THERMODYNAMIC PROPERTIES OF PEROVSKITE –TYPE
OXIDES WITH DIPOLE (OH) CENTERS
A.Ya. Fishman1, V.Ya. Mitrofanov1, V.I. Tsidilkovski2
Institute of Metallurgy1, Institute of High-Temperature Electrochemistry2,
Ural Branch of Russian Academy of Sciences, Ekaterinburg, Russia
The low-temperature thermodynamic properties of perovskite-type proton conducting
oxides, such as АВO3-y doped with cations of lower valence, with proton-associated
dipole centers (OH) and RIII - (OH) have been investigated. The energy spectrum of
these orientationally degenerate dipole centers (DDC) has been considered and
different mechanisms of the degenerate states splitting have been analyzed. The DDC
contributions to the dielectric susceptibility, heat capacity and other thermodynamic
properties have been calculated. It is shown that the centers under consideration
significantly change the low-temperature properties of oxides and these properties
strongly depend on the ratio between the tunnel splitting and the dispersion of random
crystal field distribution. H/D/T isotope effects for the properties under study are
discussed and it is established that isotopic substitution of protons significantly
changes the DDC contribution magnitudes under appropriate conditions.
INTRODUCTION
Much interest has been focused on the defect structure and transport properties
of different perovskite oxides such as АIIВIV1-xRIIIxO3-y due to their importance as
high-temperature proton conductors. Such proton conducting oxides are now
considered as candidates for use in high temperature sensors, fuel cells, electrolysers
and other electrochemical devices, to be applied in cleaner energy technologies based
on natural gas and hydrogen [1,2]. These materials can dissolve a significant amount
of protons as defects in the presence of hydrogen or water vapor containing gases [3].
Protons in oxides are attached to the oxygen ions forming OH centers, which possess
dipole moment themselves and form more complex dipoles R3+-OH with dopant ions
R3+. There exist equivalent potential minima for protons at oxygen and for OH
centers near the dopant in the perovskite lattice ([3-5] and refs therein). Both types of
multi-well states without regard for interactions between different “dipoles” and their
interactions with other imperfections are orientationally degenerate.
Evidently, such degenerated states can significantly affect different physical
properties under appropriate conditions, and, in principle, the studies of these effects
can provide essential information for understanding the nature of the state and
dynamics of protons in proton-conducting oxides. Nevertheless, we are unaware of
any results in this field. The low-temperature dielectric relaxation in the proton
containing АIIВIV1-xRIIIxO3-y oxides was investigated in the several works of Novick et
al [6,7], but possible manifestations of (OH) dipoles reorientation were not
considered there.
This communication reports our theoretical considerations of the
manifestations of the proton-associated degenerate dipole centers (DDC) in the lowtemperature properties of proton conducting oxides. The energy spectrum of such
tunnel centers has been considered and different mechanisms of the degenerate states
splitting have been analyzed. The main attention is focused on the systems with low
content of reoriented centers. A number of low-temperature physical properties of
3 - 40
such degenerate systems at arbitrary ratio between the tunnel splitting and the
dispersion of random crystal fields have been considered. Isotope effects H/D/T for
the properties under study are also discussed. The studies were performed for the
compounds АВO3-y with the acceptor doping both in the B and A sublattices: АIIВIV1III
III
II
III
xR xO3-y and А 1-xC xВ O3-y, respectively.
THE MODEL OF PROTON-ASSOCIATED DIPOLE CENTERS
We have focused on the low temperature region when only the role of the
DDC of the first type, OH dipoles, is significant. In this region, where for the effects
under study one can neglect the protons transfer between oxide ions, the OH dipoles
reorientation is due to the protons tunneling between the local potential minima near
the same oxide ion. We are unaware of the precise data for these potential minima
positions and energetics. Nevertheless, there exist different results demonstrating that
the height of potential barriers between these minima is about 0.1 to 0.2 eV, see, e.g.
[5]. A set of positions for protons near the single oxide ion, suggested in [4], is shown
