Study of intrinsic point defects in oxides of the perovskite family: II. Experiment

Article abstract of DOI:10.1088/0953-8984/10/36/012

The electrical conductivity of ATiO₃ and A¹⁺B⁵⁺O₃ perovskite-type oxides converted by vacuum annealing into states with weak and strong compensation between donors and acceptors has been measured. It was shown t

Full text of DOI:10.1088/0953-8984/10/36/012

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J. Phys.: Condens. Matter 10 (1998) 8015–8032. Printed in the UK
PII: S0953-8984(98)89571-4
Study of intrinsic point defects in oxides of the perovskite
family: II. Experiment
I P Raevski, S M Maksimov, A V Fisenko, S A Prosandeyev, I A Osipenko
and P F Tarasenko
Department of Physics, Rostov State University, 5 Zorge Street, 344090, Rostov on Don, Russia
Received 26 November 1997, in final form 14 April 1998
Abstract. The electrical conductivity of ATiO3 and A1
+
B
5+
O3 perovskite-type oxides
converted by vacuum annealing into states with weak and strong compensation between donors
and acceptors has been measured. It was shown that the reduced samples had both shallow
and comparatively deep levels which were attributed to the neutral and singly-charged oxygen
vacancy, respectively. The experimentally determined activation energies are consistent with
the results of the theoretical computation reported in Part I.
1. Introduction
The theoretical treatment carried out in the first part of the present work [1] showed
that, in oxides of the perovskite family (OPFs), the lowest energy level of the neutral
oxygen vacancy, VO, is very shallow; it is several hundredths of an electronvolt. The
singly-charged vacancy has a deeper level; its energy is several tenths of an electronvolt.
Activation energies of the order of hundredths of an electronvolt, that can be attributed
to shallow levels of the neutral VO, were observed in several heavily reduced perovskites
[
2–7]. However, experimental data concerning the activation energy of the singly-charged
VO are contradictory. The range of the observed values is spread over the interval from
.3 to 1.7 eV, even for the most studied OPFs [2–4]. This discrepancy is most likely to be
0
a consequence of the fact that, in OPFs, the VO concentration and, hence, the degree
of the compensation of the donor and acceptor centres (as well as the donor/acceptor
concentration ratio) may vary significantly in the course of the experiment, especially in
◦
the high temperature region (above 500–600 C). Note that the major role of oxygen in the
reaction between the gaseous phase and the solid is supported by the partial vapour pressure
values obtained in [9–18]. For ATiO3 perovskites (A = Ba, Sr, Ca), at temperatures below
approximately 1200 K, the equilibrium vapour pressures of the A-cations and Ti appear to
be much lower than the oxygen pressure.
One of the ways of getting reproducible experimental curves is to keep the concentration
of point defects constant. To achieve this we proposed a special sample treatment and a
special measuring procedure. To use our mode one needs to apply the usual equipment
for electric measurements. The main idea for the approach is that a sample that is reduced
at a high temperature cannot be oxidized in vacuum at lower temperatures because of the
absence of the desired oxygen in a vacuum. Further reduction of the sample cannot occur
as well because decreasing the temperature can only lead to a decrease of the equilibrium
concentration of VO but not to an increase.
0
953-8984/98/368015+18$19.50 ꢀc 1998 IOP Publishing Ltd
8015

8016
I P Raevski et al
At definite pressure and annealing temperature, Ta, there exists a definite equilibrium
concentration of VO. We emphasize that this concentration can be thought to be in
equilibrium if Ta is high enough to allow the diffusion processes to end in a few hours
during which one makes the thermal treatments of the sample. When further decreasing
the temperature in vacuum the concentration of VO cannot change because of the above-
mentioned circumstances and, because of this, it remains constant. Hence one may reduce
a sample at a high temperature and, further, measure the conductivity at lower temperatures
in vacuum. In this case the concentration of the main point defects, VO, would be a constant
and the experimental curves should be reproducible. We will show here how it is possible
to evaluate the energies of the trap levels connected with VO.
It should be noted that in our procedure much attention has been concentrated on
the measurement time of each of the points of the temperature–conductivity plot. Indeed
measuring the points is accompanied by some restoration processes. These processes are
of two types. The first is connected with the relaxation of the electron subsystem and local
lattice distortion near point defects (e.g., VO). The second can stem from changing the
concentration of the point defects. It is obvious that the first process is much faster in
comparison with the second. Indeed, at 900 K, SrTiO3 and BaTiO3 ceramics require (1–
5
3
)×10 s for full equilibration [11] whereas, at the same temperature, the period of time for
2
the conductivity measurement does not exceed 10 s after an equilibrium VO concentration
has been achieved at this or a higher temperature. The former figure is the result of a
careful experimental investigation of oxygen diffusion in the perovskite-type lattice. The
latter figure is the result of our experience. It is based on the evidence that, after the
above-mentioned period of time, conductivity does not change within the experimental
error (usually 3–4%). As the temperature lowers the difference between the measuring and
defect equilibrium restoration times increases dramatically. While, for example, at 600 K,
3
the former does not exceed 10 s, the latter (estimated using the oxygen diffusion coefficient
value obtained by the extrapolation of data from [10] to T = 600 K) is of the order of
11
10
s. This estimation is consistent with the conclusion of [9] that, in the temperature
range below approximately 600 K, the changes of charge carrier concentrations in titanates
are only the result of changes in the degree of the ionization of defects but not the result
of changes in the total concentrations of the defects.
If full equilibration was achieved at each of the measuring points then the slope of the
logarithm of conductivity against reverse temperature would correspond to the sum of the
electron trapping energy, Wt , formation energy of VO, Wf , and the activation energy of mo-
bility, Wµ. It will be shown later, that in the OPFs studied, Wµ can be neglected as compared
to Wt . As in the course of the experiment the concentration of VO does not change, the en-
ergy of the formation of VO should not be taken into account. If one took it into account then
the total activation energy of carriers in the conduction band would be of the order of a few
electronvolts (the known values of Wf are almost the same for all the perovskites under con-
sideration, namely 5.4–6.0 eV for SrTiO3 [16, 17], BaTiO3 [9, 18] and KTaO3 [19]). In fact
we observed only small activation energies of the order of a few tenths of an electronvolt.
This indirectly supports our assumption that the time required for the restoration of the equi-
librium concentration of VO is much longer than the measuring time under the experimental
conditions used. However, we should emphasize that the measuring time is longer than
the electron and local lattice relaxation time. Additional support for this assumption gives
evidence of the depletion of the donor’s levels in our experiment. It is obvious that if the con-
centration of the point defects was not a constant then the depletion region would not be seen.
It should be mentioned that, for the present investigation, only the donor/acceptor
compensation ratio but not the absolute values of either the donor or acceptor concentration

