext Hxc
VVVV
(2)
efext xc
BBB (3)
where
H
Vand
x
c are Hartree and exchange-correlation
potential, respectively. In the local Spin Density Ap-
proximation (LSDA),
V
H
V and
x
c
V are defined as fol-
lowing:
 
xc
xc xc
n,
Vn,n
n





m
m (4)


xc
xc
n, ˆ
n
m
 

m
Bm
1
(5)
where and are magnetic dipole moment and spin
density of electrons, respectively. In the spin space, due to
Pauli spin operators, Pauli Hamiltonian is a matrix
and wave function is a two component vector (spinor):
ˆ
mn
22
1
22
p
H





 

 
 
(6)
where 1

and 2

are spin-up and spin-down compo-
nents. For non magnetic compositions, spin-up density of
electrons is equal with spin-down density of electrons.
Therefore, total magnetic dipole moment of electrons is
zero. Since in magnetic compositions, spin-up density of
electrons is not equal with spin-down, the total magnetic
dipole moment of electrons is not zero and spin splitting
of electrons is occurred by exchange splitting, which is
shown in Figure 1 [3].
2. Method of Calculations and Crystal
Structure
In performance our calculations we have used the
Full-Potential Linearized Augmented Plane Wave plus
Local Orbital (FP-LAPW+LO) method. It is based on the
density functional theory (DFT) as implemented in
WIEN2K code [4]. The Local Spin Density Approxima-
tion (LSDA) is applied for the exchange-correlation in-
teractions.
The LAPW+LO method for Cr, Fe and Ti elements is
Figure 1. Spin density of electrons for (a) non magnetic
compositions (b) magnetic compositions.
suitable because these impurities have localized d states.
The structure of α–Al2O3 is hcp that consists of
close-packed planes of oxygen and aluminium. Its space
group is R-3c with number 167. There are 12 Al atoms
and 18 O atoms (30 atoms) in the unit cell of
α–Al2O3.The Cr, Fe and Ti atoms, which have the same
valence with Al (+3), are substitute at octahedral sites
within the alumina structure in the Al sites (Figure 2).
For calculation of lattice constants, lattice energy
variation as a function of the deviation of c/a (ratio, %)
in constant volume have calculated from experimental
values for α–Al2-xTxO3 (T: Cr, Fe, Ti; x = 0.5). These
results are shown in Table 1. It can be seen that our re-
sults good agreement with experimental results for
α–Al2O3. Then with calculated lattice constants, relaxed
atomic positions were obtained for α–Al2O3 and
α–Al2-xTxO3 (T: Cr, Fe, Ti; x = 0.5).
3. Results and Discussion
Alumina is a non-magnetic composition with high dielectric
Figure 2. The unit cell of α–Al2-xTxO3 (T: Cr, Fe, Ti; x = 0.5)
used in this study.

H. A. R. ALIBAD ET AL. 160
Table 1. Calculated lattice constants for α–Al2O3 and α
-Al2-xTxO3 (T: Cr, Fe, Ti; x=0.5).
Compositions This work(Å)Others(Exp., Å)
α–Al2O3
A = b = 4.75699
C = 12.98769
a = b = 4.765 [5]
c = 13.001
α–Al1.5Cro.5O3
a = b = 4.75899
c = 12.97680
---
---
α–Al1.5Feo.5O3
a = b = 4.75852
C = 12.97936
---
---
α–Al 1.5Tio.5O3
a = b = 4.76453
C = 12.94666
---
---
constant. Symmetric direction for calculation of band
structure in the first brillouin zone of α–Al2O3 and
α–Al2-xTxO3 (T: Cr, Fe, Ti; x = 0.5) is shown in Figure 3.
Energy zero was set at the top of valance band. Energy
scale is in eV and the origin of energy was arbitrarily set
to be at the maximum valance band. The calculated band
structures and densities of states of α–Al2O3 are shown in
Figure 4. Focusing on α–Al2O3, the bands with the
widths 0 - 7 eV and 16-18.5 below the Fermi level are
due to O 2p and 2s states. Al 3p and 3s states are ap-
peared above the Fermi level in conduction band. The
distance between the maximum of the conduction band
and the minimum of the valence band (at the Γ point)
will be denoted as band gap (Eg). There are a large num-
ber of localized states at the top of the valance band.
These states are originating mainly from the O 2p atom.
The valence band edges near the Fermi energy for O atom
are quite sharp, while the conduction band edges near the
Fermi energy are not. The valance band is composed of
the O-2p orbital hybridized with the Al-3s, 3p and 3d
orbitals. The lower valance is formed predominantly by O
2s atom. The contribution of Al 3d and Al 3p in the val-
ance band are rather small.
A band gap of ~6.33 eV (without empirical correction
factor) was obtained for α–Al2O3. That is fairly closed to
experimental values. For comparison, the band gap re-
sults obtained from experimental measurements and
other theoretical methods have been summarized in Ta-
ble 2. It can be seen that our results agree qualitatively
with experimental results. There is also a well agreement
with the other theoretical calculations.
Figure 3. Considered symmetric direction in first brillouin
zone for calculation of band structure of α–Al2O3 and α
-Al2-xTxO3 (T: Cr, Fe, Ti; x = 0.5).
Figure 4. Calculated density of states and band structure
for α–Al2O3.
Table 2. Obtained band gaps for Al2O3 by various methods.
Band gap (eV) Methods
~10.8
~8.8
~8.7
~5.3
~6.2
~6.33
Experimental
α–Al2O3 [6]
α–Al2O3 [7]
Amorphous alumina [8]
Theory(LDA)