in Fig. 1. The authors of [4], on the basis of atomistic computer modeling results for
LaMnO3, stated that protons are mainly located in the positions i and ii, see Fig. 1. A
precise experimental determination of proton positions is difficult. Some experimental
results obtained by different experimental techniques, however, have shown some
consistencies and confirm a model in which protons lie in the potential minima
between the two neighboring oxygen ions, ii and iv positions, see [5] and refs therein.
Fig 1. Possible types of proton positions in the cubic perovskites with dopants in
different sublattices.
Let us consider in brief the degeneracy of the energy levels, without regard for
tunneling, of the OH groups located in the nearest neighborhood of dopant, and far
from it: “bounded” and “free” states, respectively. The splitting of DDC states caused
by the DDC-dopants (and/or with other imperfections) interactions for the
intermediate cases will be taken into account via interaction of DDC with random
fields of appropriate symmetry.
3 - 41
For the sake of simplicity, we restrict the considerations below to cubic
perovskites. It is clear, see Fig. 1a, that free DDC states for the i and iii positions are
4-fold degenerated and the other are 8-fold degenerated (4 and 8 equivalent potential
minima of each type for protons in the lattice). For the protons located in the
neighborhood of the dopant in the B sublattice the “bounded” 8-fold degenerated
states, due to the interaction with the dopant, are split into two 4-fold degenerated
levels. It is easy to see that the bounded states located in the neighborhood of the
dopant in the A sublattice are split as follows. The DDC levels for the i and ii type
positions are split into two 2-fold and 4-fold degenerated levels, respectively, and the
iii state is split into ground singlet and excited doublet and singlet. Thus, the ground
state of the proton-associated dipole centers is degenerated (except the case of the
bounded DDC of the type iii near the dopant in the A sublattice).
Next we concentrate on the consideration of the bounded states of protons,
when OH groups are located near the dopant R3+ in the B subblattice (see Fig. 1a).
This case is most general and the results for other situations could easily be obtained
using the ones derived below. In particular, the contribution of the “free” states of
OH groups would not change qualitatively the results reported.
For the sake of definiteness let us consider the potential minima located
between the two neighboring oxygen ions, ii and iv positions (see Fig. 2).
Fig. 2. Four equivalent
positions of the type ii for
protons in R-OH center.
Then the Hamiltonian H of OH centers (for the 4-fold symmetry axe Z) in the space
of wave functions respondent to the irreducible representations (A1+B1+ E) of the
group C4V may be written as follows
(1)
H=
A1
Ex
Ey
B1
A1
t
h3 + V3exz  pEx
h4 + V4eyz  pEy
h1 +V1(exx  eyy)
Ex
h3 + V3exz  pEx
h1 +V1(exx  eyy)
h2 + V2exy
h3 + V3exz  pEx
Ey
h4 + V4eyz  pEy
h2 + V2exy
h1  V1(exxeyy)
h4 V4eyz + pEy
B1
h1 +V1(exx  eyy)
h3 + V3exz  pEx
h4 V4eyz + pEy
t
Here t is the splitting parameter for the OH center due to the proton tunneling
between adiabatic potential minima near the single oxygen ion; Vi are the interaction
constants of the OH center with strains e; p is the dipole moment of the OH when
proton is localized in one of the potential minima; Ex and Ey are x and y components
of the electric field intensity, and hi are the components of many-dimensional crystal
fields which transform under symmetry transformations as proper components of the
strain tensor.
3 - 42
Note that the Hamiltonian (1) is quite general, and also allows considering the
properties of DDC for the A-substituted perovskites, Fig. 1b, if the appropriate low
symmetry field, caused by a substitution in the cation sublattice A, is specified.
DDC CONTRIBUTION TO THE CRYSTAL PROPERTIES IN THE ABSENCE
OF RANDOM CRYSTAL FIELDS
One can consider possible orientations of R3+-OH groups in the crystal as
equiprobable. Then, the energy spectrum of DDC in the absence of random crystal
fields can be written as follows 1
E(A1)= t , E(B1)= t , E(Ex)= E(Ey)= 0 .
(2)
It is seen that due to the tunneling effect the 4-fold orientational degeneration is
removed and several low lying excited states appear in the energy spectrum of DDC.
As a result, the contribution of the DDC to the heat capacity С, elastic moduli С11,
С12, С66 and the dielectric susceptibility  is as follows:
2
 