Intrinsic point defects in oxides of the perovskite family: II
8017
are of importance. So the initial defect structure of a specimen as well as the non-controllable
impurity concentration are of minor significance. We used polycrystalline samples of OPFs
because they were much easier to prepare and the kinetics of defect equilibrium restoration
in ceramics is usually faster than in a single crystal [11].
A few words should be said regarding the use of expressions for the conductivity
treatment having kT and 2kT in the Boltzmann distribution function. It is well known [8]
that, in semiconductors having both the donor and acceptor states, the following relations
between the donor and acceptor concentrations can be realized:
1
2
.
.
2Nd < Na
0 < (2Nd − Na) < Nd
2
2
.1. 0 < (2Nd − Na) ꢁ Nd
.2. 0 < (2Nd − Na) ≈ Nd
3.
Nd < (2Nd − Na)
where, in our case, Nd is the VO concentration while Na is the concentration of singly-
charged (for the sake of definiteness) acceptors.
In the first case electric conductivity is p-type, while in the second and third cases it is
n-type. In the second case, the first (deep) VO level is partially occupied while the second
(shallow) level is empty. In the third case the latter is already partially occupied. It should
be noted that there are two extremely different situations in the second case (2.1 and 2.2).
The first one corresponds to the strong compensation of donors and acceptors while the
second situation deals with the weak compensation.
It is well known that, when the strong compensation occurs, the electron concentration
in the conduction bands is given by the following expression
ꢀ
ꢁ
W1
n ∼ exp −
(1)
kT
where W1 is the thermal ionization potential of a point defect. The other dependence is
valid when the compensation is weak
ꢀ
ꢁ
W1
n ∼ exp −
.
(2)
2
kT
Thereby, while going from the case 2.1 to 2.2, the effective ionization potential decreases
twice. In the following it will be shown that all these dependences can be observed
experimentally in OPFs if one uses the special treatment procedure.
In the present work we study the temperature dependence of the electric conductivity
σ, in several OPFs including the classic objects of theoretical calculations such as SrTiO3
and KTaO3. The polycrystalline samples were converted into states with strong and weak
degrees of the donor/acceptor compensation ratio by means of the reduction and oxidation
treatment. In either of these states the activation energy of the donor level corresponding
to the singly-charged VO was determined from the temperature dependence of σ with high
degree of reliability.
The structure of the paper is as follows. The second section is devoted to a
characterization of the samples and experimental methods in use. The results of the
experiments are represented in section 3. This section is divided into three subsections:
experiments on titanates, a theoretical treatment of the data for titanates and experiments
on A1
+
5+
B O3. Then we discuss the results obtained in the light of the model proposed in
Part I [1].

8018
I P Raevski et al
2. Preparation and characterization of the samples
The ceramic ABO3 samples were prepared by using high-purity materials. The calculated
quantities of the A-element carbonate and the B-element oxide were mixed thoroughly in
an agate mortar in the presence of ethyl alcohol. Then the synthesis was carried out at 1100
◦
◦
1+ 5+
to 1200 C for titanates and at 750 to 850 C for A B O3 compounds. The synthesis was
performed in two 4 h long stages. After each stage the product of the reaction was crushed
and mixed in an agate mortar as described above.
The ceramic samples were pressed in the form of disks of 5 to 15 mm in diameter at
◦
1
00 MPa. The sintering was carried out at 1300–1400 C. Pellets were placed on a Pt foil.
The obtained ceramics had a density of about 93–95% of the theoretical density. The x-ray
analysis revealed the absence of non-perovskite phases.
Rectangular plates (typically 4 mm × 4 mm × 1 mm) were cut from the pellets for the
electric conductivity measurements. Electrodes were applied to the large edges by firing
the Pt paste or by rubbing the In–Ga alloy.
◦
The thermo-emf measurements were carried out in air at temperatures above 400–500 C
when the sample’s resistance became low enough. Thermo emf was measured by an
◦
electrometer at a temperature difference between the opposite edges of 20–30 C. To
eliminate the effect of thermostimulated current, prior to measurements, the samples were
heated up to the maximum temperature of the measurements and the emf was measured
in the course of the consequent cooling. The temperature at each measuring point was
◦
stabilized within ±0.5 C accuracy and the thermo-emf signal was detected repeatedly three
to five times at different signs of the temperature gradient until stable steady-state values
were obtained.
The results of thermo-emf measurements (figure 1) show that, for the ATiO3 samples
◦
above 500 C, the Seebeck coefficient α is positive and approximately linearly decreases
with temperature. In A1
the α(T ) dependence is more complex. In the high-temperature region, all the A
+
B
5+
O3 compounds, α is substantially smaller than in titanates and
1+
5+
B O3
Figure 1. The temperature dependences of Seebeck coefficient α, for the investigated OPFs.
◦
◦
1
2
. NaNbO3 reduced in vacuum at 1000 C for 2 h and then oxidized in air at 800 C for 2 h.
. Na0.95NbO2.75 ceramic obtained by the sintering of non-stoichiometric mixture of components.

