Indian Journal of Pure & Applied Physics
Vol. 37, April 1999, pp. 306-312
Schottky clefect enthalpies of alkaline earth oxides
Munima B Sahariah l ,Radha 0 Banhatti 2 . Usha Bharali l • C Kadolkar l & Y V G S Murti l •
10epartment of Physics, Indian Institute of Technology, Guwahati 781001
2Institut fur Phys ikalische Chemie, Westfalische Wilhelms Universitat.
Munster, Sclossplatz 4, 0-48149 Munster, Germany
Received 3 February 1999
The Schottky defect formation process in MgO, CaO, SrO, and BaO i~ investigated using our recently introduced
EPPI modeL3 We find that inclusion of quadrupolar interactions makes a substantial contribution to the formation energy
by values ranging from 1.2 eV to 2.9 eV. The EPPI values of the enthalpies (hI) are 5.9 ,5.8 ,4.7 and 1.4 eV respectively
for MgO, CaO, SrO, and BaO.The results of cation self diffusion experiments in MgO are reanalyzed to include the effects
of impurity-vacancy association reaction. The analysis suggests the previous interpretation of the experimental data to be
an oversimplification. Since the binding energy of an impurity - defect complex in MgO could be quite large, the effects of
the association reaction on the defect densities are appreciable. On the basis of near complete association of the dominant
tetravalent cation impurity , hI in MgO is found to be 5.8 eV which is in .good agreement with the theoretical value of the
present work.
Introduction
In recent years point defects in oxide materials have
attracted attention, firstly because the family of ceramic
oxides MgO, CaO, SrO, BaO have the rocksalt structure,
are essentially ionic and have very high melting points.
Secondly, the transport properties in these oxides exhibit
some puzzling features in respect of anion diffusion .2
Thirdly, oxygen atomic ordering and oxygen vacancies are
believed to playa significant role in the superconducting
transition of high Tc oxide materials 3
The existing picture on the point defect energetics in
oxides is quite unsatisfactory . It is argued that owing to the
bivalency, Coulomb forces tend to prevail over other
interactions resulting in rather large Schottky formation
energies (7-8 eV) .3 On the other hand, a much smaller
Schottky defect formation energy is reported by Harding
and Price 4 on the basis of their cation self-diffusion
measurements . The shell model calculations that use a
HADES software package led to values of h/ which are
much larger (7.5 eV ego in MgO).5 These authors have
ignored the conclusions drawn from se lf-diffusion
experiments. Our recent studies 6 •8 have indicated c1eafly
that it is important to take into account the inhomogeneous
electric fields and the induced quadrupoles in the defect
environment, if one is to satisfactorily mode l the point
defects especially for systems whose ions are highly
. polarizable.This is certainly the case for oxygen ions . It is
thus important to explore the nature of the Schottky
disorder in these systems afresh.
Furthermore, the EPPI model was shown to be
successful and in fact essential to resolve the many
anomalies of the transport behaviour relating to Schottky
and Frenkel defects in the case of the silver halides . In this
paper we plan to apply this model for the case of alkaline
earth oxides with a view to seeing the effect of the induced
quadrupoles on the Schottky defect process. We are also
reporting here the first results of the first use of the beta
version of MUDRA (a defect modelling software under
development at IITGuwahati).
2 Potential Model
The short range potential for a pair of ions of type i and j
at separation rij is taken to be of the form
-r
m
't'
= A 1/.. exp-'-~ .
Pu
C
--':!....
6
rij
. .. (I)
where Aij 's are the strength coefficients fo r the overlap
repulsion potential and Pij 's are the hardness parameters.
The va lues of these constants for all the pairs (Mg-O, Ca0 , Sr-O, Ba-O, Mg-Mg, Ca-Ca, Sr-Sr, Ba-Ba and 0-0)
taken from Stoneham'S compilation 9 for separations
appropriate for these crystals, are listed in Table I. The Cj
's are the constants for the dipole-dipole van der
Waals(vdW)
interactions.These
coefficients
are
notoriously hard to estimate either from its ab-initio
calculations or by semiempirical methods . Here we adopt
an approach similar to the one 6 we had successfully
followed for Agel and AgBr. The constants Ccc and C aa .