1
C  N 0 k B  t
2
 k BT

 
 cosh 2  t

 2k BT
C11  2C12  
 V2
1
N0  1
 k BT
3

 V 2
1

C 66  
N 0  2

24
 k BT

 ,


 cosh 2   t
 2k T

 B


 cosh 2   t
 2k T

 B

  2k BT
  
  t
 V32,4

  4
 t


 
p2
 xx   yy   zz  23 N 0
tanh t
t
 2k BT

 
 tanh t
 2k BT


 tanh  t
 2k T

 B


 ,


 ,

(3)

 .

where kB is the Boltzmann constant and N0 is the number of DDC (N0 in our case is
about the hydrogen content). The typical temperature dependences of the calculated
physical quantities (3) are shown in the Figs. 3 - 6.
Fig. 3. Temperature dependences of the Fig. 4. Temperature dependences of the
heat capacity С, in relative units
dielectric susceptibility , in relative
units.
The structure of the lowest energy states of such R3+-OH center is analogous to that
observed for Jahn-Teller ions Tb3+ in crystals with zircon structure (TbVO(4),
TbAsO(4)) [8].
1
3 - 43
Fig. 5. Temperature dependences of the Fig. 6. Temperature dependences of the
elastic modulus С11, in relative units.
elastic modulus С66 (in relative units) at
different V3/V2 ratios: V3/V2 = 0.05(1),
0.5(2), 1(3), 2(4).
It is seen that the DDC contribution can be large and drastically change the lowtemperature (kBT < t or kBT ~ t) properties of the considered systems.
THE EFFECT OF RANDOM CRYSTAL FIELDS
These low-temperature properties may change even more radically when allowing for
the effect of the random crystal fields on the DDC states [9]. It’s due to the possibility
of bringing close and even crossing of the main singlet and the lower of the splitted
doublet DDC state under the influence of these fields. Let’s illustrate it for the heat
capacity С, elastic modulus С66 and dielectric susceptibility . In order to be
short we’ll restrict ourselves here to the case of the random fields of the h2 type. Then
we obtain the following expressions for calculated values:
2
2



 




2
  h  2  dh


 
N 0 k B   t   
(h   t )
(h   t )

   
C 
,
 
  exp   
  
 (h   t )  
 (h   t )   
2   k BT 







cosh
 cosh 2k T  

2k B T   
B








  
N0
3 
p2
 
 h  t
1
tanh
 
 2k BT
   h   t

 h  t
1
 
tanh
 h  t
 2k BT
  h  2  dh

 exp   
,

     
 V 2 
 2  cosh  2  h   t   cosh  2  h   t
 2k T 
 2k T
 2k BT 
B 
B






N0

 h  t 
 h  t  
C 66  
 
 tanh



tanh
24    
 2k T 
2k BT  

B 


2

 2V3,4 

h  t
h  t









( 4)

 

 

 

  h  2  dh

.
 exp   
     





The normal distribution with dispersion  has been used here for random crystal
fields. The typical temperature dependences of С, С66 and  are shown in
Fig. 7-9.
3 - 44
Fig. 7. Temperature dependences of C:
/= 0(1), 0.5(2), 1(3), 5(4).
Fig. 8. Temperature dependences of :
/ = 0.05(1), 0.5(2), 1(3), 5(4).
Fig. 9. Temperature dependences of С66:
/ = 0(1), 0.5(2), 1(3), 4(4). The ratio
(V3,4/V2)2 = 0.5 was used.
It is seen in the Figs 7-9 that the effect of random crystal fields results in the
distinctive qualitative change in the thermodynamic parameters behavior at the
temperatures kBT  . In this case in the temperature region kBT   /t for the С,
С66 and  we obtain:
  
C  N 0 k B exp  t
  



2
3/ 2
 k B T 2
,

3
 
  
p2
   N 0
exp  t

  
2
    4

,
 ln 
  k B T  3 
2
    2 
V22
 V3,4 
 
t

 ln 
C 66   N 0
exp 
  1  2
k T


 B
 V2 
    



(5)