Intrinsic point defects in oxides of the perovskite family: II
8019
samples had p-type conductivity. The decreasing temperature led to a change of the type
of conductivity, from p to n.
1+
5+
O3 compounds, as compared to ones in the ATiO3
Relatively low values of α in A
B
oxides, are likely to be caused by a higher degree of the donor/acceptor compensation in
1
+
5+
B O3. One of the reasons for such an explanation may be the easier formation of VA
A
in A1
+
B
5+
O3. We will return to this question at the end of this section in connection with
the effect of the reduction temperature on the conductivity of OPFs.
Our results of thermo-emf measurements are consistent with numerous data from the
literature (see, e.g., [2, 12–14, 20]). All the references reported that nominally pure OPFs
contain an excess of acceptors with respect to donors. The origin of these acceptor states
is a topic for discussion [2, 20]. Nevertheless it is common to change the degree of the
donor/acceptor compensation by changing the oxygen vacancy concentration. To do this
−
1
we used the annealing in vacuum with a residual pressure of about 10 Pa.
The samples were placed in a horizontal vacuum ceramic tube. This tube was heated in
◦
−1
a furnace to the annealing temperature, Ta, at a rate of 500 C h . The annealing time was
–4 h for A1
+
B
5+
O3 and 1–2 h for ATiO3. The increase of this time did not significantly
2
change the σ values at room temperature. In order to fix the defect structure formed as
a result of annealing, the samples were quenched by pulling the ceramic capsule with the
specimens, connected by a piece of steel, out of the hot zone of the furnace with the help of
a magnet. After cooling the samples were provided with electrodes by rubbing with In–Ga
alloy and σ values were measured at room temperature. For the sake of reproducibility the
investigation of the σ(Ta) dependence for each oxide was performed by using samples cut
from the same pellet.
The results obtained are shown in figure 2. It can be seen that, for titanates, conductivity
increases with Ta although, for BaTiO3, the σ(Ta) dependence has a tendency to saturate.
The rapidly increasing portions of the σ(Ta) curves for SrTiO3 and CaTiO3 are shifted to
Figure 2. The electric conductivity measured at room temperature as a function of the vacuum
annealing temperature.

8020
I P Raevski et al
high temperatures, as compared to BaTiO3, hence the saturation portions are not observed
within the investigated temperature interval.
For A1
+
B
5+
O3 oxides the character of the σ(Ta) dependence is extremely different
from that for titanates. At low annealing temperatures σ increases with Ta, after that
◦
it reaches the maximum at Ta ≈ 800 C and then decreases. The decrease of σ above
◦
8
00 C is likely to be caused by the formation of not only VO but also VA. The latter
form acceptor levels in the band gap. This assumption is consistent with the data of
chemical, thermogravimetric and x-ray analysis showing that, during vacuum annealing,
◦
◦
KNbO3 and NaNbO3 lose only oxygen in the region 400–800 C, while above 800 C
intensive evaporation of alkali metal oxide takes place [21–23]. As a result a defect solid
solution of the [A1−x(VA)x]Nb[O3−x/2(V0)x/2] type is formed. In the case of NaNbO3 this
solid solution has the perovskite structure up to x ≈ 0.1. This explanation is consistent with
changes in the colour of A1
+
B
5+
O3 specimens which resulted from vacuum annealing. The
sintered in air ceramics of ATiO3 and A1
+
5+
B O3 are light-yellow in colour. As the vacuum
annealing temperature is increased the colour of the titanates changes to grey, dark-grey
1
+
5+
B O3 specimens changed with Ta in a similar way
and finally to black. The colour of A
◦
up to Ta ≈ 800 C. Above this temperature the colour of the specimens became less dark
and a yellowish tint appeared. The intensity of this tint increased with Ta. After annealing
◦
in vacuum at Ta = 1100 C the NaNbO3 ceramics had a yellowish-brown colour.
◦
Figure 1 shows the thermo-emf of NaNbO3 annealed in vacuum at 1000 C for 2 h
◦
and then oxidized in air at 800 C for 2 h (curve 7). This sample has the stable
p-type conductivity in the temperature region below 700 C where air-sintered nominally
◦
stoichiometric ceramics have n-type conductivity. Moreover the Seebeck coefficient of the
specimen annealed in vacuum proved to be substantially larger as compared to the non-
annealed one. This result shows that the effective acceptor concentration increases after
◦
vacuum annealing at Ta > 800 C, that is in accordance with the proposed scheme of the
defect formation.
The above explanation is also consistent with the thermo-emf data for the non-
stoichiometric NaNbO3 ceramics in which the Na2O deficiency was achieved by using the
non-stoichiometric mixture of Na2CO3 and Nb2O5 taken in the molar ratio Na2O/Nb2O5 =
0.95 [22]. In this specimen the transition from the n- to p-type of conductivity took place at
significantly lower temperature than in nominally stoichiometric NaNbO3 ceramics (figure 1,
curve 8).
The decrease of the A1
+
B
5+
O3 conductivity caused by the vacuum treatment at high
Ta may be compensated by appropriate donor doping. Figure 3 shows the effect of vacuum
annealing on the conductivity of Ca-doped NaNbO3 ceramics. Though the increase of Ta
leads to the decrease of the σ values both in the doped and undoped ceramics, σ increases
dramatically with Ca content. This fact supports the assumption that Na2O evaporation
increases the acceptor’s concentration.
3. Investigation of the temperature dependence of the electrical conductivity
3.1. Titanates
The conductivity of the samples with fired platinum electrodes was measured in a weak
−1
electric field (E 6 10 V cm ) with the help of an electrometer. At every measuring point
the linearity of the volt–ampere characteristic and fulfilment of the Ohm low was checked.
To avoid the influence of the thermostimulated current on the result, prior to measurements,
the samples were heated up to the maximum temperature of our study and the temperature