307
SAHARlAH et at. : SCHOTTKY DEFECT OF EARTH OXIDES
Table I Salts
MgO
CaO
SrO
BaO
:Short range overlap repulsion parameters
Acc(eV)
18147.9
54898. 1
91113 .2
588445.0
Ac.(eV)
22 14.39
1996.35
2187 .61
1994.97
A•• (eV)
871.489
583.532
538.788
444.760
are related to the cation-anion .coefficient Ce. through the
relations based on London's one-level approximation for
the van der Waals dipol e-dipole potential:
c = (~)(Ec + ElI)C
au
cc
2E"
Ec Eu)C
all
3
rc
)
=(
r"
co
. .. (3)
Ec and E, are the second excitation energies of the cation
and that of anion respectively. The coefficient Ce. is
obtained by using the equilibrium condition of the lattice
at 0 K.
dWI. =0
... (4)
dr"
where r 0 is the anion-cation shortest separation and th e
lattice energy is W L given by
... (5)
. .. (8)
)
potential (vdW and overlap repul sion) and rc ' ru are the
radii of the cation and anion respectively.This procedure
ensures a correct simulation of the static dielectric
response of the material so important for the defect
modelling." Table 2 gives th e van der Waals param eters.
3 The Physical Model
The calculations are based on a phys ical model in which
the defect field is sp lit into two parts: a region I
comprising of the defect and its nearest neighbours, with
the rest of the lattice term ed as region 2 (Fig. I). The
modelling combines both region s in a consistent way but
with the di splacements and pol arization s eva lu ated
di ffe rentl y. The relaxation s and dipole mom ents of all the
ion s of th e region 2 are solved by a se mi contin uum
approach in which the defect charge and the crystal
dielectric constant essentially simulate the response. The
-
Region II'
where a
M
p••(nm)
0.03679
0.04026
0.04 103
0.04260
where K is th e force constant of the total short range
. .. (2)
ell
C = (~)( +
aa
2E
ac
c
ac
(
pcc(nm)
0.01452
0.01896
0.02068
0.02070
pc.(nm)
0.02756
0.03180
0.03380
0.03620
-,
"
is th e Madelung constant.
As for the polarizabilities, we are using the
displacement polarizability a d as fix ed by the short-range
potenti al and the static dielectric constant Es to find a c and
a •.
a
e2
d
=-
... (6)
K
c ud (3r2 c.c, --1)
2
3
a +a +2a
= _,_, ) ( _ .' -1C
... (7)
Fig. 1 -- Sp li tt in g of th e whole crystal into two regions
acco rdin g to ML s c~eme .
corresponding solutions for the ions of region I are
derived by a fully atomistic method. In addition', in the
INDIAN J PURE & APPL PHYS, VOL 37, APRIL 1999
308
Table 2 Salts
MgO
CaO
srO
BaO
van der Waals parameters
Cee
6
(eYA )
6.822
46.164
103.72
176.01
Ce•
6
(eYA )
11.076
31 .642
54.710
67 . 188
ae
C••
(eYA 6 )
60. 182
50.91 I
60.398
45.203
EPPI model, the distortions of the ions of this region I, are
taken to the quadrupolar approximation. See Ref 'o for a
recent review.
a.
3
aed
(A 3 )
0 .244
0.628
1.146
1.764
(A )
0.024
0 .222
0.603
1.582
3
(A )
1.541
2. I 51
2.4 I I
2.928
moments and quadrupole moments of the ions
neighbouring the defect with all the necessary input data
fed in on prompt, MUDRA also yields th e energies of the
anion vacancy E u ,' , cation vacancy energy
3.1 Theory of defect energetics
The energy ED of the crystal lattice distorted by ion
displacements and deformations of the electronic charge
distributions of the ions, is given by
... (9)
where
E, (13)
is the energy of region 1
energy of region 2 ,
E2 (/3, y)
,E3 (y)
is the
gives the interaction
energy between region 1 and 2 . P represents the complete
set of ion displacements, induced dipole and quadrupole
moments of ions in region I. y together represe nts the
ionic displacements and induced dipole moments of region
2.
.. (12)
In contrast to the more elaborate scientific defect
HADES - type packages , which requ ire a heavy
computational facility, MUDRA is a PC- based code with
relatively much higher efficiency.
3.2 Outline of EPPJ model
In EPPI
model , the
contribution
from
the
quadrupoles to the defect enthalpy has been taken into
account. Here the equations for the d ipole (p) and
quadrupole moments (Q) are defined by
Q = a'lCI"c
Eo = E,(/3) + E,(/3,y,,)- ~~2I r Y"
wh ere
... (10)
p
= a d J"r.. .
/ O(
a"
.. . ( 13)
... ( 14)
is
the
dipole
quadrupole polarizability,
For the region I however, the equilibrium re laxations are
detennined by
lattice
energy (WI. ), and the energy of formation of Schottky
defects given by
Now , the distortion of regi on 2 is relatively weak and
is treated in the harmonic approximation . The equilibrium
displacements and dipole moments Yo of this region are
taken to be known via the Mott-Littleton (ML) scheme.
Then , it follows that the defect energy can be written as
.
En. ,the
and
C",,·
F loc
polarizab ili ty
aq
is
the
is the local electric field
is the local field gradient.