 1

 6 

.
Thus at low temperatures the DDC contribution to the heat capacity varies
linearly with temperature and is reciprocally proportional to the concentration of the
random fields sources. The maximum slope in the C(T) dependence occurs at
 /t.=2. The susceptibility is featured by the weak, logarithmic, increase with the
temperature lowering in the region kBT   2/t.. The behavior of the elastic modulus
С66 in the low-temperature region is the same (as for ) if V2/ V3,4 1, and if
V2/ V3,41 the modulus С66 is nearly independent of T.
At the temperatures kBT >  the obtained expressions for С, С66 and 
result in the behavior of these parameters qualitatively close to that given by the
expressions (3).
3 - 45
H/D/T ISOTOPE EFFECTS FOR THE DDC-CAUSED PROPERTIES
Finally, it is notable that the H/D/T isotope effects for the properties under
study can be pronounced enough. The isotope effects studies are of particular interest
for proton conducting oxides due to their possible impact on the understanding of the
state and dynamics of protons. The main references to the previous isotope studies
and some new predictions for the abnormal behavior of thermodynamic isotope effect
in proton conducting oxides can be found in [10].
For the properties considered above, H/D/T isotope effects are caused by the
strong dependence of the tunnel splitting parameter t on the mass of tunneling
particle (H, D or T). For example, with making use of the quasiclassical
approximation for the parameter t in the symmetric double-well potential U(x) (see,
eg [11]), the ratio of the splittings for protons and D or T ions can be written as
follows:
it H
t 

 
mi
mH
2 a
 

exp  A
 1 J  , J 
 2m H (U ( x)  E0 )dx ,
mi
m
h




H


a


(6)
where a and –a are the boundaries of the region of sub-barrier motion, i denotes H, D
or T, E0 is the ground state energy in a potential well, and the factor A (resulting from
the different E0 values for H, D and T) slightly affects the results in our case. The
estimations for the characteristic parameters for the oxides considered (barrier
height ~ 0.2 eV, E0 ~ 0.05 eV and 2a ~ (0.5 to 0.7)x10-8cm) give the typical J values
about 4.5 to 7 for protons. Thus, one can expect significant isotope effects at low
temperatures.
The example of such isotope effect for the heat capacity can be seen in the
Fig. 10.
Fig. 10. The temperature dependences of
H/D isotope effects for the DDC
contributions to the heat capacity at
negligible random crystal fields.
tH /tD = 10(curve 1); 20 (curve 2).
Fig. 11. The temperature dependences of
the DDC contributions to the heat
capacity for the noticeable magnitude of
random crystal field:
/tH = 0.25, tH /tD = 5
It is important to note, that the substitution of protons by deuterium or tritium
ions can change the behavior of thermodynamic values not only quantitatively, but
also qualitatively (because the ratio between the tunneling parameter and the
dispersion of random crystal fields also changes with this substitution). For example,
the exponential temperature dependence of the heat capacity, which is expected for
3 - 46
protons at low temperatures and tH, can change into the linear one for D and T,
given the condition  tD,T (Fig. 11).
CONCLUSIONS
The contributions of the proton-associated orientationally degenerate dipole
centers, DDC, to dielectric susceptibility, heat capacity and elastic constants have
been calculated. It is shown that DDCs significantly change the properties of oxides.
In particular it has been shown that the DDC contribution to the heat capacity may be
significant at low temperatures and exhibits unusual temperature dependence: it has a
Shotky anomaly and may linearly vary with T decrease. The temperature dependence
of the DDC contribution to the elastic constants and dielectric susceptibility may also
be unusual: in some cases it may increase as lnT at low temperatures. Isotope effects
H/D/T for the properties under study have been also considered and it was shown that
isotopic substitution of protons can significantly change the DDC contribution
magnitudes at appropriate conditions.
The work was supported by the Russian Foundation for Basic Research (grant
04-03-32377) and by the Program for Basic Research “Hydrogen Energy” (N 26) of
Russian Academy of Sciences.
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