Intrinsic point defects in oxides of the perovskite family: II
8021
◦
Figure 3. The effect of vacuum annealing for 2 h at 1000 and 1100 C on the electric conductivity
of Na1−xCaxNbO3 measured at room temperature.
dependence of σ was measured in the course of cooling. At every measuring point the
◦
temperature was stabilized within ±0.5 C accuracy and steady-state values of the current
were used for the calculation of σ.
After measuring the σ(T ) in air the samples were placed into a vacuum measuring
◦
furnace. The initial heating was usually performed up to 750 C in vacuum with the
−1
residual pressure 10 Pa. The samples were annealed at this temperature for 1 h, then
the temperature dependences of σ were measured in vacuum in the course of cooling and
subsequent heating. The curves obtained during cooling and further heating were found
to be identical, that is the log σ = f (1/T ) dependences were reproducible. This fact
indicates that the ‘frozen’ non-equilibrium concentration of oxygen vacancies does not
change during measurements, i.e. the condition of constant composition is fulfilled. New
stable defect states of the specimens (with higher VO contents) were achieved by increasing
the annealing temperature by 50 K steps. After each of the steps the σ(T ) dependence
was measured again under cooling and subsequent heating. As a result for each sample a
set of log σ = f (1/T ) curves was obtained with different degrees of the donor/acceptor
compensation (figures 4–6).
It should be mentioned that, within the described method, the reproducibility of
log σ = f (1/T ) curves was observed only at temperatures below approximately 900–
◦
9
30 C (full curves in figures 4–6). At higher temperatures the equilibrium between the
sample and ambient is likely to be achieved very quickly and one cannot obtain a non-
varying ‘frozen’ concentration of oxygen vacancies during measurements, i.e. to fulfil the
condition of constant composition. Indeed, using the oxygen diffusion coefficient data from
2
[
10] gives a value of about 10 s for the equilibrium restoration time of the VO concentration
at T = 1200 K which is of the same order as the conductivity measuring time.
The preliminary investigation showed that the increase of the degree of the vacuum
reduction causes a shift of the high-temperature linear portion of the log σ = f (1/T )

8022
I P Raevski et al
Figure 4. log σ = f (1/T ) dependences for SrTiO3 measured in air (1, 5, 6) and in vacuum
◦
(
2–4). 1. As-sintered sample. 2. After vacuum annealing at 750 C for 1 h (strong degree of
−
w1/kT
◦
donor/acceptor compensation σ ∼ e
). 3. After subsequent vacuum annealing at 800 C
−
w1/2kT
for 1 h (weak degree of donor/acceptor compensation σ ∼ e
). 4. After subsequent
◦
vacuum annealing at 1300 C for 3 h (heavily reduced sample). 5. After oxidizing the heavily
reduced sample by heating it in air up to 900 C (weak degree of donor/acceptor compensation
◦
−
w1/2kT
σ ∼ e
). 6. After subsequent oxidation of the same sample by heating it in air up to
◦
−w1/kT
).
7
00 C (strong degree of donor/acceptor compensation σ ∼ e
dependence to lower σ values as compared with the similar dependence measured in air
for as-sintered samples (curves 1 and 2 in figures 4–6). This shift is accompanied by the
increase of the slope of the high-temperature portion of the log σ = f (1/T ) dependence. At
higher degrees of reduction both the σ value and the slope of the high-temperature portion
did not change significantly (curves 2–4 in figures 4–6).
The values of σ and the slope of the high-temperature portion of the log σ = f (1/T )
curves measured in vacuum agree well with the log σm = f (1/T ) dependences obtained
for BaTiO3 and SrTiO3 in [12–14], where σm is the minimum value of the equilibrium
conductivity taken from conductivity–partial oxygen pressure dependence. Thus σm is the
σ value corresponding to the case of the strong donor/acceptor compensation. These results
are consistent with the theoretical predictions [8] that the temperature of the transition
from extrinsic to intrinsic conductivity decreases with the increase of the donor/acceptor
compensation degree. Thus the high-temperature portions of the log σ = f (1/T )
dependences measured in vacuum are likely to correspond to the intrinsic conductivity
and their slopes correspond to one half of the band gap (Eg) value. The Eg values for
titanates, shown in table 1, determined from the data shown in figures 4–6, agree well with
the values obtained from experiments on optical absorption, photoconductivity [2] and from
the log σm = f (1/T ) dependences [12–14].
It is of interest to note that the results obtained show that the state with high degree of the
donor/acceptor compensation is stable enough, though according to theoretical calculations
[24] the condition of the strong compensation is very rigid.

Intrinsic point defects in oxides of the perovskite family: II
8023
Figure 5. log σ = f (1/T ) dependences for BaTiO3 measured in air (1) and in vacuum (2–4).
◦
1
. As-sintered sample. 2. After vacuum annealing at 750 C for 1 h. 3. After subsequent vacuum
◦
−w1/kT
).
annealing at 850 C for 1 h (strong degree of donor/acceptor compensation σ ∼ e
◦
4
. After subsequent vacuum annealing at 950 C for 1 h (weak degree of donor/acceptor
−
w1/kT
compensation σ ∼ e
).
Figure 6. log σ = f (1/T ) dependences for CaTiO3 measured in air (1) and in vacuum (2–4).
◦
1
. As-sintered sample. 2. After vacuum annealing at 700 C for 1 h. 3. After subsequent vacuum
◦
−w1/kT
).
annealing at 750 C for 1 h (strong degree of donor/acceptor compensation σ ∼ e
◦
4
. After subsequent vacuum annealing at 850 C for 1 h (weak degree of donor/acceptor
−
w1/2kT
compensation σ ∼ e
).