The total electric field F is given by
=0
... ( I I)
The software MUDRA is coded in C++ to evaluate
ED as
a function of the rela xations of th e ions nearest to
the defec t and alsu m illlmize the energy. The calculations
are perfonned for both components of the Schottky
disorder viz., an ion and cation vacancy. The output wili
consist of equilibrium values of relaxations, dipole
F = F,,: + F) + F,i + F,;
.. . ( 15)
here F,,~ gives the fi eld due to monopoles including the
real and v irtual charges of reg ion I.
F;; -I- F~;
g ives th e
field due to dipoles in reg ion s I and 2, and ~' g ives the
field due to quadrupoles of th e region I .
309
SAHARIAH el 01. : SCHOTTKY DEFECT OF EARTH OXIDES
Wi
q
Table 3 -
srO
BaO
I
... (20)
III
Theoretical Schottky defect enthalpies
Salts
MgO
CaO
=_(l)QG
4
2.11
2.41
2.58
2.76
hIs
h{ (eV)
(eV)
(PPI)
(3)
8.76
7.29
5.96
3.38
(EPPI)(4)
e
the expression of F is
1.6642
4(I+s)
2.3713
r~'(I+sf
r.2 - (I+s) ' - (2+s)' - {(1+s)' + I} IS
gives the additional contribution to the
... (16)
arising
•
h{
•
Quadrupolar contribution to
•
h{
from
quadrupole
Values decreases in the order of increasing
h{
r" .
is quite significant.
Values (EPPI) for all the oxides are much lower
than the Coulomb estimates.
In Table 3 we list the values of the computed
Schottky formation enthalpies according to the PPI and
EPPI models for the four oxides.
4
e {
}
0.8893
+ 2"" odds(s)Mc + evens(s)Ma - )
4 Q
'0
'" (I + s)
Wql
vacancy formation energy
interactions.
The main fmdings are:
2.9
1.5
1.2
1.9
5.88
5.79
4.72
1.43
In case of anion vacancy
obtained as
Hence
Diff(eV)
(3)-(4)
Results
In Fig.2 regions I and II are regions of intrinsic and
extrinsic diffusion (D) respectively while region III is the
precipitation - influenced region according to Harding and
Price 4
who interpreted the slope of region I as
1 / 2h{ + hili and that of region II as hili .Thus they
.. . (17)
and s is the fractional displacement of region I ions.
reported an enthalpy of Schottky defect formation for
MgO as 3.8 ± 0.3 eV.Harding and Price however did not
take into account of any impurity - vacancy association
effects. This is an oversimplification in their interpretation
of experimental results especially since the binding
energy terms turn out to be substantial.
Simi(arly the expression of G is
G
=
2e
)-
(1 + s) ro
e {
4e{2(I+s)2_1}
-)
'0(1 + s)
{(1 + S)2 + I} . ,;
}
+") odds(s}Mc + evens(s}Ma -
'0
3.5570p
2 ..957Ie
25
)+
2.7066
s
r;, (I + s)
s
4
'0(I + s)
4
Q
... ( 18)
Here odds(s) and evens(s) represent the magnitudes of
appropriate lattice sums for the field at the site (l+s,O,O)r0
due to the dipole moments induced in ions of region 2
corresponding to cation and anion shells surrounding the
defect, respectively. The associated electronic polarization
energy which is a major contribution to the removal of the
ion from the lattice is given by
r
m
. . . (19)
The quadrupolar polarization energy is given by
--'" Iff
Fig.2 -- Arrhenius plot for cation self diffusion in single
crystal MgO
310
INDIAN J PURE & APPL PHYS, VOL 37, APRIL 1999
4.llmpuriry - \'8cancy association effects
We discuss the impurity - vacancy association effects
taking the following impurities:
cr.
Si4+ ,AI 3+,Fe 3+ and
4
In the case of Si + (Fig .3a) we have the following
reaction for the impurity - vacancy pairs:
g b + X <=>
~
+ C,
... (21 )
where X is the impurity - vacancy ensemble and
gh
is
the free energy of binding for the impurity - vacancy pair.
~
0
0
•
•
•
0
•
•
0
0
•
--""--- = ze
of distinct orientations in the lattice. k IJ is the Boltzmann
.. .(23)
•
where
•
0
'7=_IJ-
Vc
0
kT
•
•
•
0
For near complete association
CO
.. . (25)
•
For the ions AI +, Fe + and
(Figs.3b & 3c) we
encounter an impurity - vacancy triad
3
g h + T = V,. + C; + C;
0
T
is the concentration of associated triads.
•
•
cr
we get
•
+le
(C)
(el
= fJc
(27)
C i [110]
Vc
0
... (26)
0
0
-2e
0
cr
3
0
•
0
0
•
Near-complete
association
corresponds
to
(f3 ;:::: 1)corresponds to
2
Fig. 3 - Crystal MgO with impurities (a) Si 4 + (b) All +. Fe l +
l
),we have ,
[110]
~.