–Al2O3 [9]
α–Al2O3 [10]
α–Al2O3 ( this work)
The calculated electronic band structure and density of
states of α–Al1.5Cro.5O3, α–Al1.5Feo.5O3 and α–Al1.5Tio.5O3
are shown in Figures 5-7. By comparing Figure 4 and
Figures 5-7, it can be seen that the substitution of Cr, Fe
and Ti for Al in α–Al2O3 structure results in reducing the
band gap. The band gap decreases mainly due to Cr, Fe
and Ti-d state in the conduction band in different energies.
The obtained results from Figures 5-7, show that for
all compositions, the energy gaps are smaller for spin-up
states than spin-down. Therefore, electrical properties are
different for both states and they can be applicable in
spintronic devices. As can be seen in Table 3, among
these impurities the effect of Ti on size of the energy gap
is smaller in comparison with Cr and Fe. These are re-
lated to exchange energy splitting of d-states and mag-
netic properties of these materials.
4. Conclusions
The results of obtained show that the alumina energy gap
was decreased by substitution of Cr, Fe and Ti impurities
on the Al sites in α–Al2O3. It was found that energy gaps
were different for spin-up and spin-down states for all
α–Al2-xTxO3 (T: Cr, Fe, Ti; x=0.5) compositions. There-
fore, electrical properties are depended on spin polariza-
tion of electrons. Spin splitting effect were appeared from

H. A. R. ALIBAD ET AL. 161
exchange interaction among the electrons in d-states.
These results may be useful to obtain a physical picture of
electrons spin polarization in d-states of transition metals.
Among these impurities, the Cr element has the stronger
spin splitting effect than the other impurities (Fe and Ti)
in Alumina.
5. Acknowledgements
We thank Professor P. Blaha, Vienna University of Tech-
nology Austria, for help with technical assistance of using
Wien2k package.
Figure 5. (a) Band structure of α-Al1.5Cro.5O3 (b) Density of
states for α–Al1.5Cro.5O3.
6. References
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[3] R. Laskowski, “Magnetism and Soc in Wien2k,” Vienna
University of Technology, Institute of Materials Chemis-
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[4] P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka and J.
Luitz, Institute of Materials Chemistry, TU Vienna, 2011.
http://www.wien2k.at/
Figure 6. (a) Band structure of α–Al1.5Feo.5O3. (b) Density of
states for α–Al1.5Feo.5O3.
[5] S. M. Hosseini, H. A. Rahnamaye Aliabad and A. Kom-
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Figure 7. (a) Band structure of –Al1.5 Tio.5O3. (b) Density of
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Band gape (eV)
Spin-down Spin-up
Compositions
6.33 α–Al2O3
4.13 1.77 α–Al1.5Cro.5O3
2.04 1.56 α–Al1.5Feo.5O3
4.94 4.13 α–Al1.5Tio.5O3
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