8024
I P Raevski et al
Table 1. Experimental band gap Eg values, experimental (1We) and calculated in [1] (1Wc)
values of a singly-charged oxygen vacancy level activation energy and values of polaron energy
1
Ep (from [33]) for the investigated OPFs.
OPFs
Eg (eV)
1We (eV)
1Wc (eV)
1Ep (eV)
BaTiO3
SrTiO3
CaTiO3
NaNbO3
KTaO3
3.2–3.3
3.3–3.4
3.1–3.2
0.5–0.6
0.3–0.4
0.1–0.2
0.5–0.6
0.1–0.2
0.15
0.14
0.14
0.11
0.10
0.35
0.15
0.4
The change of the donor/acceptor compensation degree due to vacuum treatment of the
titanate ceramics (the transition from the strong to weak compensation) is confirmed by the
change of the slope of the low-temperature portion of the log σ = f (1/T ) dependences by
a factor of two (figures 4–6).
For SrTiO3 the effect of oxidation on the slope of the log σ = f (1/T ) dependence
was also studied. The temperature dependences of σ were measured in air at relatively
moderate temperatures where oxidation was not observed. Special attention was paid to the
elucidation of the formation of barrier layers near the electrode surface of the specimen.
In some cases the linearity of the volt–ampere characteristic after oxidation was restored
by applying a short pulse of high voltage (300–600 V). The results obtained are shown
in figure 4. Curve 4 represents the temperature dependence of σ for a heavily reduced
specimen (the initial state). Curve 5 was measured after oxidation of the specimen by
◦
heating it in air up to 900 C and curve 6 was produced after subsequent heating in air
◦
up to 700 C. As a result of oxidation the slope of the low-temperature portion of the
log σ = f (1/T ) dependence increased by a factor of two (figure 4, curves 5 and 6). The
activation energy calculated from the slope of either dependences 3 and 5 by using formula
(2) or dependences 2 and 6 by using expression (1) is practically the same (0.3–0.4 eV) and
agrees well with the calculated value W = 0.2–0.4 eV (see [1]).
3.2. Theoretical treatment of the experimental data for titanates
The experimental results obtained allowed us to evaluate a number of parameters for SrTiO3.
The calculation was performed in the following manner. The slopes of the low-temperature
portions of log σ = f (1/T ) dependences in figure 4 give the activation energy values of
about 0.33 eV. The increase of the log σ = f (1/T ) curve slope can be thought to be a switch
from the case of the weakly compensated donors to the case of the strongly compensated
ones.
The horizontal portions of log σ = f (1/T ) dependences can be considered as a result
of donor depletion. All the carriers other than those compensated by acceptors are thermally
excited to the conduction band. If the region of depletion was observed in a fixed sample
the evaluation of Na and Nd would become available in a simple way.
The VO can accept two electrons. As a consequence the system with the vacancy
concentration Nd has 2Nd additional electrons in the unity volume. On account of the
compensation by acceptors 2Nd − Na electrons are localized on VO at T = 0. If the
inequality
Na < (2Nd − Na) < Nd
(3)
is valid then the lower level of VO with a depth of several tenths of an eV is filled. If the

Intrinsic point defects in oxides of the perovskite family: II
8025
(4)
following is valid
2Nd − Na) > Nd
(
then the upper level of VO becomes filled. The depth of this level is about 0.01 eV. So one
could expect that the thermal activation of this level takes place at low temperature.
In such a case (the strongly reduced sample) the carrier concentration is a constant
and equals Nd − Na. Consequently, the ability to observe the temperature dependence
of conductivity controlled by the second ionization energy is restricted by the very rigid
restriction (4). Considering the region of depletion (figure 4, curves 4, 5 and 6) one can
obtain the electron’s concentration, n, in the conduction band by the expression
σ = |e|µnn
(5)
where e is the electron charge and µn is the electron mobility. Regarding the temperature
dependence of the electron mobility it should be noted that, according to experimental data
[25], the activation energy of the electron mobility is very small: Wµ is of 0.023 eV in
n-type barium titanate and is likely to be even less in n-type strontium titanate. Due to this
fact we neglected the temperature dependence in our computation. Assuming µn to equal
−
4
2
−1 −1
5
× 10 m V
s
[15] we found
2
2
1
2
1
9
−3
−3
−3
n(4) = 2.6 × 10
n(5) = 6.0 × 10
n(6) = 5.4 × 10
m
m
m
(6)
(7)
(8)
.
The numbers in the brackets correspond to the curve numbers in figure 4. The data obtained
can be expressed via the defect’s concentrations as follows
n(4) = Nd − Na.
Here the higher level of VO is partially filled at T = 0.)
n(5) = 2Nd − Na ≈ Nd.
(9)
(
(10)
(
The lower level of VO is weakly compensated while the higher level is empty at T = 0.)
With the use of (3) to (9) we evaluated the acceptor concentration and found it to equal
21
−3
Na ≈ 6 × 10
m .
To justify this evaluation we performed the computation of the log σ = f (1/T ) dependence
over a wide range of temperature and compared the results with those obtained with the
experiment.
The calculation was carried out in the framework of the statistics of carriers on the
multiple-charged centres [8]. The following is devoted to a brief description of the theory.
It was supposed that
X
σ = |e|
µini
(11)
i
where µi is the mobility for carriers of the type i and ni is the concentration of these
carriers.
The concentration of electrons in the conduction band and holes in the valence band
are given by
Z
n =
p =
ρc(E)f (E, T ) dE
(12)
(13)
conduction
band
Z
ρv(E)fp(E, T ) dE
valence
band