0
(f3 ;:::: 1
1
•
+ Ie
.. .(24)
gh
(a)
Putting Xl
i
= fJc we get
constant. With x k
where x
•
C
(22)
Here c is the total impurity concentration and z is the no .
0
(b)
0
•••
C·1
'..I 2r o
•
0
H
Xk(C-X k )
[110]
O
•
•
ki
•
0
-2e
0
g.
xk
C; , is the impurity ion.
is a cation vacancy,
•
From Eq . (21), we can derive the equation for the
( )
concentration x; of the associated pairs as
X,
= C;
;::::
_~
1
2 -3 c 3e
3kHl
. .. (28)
Thus taking into account the vacancy - impurity
association we see that we cannot si mpl y interpret the
slope of region
II
as
hl/l
Instead
it should be
311
SAHARIAH et at.: SCHOITKY DEFECT OF EARTH OXIDES
Table 4 Impurity
hb
(eV)
2.0
1.7
2.4
Si 4+
Fe 3+
AI 3+,
cr l
1 / 2hb + h
Experimental Schottky defect enthalpies
c
(ppm)
~
Degree of
Association
h m (eV)
hI (eV)
400
200
200
0.960
0.003
0.253
high
low
medium
0.56
0.99
0.75
5.8
3.8
(3.8-5.4)
dlQlV. of ~cillion : MriD:!
m
in the case of impurity- vacancy pairs and
1/ 3h h + hlll in case of triad under consideration of near
complete association.To estimate the binding energy of the
impurity-defect clusters in the three cases (Figs. 3a, 3b and
3c) we use the simple electrostatic approximation
o.g AB8ociliion
qlnppm
0.8 cornplll8
20
0.7
2 100
3 300
0.8
4 : 700
5 : 1500
J05
0'"'
hh "
- ~
qiqJ
... (29)
0.3
i.J £Ji.J
0.2
0.1
where qi 's are the effective charges of the defect
impurity in the cluster and the summat ion is over all the
pairs of point charges in the cluster.
8D~2:--~--:~--:~-==~-=-~~e=~~~
Table 4 presents the binding enthalpy (hh ) and the values
of Schottky fonnation enthalpy for the different impurity
types.
The degree of association as a function of the
k8 T
parameter '7 (= - - ) is shown in Figs 4 & 5 for the
gh
dlQf9.'" ~cillion : MJlir
o.g
case of the diad and triad complexes.The dominant
impurity in the crystal samp le of Harding and Price is Si 4+
(400 ppm) which suggests the most probable value of
qin ppm]
0.8
1: 20
2 : 100
0.7
3 : :!(ID
0/1 : &lD
0.8
h{ to be 5.8 eV for MgO in excellent agreement with the
theoretical value of the present work with EPP! model
(5.88 eV).
5 : goo
e:1500
Jos
0'"'
5 Acknowledgement
o.~
0.2
0.1
DO
0.1
0.5
The autho(s are grateful to the Department of Science and
Technology, Government of India,for funding the project
to develop the defect-modelling software MUDRA
References
FigA -- p vs. k BT/~ with different total impurity concentrations
(c) for impurity-vacancy diad .
Banhatti R 0 & .Murti Y V G S, Phys Stat Sol (b) . 166
(1991) 15-23 .
2
Nowick
A S, MRS Bul/etin, November issue, (1991) 38-
41.
3
Zhu Yimei, Tafto J & Suenga
November issue, (1991) 54-59.
M, MRS Bul/etin,
312
INDIAN J PURE & APPL PHYS, VOL 37, APRIL 1999
4
Harding B C & Price 0 M, Phil Mag, 26 ( 1972) 253260.
8
Murti Y V G S & Banhatti Radha 0, Bull Mat ScI, 20
(1997) 435-440.
5
Mackrodt W C & Stewart R F, J Phys C Solid State
Phys, 12 (1979) 431-448.
9
Stoneham A M, Handbook of Interatomic Potentials, I
Ionic Crystais, (AERE Report 9598), 1979.
6
Banhatti Radha 0 & Murti Y V G S, Phys Rev, B48
(1993) 6839-6853 .
10
7
Banhatti Radha 0 & Murti Y V G S, Radiation Effects
and Defects in Solids, 134 (1995) 157-159.
Murti Y V G S & Banhatti Radha 0 , Proc Int Conf on
defects in Insulating Materials, Eds O.Kanert and J-M
Spaeth, (Wrold Scientific, Singapore) , 1993, p.198 .
II
Murti Y V G S & Usha V, Physica, 838 (1976) 275 .