8026
I P Raevski et al
where ρc and ρv are the density of states in the bands; f and fp are the Fermi–Dirac
distributions for the electrons and holes
ꢂ
ꢀ
ꢁꢃ−
1
E − F
f (E, T ) = 1 + exp
(14)
(15)
kT
fp = 1 − f
F is the electrochemical potential and k is the Boltzmann constant. The F value can be
found from the electroneutrality condition at T = constant
n(F, T ) + nt (F, T ) − p(F, T ) − pt (F, T ) = 0
(16)
where n and p are the electron and hole concentrations in the conduction and valence bands
while nt and pt are the concentrations of the electrons and holes bound to the donor and
acceptor centres, respectively. We assume the donor centres (the oxygen vacancy) to have
both the shallow and comparatively deep levels. Because of this circumstance we are to
use the grand canonical distribution together with the mean field theory. Within this theory
the probability for j electrons to be on a multiple charged centre is given by the expression
(
j)
jF − Em
(
m
j)
f
= A exp
(17)
kT
(
j)
where Em is the energy of the mth quantum state of the centre. The constant A can be
calculated from the normalization rule
m
(j)
X X
1
jF − Em
=
exp
(18)
A
kT
j=0
m
where M is the maximal number of electrons that the defect can accept. If the centre with
j electrons has excited states, it is convenient to write
(
m
j)
(j)
= E + εjm
E
(m = 1, 2, . . .).
(19)
(
j)
Here E is the energy of the centre with j electrons in the ground state, εjm is the energy of
the mth excited state (with respect to the ground state). The energy level of the point-defect,
Ej , is related to the many-electron energy, E , as follows
(j)
(
j)
E
= E1 + E2 + E3 + · · · + Ej .
(20)
The probability of finding the centre in a state with j electrons is defined by the equation
X
(
j)
(j)
fm
f
=
.
(21)
m
Let us introduce the degree of the degeneration, βjm, for various excited states of the centre
with j electrons. After that, from (17)–(19), one has
j
gj exp((jF − E )/kT )
(
j)
f
= P
(22)
M
(
j)
gj exp((jF − E )/kT )
j=0
X
m
where
ꢄ
ꢅ
εjm
gj = βj +
βjm exp −
.
(23)
kT
Here βj is the degeneration of the ground state.
For the multiple-charged donors having M electrons in the electroneutral state the
positive charge can be defined as
M−1
X
(
j)
pt = Nd
(M − j)f
.
(24)
j=1

Intrinsic point defects in oxides of the perovskite family: II
8027
Similarly, for the multiple-charged acceptors which can accept M holes to become
electroneutral the negative charge is
M
X
(
j)
nt = Na
jf
(25)
j=1
where Nd and Na are the donor and acceptor concentrations, respectively.
Equations (12) to (25) are sufficient for determining F and the carrier concentrations.
After obtaining these quantities one can calculate σ(T ) on the basis of equation (11). Below
we represent the parameters and results of the particular calculation for SrTiO3.
The density of states in the conduction band of OPFs is well known [26]
6
2
|E|
K (k)
2
0
ρc(E) =
(26)
π D
where K(k) is the full elliptic integral of the first kind,
p
2
2
2
4
D − E + 1
0
2
K (k) = K( 1 − k )
k =
.
2
4
D
Here D and 1 are the (pdπ)-type hopping integral and the half width of the forbidden gap,
respectively.
From (16) and (12) one can obtain the following evaluation for Nc
3
1kT
3
Nc =
.
2
πa D
Concerning Nv, it can only be mentioned that it must be somewhat higher then Nc due to
the existence of non-bonding states at the top of the valence band. Knowing the number of
the non-bonding states (six) one can write the inequality
Nv < 6/a3
(27)
where a is the lattice constant. In the computation we used the following assumptions
−
4
2
−1 −1
µ = µ = 5 × 10 m V
s
n
p
ρv(E) = 2ρc(−E)
Ma = 1
Md = 2
ga = 1
gd = 12
2
a
d
d
E = −1.0 eV
E = 1.35 eV
E = 1.65 eV
D = −1.12 eV
1
1
1
= 1.70 eV.
Here the upper index refers to the acceptor (a) and donor (d) levels, respectively. With
2
2
−3
m (curve 4, figure 4);
these assumptions taken into account we have Nd = 3.2 × 10
21
−3
21
−3
Nd = 6.0 × 10
m
(curve 5) and Nd = 3.0 × 10
m (curve 6).
All acceptors were supposed to be singly-charged and non-degenerated and their energies
were taken rather arbitrarily. Only the total acceptor concentration, Na, had an impact on
the result (we only studied the case of the n-type conductivity which is true for the reduced
samples of strontium titanate). The Na and Nd obtained were found to satisfy equations (4)
to (10). The energy parameters for VO in SrTiO3 were taken in the same manner as in [1]
and in [24]. It must be noted that, in the references cited, the energy levels of the VO were
calculated with respect to the boundary of the conduction band. In the present notation it
d
i
corresponds to the quantity 1−E , i.e. in our case 0.35 eV and 0.05 eV. At the parameters
described above the solutions of (12) to (16) faithfully reproduce the experimental curves 4,
5
and 6 in figure 4. So the use of the theoretical data regarding the electronic structure of
VO allows the defect concentrations in perovskites to be evaluated rather exactly.

8028
I P Raevski et al
.3. A1
+
5+
B
O3 compounds
3
◦
1+ 5+
B O3 compounds
As already mentioned, during vacuum annealing at Ta > 800 C the A
lose not only oxygen but also alkali metal ions. As the result of this fact the procedure
for obtaining the set of the log σ = f (1/T ) curves used in the case of titanates is not
applicable to the A1
+
B
5+
O3 oxides. Therefore in order to determine the activation energy
of the VO levels in the A1
+
B
5+
O3 compounds we investigated the σ(T ) dependences of
ceramic samples annealed in vacuum at temperatures not exceeding the temperature of the
maximum in the σ(Ta) dependence. Measurements were carried out in air using electrodes
from rubbed In–Ga alloy.
According to differential thermal analysis data [22, 23], the oxidation of reduced
◦
−1
NaNbO3 ceramics in air begins at 200–250 C at a heating rate of 15 K min , i.e. at
substantially lower temperatures than in titanates. One may expect that at lower heating rate
oxidation begins even at lower temperatures. In order to elucidate the influence of oxidation
◦
on the result, the upper border of the measured interval was limited by 100–150 C. Prior
to first measurement a sample was heated up to the maximal temperature. The σ(T ) curves
were reproducible both for heating and cooling the sample.
Figure 7 shows the log σ = f (1/T ) dependences for KTaO3 ceramics. The slope of
the log σ = f (1/T ) dependence decreases by a factor of two with the increase of Ta from
◦
7
00 to 800 C (figure 7 curves 1 and 2).
The slope of the log σ = f (1/T ) dependence for the most conductive vacuum-reduced
sample of nominally pure NaNbO3 was practically the same as in the less conductive samples
(figure 8, curve 1). We supposed that under the experimental conditions employed it is
impossible to obtain a concentration of donor centres in NaNbO3 large enough to achieve the
state with the weak degree of the donor/acceptor compensation. Therefore we investigated
the log σ = f (1/T ) dependences of Ca-doped NaNbO3 ceramics Na1−xCaxNbO3 reduced
Figure 7. log σ = f (1/T ) dependences for KTaO3 measured in air after vacuum annealing for
◦ ◦
h at 700 C and 800 C.
4

Intrinsic point defects in oxides of the perovskite family: II
8029
Figure 8. log σ = f (1/T ) dependences for NaNbO3 (1) and Na1−xCaxNbO3 (2–5) measured
◦
◦
in air after vacuum annealing for 4 h at 800 C (1) and 1000 C for 2 h (2–5). 2. x = 0.01;
3
. x = 0.0125; 4. x = 0.0175; 5. x = 0.035.
◦
in vacuum at 1000 C for 2 h. For such ceramics the σ values at room temperature increase
dramatically with x (figure 3). For the samples with 0 < x 6 0.015 the increase of x is
not accompanied by changes of the log σ = f (1/T ) slope (figure 8, curves 2 and 3). The
activation energy value calculated using expression (1) is about 0.55 eV, i.e. is the same
as in the non-doped ceramics. The slope of the log σ = f (1/T ) dependence for a sample
with x = 0.0175 is smaller by a factor of two as compared to the samples with the lower
x values. Such behaviour is likely to be caused by the transition to the state with the weak
degree of the donor/acceptor compensation. For the samples with x > 0.030 the activation
energy is found to be of about a few hundredths of an eV (figure 8, curve 5).
4. Discussion
The results obtained are consistent with the data from other investigators and show that
there are two types of donor levels in reduced OPFs. The shallow one has the activation
energy of several hundredths of an eV while for the deeper one it is of about several tenths
of an eV. Based on the results of computations published in the first part of this work [1]
we attribute the shallow level to the neutral VO and the deeper one to the singly-charged
VO capturing an electron.
As we have mentioned the activation energies of several hundredths of an eV have
been reported for heavily reduced perovskites in many works [2–7, 27]. Besides, there
are several works in which the activation energies observed were similar to ours. For
◦
◦
example, according to [28], BaTiO3 ceramics reduced in vacuum at 700 C and 800 C
had conductivity activation energies of about 0.5 eV (deduced using expression (1)). For
◦
the same ceramics reduced at 900 C the log σ = f (1/T ) slope was approximately half
that which is likely to be caused by a transition between the states with the strong and

8030
I P Raevski et al
weak degrees of the donor/acceptor compensation ratio. The activation energies of oxygen
vacancy levels determined in [29] from the temperature dependence of the secondary
electron emission current were of the order of 0.35–0.50 eV for both SrTiO3 and BaTiO3
vacuum-reduced single crystals. Investigations of thermostimulated currents spectra and
volt–ampere characteristics in Ba0.4Sr0.6TiO3 thin films revealed the trap energies distributed
at 0.4–0.5 eV which were ascribed to oxygen vacancies [30]. In KTaO3 crystals with
different degrees of reduction, the investigation of thermostimulated currents revealed
activation energies, the values 0.08 eV and 0.13–0.18 eV of which were supposed to
correspond to the neutral and singly-charged VO, respectively [31]. An activation energy
of 0.55 eV was detected in NaNbO3 crystals by the same method [32].
In the present study samples of several perovskites were prepared so that they proved to
be in the states of both the strong and weak degree of the donor/acceptor compensation. A
study of the temperature dependence of conductivity has given us the possibility to determine
the energy of the lowest electronic level in the singly-charged VO.
Table 1 collects the experimental values of the forbidden gap, Eg, the activation energies
of the donor centres obtained with the use of the experimental treatment of the samples
described in the present work (1We = 1W1) and the activation energies calculated in
[1] without taking into account the polaronic effect (1Wc).The last column represents the
polaron energies for several OPFs obtained experimentally in [33].
It is worth noting that for the given OPFs the same 1We values were obtained
irrespectively of the prehistory and initial defect structure (including impurities) of the
samples. For example in the case of SrTiO3 both the reduction of the as-sintered dielectric
sample having p-type conductivity (curves 2 and 3 in figure 4) and oxidation of the heavily-
reduced sample having n-type conductivity (curves 5 and 6 in figure 4) gave the same 1We
values. The same was true for nominally pure NaNbO3 annealed in vacuum at temperatures
where predominantly VO were formed (curve 1 in figure 8) and for donor-doped NaNbO3
◦
(
Na1−xCaxNbO3, 0 < x 6 0.015) annealed in vacuum at 1000 C where both VO and
VA were formed, the latter being acceptors (curves 2 and 3 in figure 8). These facts
are in agreement with usual theory of semiconductors [8] according to which only the
donor/acceptor ratio (not the absolute values of either the donor or acceptor concentration)
has an impact on the temperature dependence of conductivity.
Experimentally determined 1We includes both Wn and Wµ, the activation energies of
carrier concentration and mobility, respectively. In turn, Wn includes both Wt , the desired
trapping energy of electron at VO, and Wf , the formation energy of VO (activation energy
for carrier formation by reduction). We have already mentioned that, under the experimental
conditions used, the condition of constant composition was fulfilled and thus the influence
of Wf was circumvented. Besides the reproducibility of the log σ = f (1/T ) dependences
in the course of heating and cooling there are some additional arguments supporting the
fulfilment of the condition of constant composition in our experiments. The portions of
the log σ = f (1/T ) dependences with the desired slopes are observed at temperatures
substantially lower than 600 K where, in titanates, changes of the total concentrations of the
defects may be neglected [9]. Wide horizontal portions of log σ = f (1/T ) dependences
observed in some SrTiO3 and BaTiO3 samples are likely to be a result of donor depletion.
The existence of donor depletion regions on the log σ = f (1/T ) curves may also serve
as evidence of the condition of constant composition fulfilment. The main reason for
the constant composition is that the samples were, at first, thermally treated up to an
equilibrium VO concentration and, then, the measurements were made at lower temperatures
in vacuum. This procedure does not allow the VO concentration to change at cooling
because of the circumstances mentioned in section 1. In other words, within the framework

Intrinsic point defects in oxides of the perovskite family: II
8031
of our experiment, the period of time needed for changing the VO concentration is much
higher than the measuring time while the measuring time is much higher than the electron
and local lattice relaxation time. The fact that the curves obtained at cooling and heating
coincided each with the other in some temperature interval supports the assumption that the
VO concentration did not change not only at cooling but also at heating.
Let us try to estimate the Wµ contribution into the measured 1We values. In the OPFs
studied the current carriers are likely to be small polarons [7, 25, 33–35]. According to
existing models [34–36] at low temperatures small polarons move via a band mechanism
and their mobility temperature dependence is non activated. At temperatures exceeding
2
D/2, where 2D is the Debye temperature, the transport mechanism changes to thermally
activated hopping. The mobility activation energy in this case equals one half of the
polaron energy. For NaNbO3 2D is about 1400 K [34], so at temperatures below 450 K,
where the conductivity of NaNbO3 was measured in the present study (see figure 8), the
carrier mobility in this OPF seems to be non activated. The extensive studies of charge
transport in BaTiO3 and some other OPFs have shown [25, 27, 36] that in the samples
with n-type conductivity the activation energy of the electron mobility is very small in
comparison with the polaron energy extracted from optical measurements. The nature of
this discrepancy is complex and was under discussion in [25, 27, 36]. Thus one may expect
that the Wµ values for the reduced samples of all the OPFs investigated are much lower
than the experimentally measured conductivity activation energies 1We which are therefore
close to the desired electron activation energy from the singly-charged oxygen vacancy level
(assuming the the condition of constant composition is fulfilled and the formation energy
of the VO contribution to Wn may be neglected). However, when calculating the electronic
structure of VO one should take into account the polaronic effect due to the well localized
nature of an electron at a singly-charged VO. This was a major point of Part I [1]. It can
be seen that the difference between the values given in the third (1We) and fourth (1Wc)
columns of table 1 is nicely fitted by the values in the last column (1Ep). This confirms the
idea presented in Part I, that the interaction between the lattice polarization and the electron
trapped by the vacancy, leads to a lowering of the energy levels of the singly-charged VO.
Indeed, the polaronic contribution vanishes if the electronic state of VO is well delocalized.
Besides the electric measurements already mentioned, one can find some supporting
data in optical measurements. For example in [37] it was shown that the reduction of cubic
KTaO3 leads to the appearance of local dipole distortions that can be associated with the
dipole state of VO. The same tendency is observed in the luminescence spectra.
5. Conclusions
The experimental study of OPFs carried out in the present work confirms the results obtained
in Part I [1] theoretically. A special treatment of the as-sintered samples having the p-type
of conductivity, i.e. containing an excess of acceptors, was employed. The oxygen vacancy
concentration was varied by vacuum annealing so as to obtain the samples in the state with
either a strong or weak degree of the donor/acceptor compensation.
The present experimental investigation has shown that the neutral VO has a shallow level
of several hundredths of an electronvolt. Owing to this the neutral VO has to be ionized at
not very high temperature, often even at room temperature. The singly-charged VO already
has a comparatively deep level with an energy of several tenths of an electronvolt.
The results of the investigation of the influence of the vacuum annealing temperature on
the conductivity of the A1
+
5+
B O3 samples are consistent with the results of the calculation
carried out in Part I of this work which predicted the acceptor character of the energy levels

8032
I P Raevski et al
formed in OPFs by the A-sited vacancies. Note that this point of view does not contradict the
results of the computation performed in the framework of the shell model [20]. According
to this computation holes are comparatively weakly bound to the Ba-vacancy in BaTiO3.
The same result was obtained in [1].
In conclusion we want to stress that the main object of our study was VO. As the result
of this we were generally interested in the n-type conductivity samples. Obtaining reliable
+
information concerning VO was possible because these point defects are main in the n-type
OPFs. The question regarding the nature of the main point defects in the p-type OPFs (or
main carriers) is not so clear. New studies are required to answer this question.
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