Positive temperature coefficient of resistivity for BaTiO3-based materials

Ceramics International 17 (1991) 227 241

Positive Temperature Coefficient of Resistivity for BaTiO3-based Materials J. N o w o t n y Australian Nuclear Science & Technology Organisation, Advanced Materials Program, Lucas Heights Research Laboratories, Menai, NSW 2234, Australia

& M . Rekas Technical University of Clausthal, Institute of General Metallurgy, 3392 Clausthal-Zellerfeld, FRG (Received 3 September 1990; accepted 30 October 1990)

Abstract: In this paper electrical properties of BaTiO 3 and its solid solutions are reported at room and moderate temperatures in the range 300-500 K. The electrical properties are mainly analyzed in terms of the positive temperature coefficient of resistivity (the PTC effect). Experimental data on the PTC effect have been reviewed and analyzed against several theoretical models. The Heywang-Jonker model and its further modifications are discussed in more detail. The models involve different explanations of the nature of the potential barrier which is formed across grain boundaries of polycrystalline BaTiO3. The grain boundary chemistry of BaTiO3 is considered, involving phenomena such as segregation, oxygen chemisorption, oxygen incorporation into the grain boundary layer and the formation of surface states. The effects of these processes on the temperature characteristics of resistivity are considered, with particular emphasis on the preparative procedure and postpreparative treatments such as annealing under gaseous atmospheres of controlled composition and cooling at different rates.

1 INTRODUCTION

thermistors, varistors as well as sensors and actuators. It has been documented that the PTC effect is closely related to the specific properties of grain boundaries and of the grain boundary region. On the other hand, there has been an accumulation of both experimental and theoretical material indicating that properties of grain boundaries of nonstoichiometric oxide materials in general and BaTiO3 in, particular are essentially different from those of the bulk phase. In contrast to the bulk properties of BaTiO3, which have been relatively well explained in terms of defect chemistry,1-7 little is known about

Barium metatitanate (BaTiO3) exhibits interesting electrical properties. One of the most spectacular properties is its positive temperature coefficient of resistance, termed the P T C R or the PTC effect. This effect involves a substantial nonlinear change of resistivity with temperature and is displayed around the Curie temperature (T~). The PTC effect and other electrical properties of BaTiO 3 and its solid solutions have generated an intensive interest in this compound. The applications of BaTiO3 include its use for capacitors, 227

Ceramics International 0272-8842/91/$03.50 © 1991 Elsevier Science Publishers Ltd, England. Printed in Great Britain

228

J. Nowotny, M. R e k a s

the properties of the grain boundary region, its structure, chemical composition and related defect structure. Accordingly the PTC effect, which has been widely described in the literature, is still not well understood. The purpose of the present paper is to overview electrical properties of BaTiO3 and its solid solutions, with emphasis on the PTC effect. Accordingly the reported electrical properties will be limited to the temperature range in which the PTC effect is displayed, i.e. in the temperature range up to 500 K. In this range the defect structure in BaTiO3 is in a quenched state and the concentration of defects corresponds to defect equilibria at elevated temperatures. Then changes in gas composition in the surrounding atmosphere do not produce any changes in the oxide defect structure. This defect structure and the related electrical properties are essentially determined by the conditions of the previous equilibrium state and the procedure of quenching or cooling. Electrical properties of oxide materials in the quenched state are not well defined, especially with regard to effects of the grain boundary region. Therefore, despite significant literature on the PTC effect, there have been an accumulation of many conflicting results and models. In the present paper the most important models will be reviewed and analyzed against available experimental data. 2 GENERAL CHARACTERIZATION BaTiO3

OF

According to the phase diagram of the BaO-TiO 2 system s (Fig. 1) BaTiO 3 may tolerate a certain excess of Ti, but this excess is substantially reduced with 1860°

1800

LIQUID

2100

Bo2TiO4 * LIQUID HEX. BoTiO3 • LIQUID \

o LU r,"

1600

~

;,

v

.HEX BoTiO3. "

Bo T O,.

2 '

n¢ LIJ

HEX

/,4----~CUB .,I- EX

//tCU

t/aon03\

BoT,O

' 1,oo

cuL io \

1400 I.IJ

B°2TiO4"

LIO~UID

CUB. BoTiO3

~UB

12oo

(~ I

"~

BoT'205

~ 1BoT'307

.--~ I P--

BoTi30 7 I

1.

-05

A

BaTi03

Io

-10 '

'E .._u _ 1.5 .C:

= Bo/Ti :0954 ] LONG. = ' "x~ a Ba/Ti =09?9 ~ BLUMENTHAL ae ~ l -2.0 -= Bo/Ti =0954 J (1971) ~ ~1o ~ O l SEUTER [1974) ~ .g ~ t~ I/ OANIELS (1976) EROR,SHYTH (1978) xa -90 -2.5 SMYTH.CHAN (1981) " ~ PRISEDSKII.TRETIAKOV (1982) x ~

"x "x • • x

• x

~"d oaAo I -11

Fig. 2.

i I -7 -3 log P02[Po2in Po ]

,~ r~ll°~

I 1

DO

" I 5

Log cr as a function of log po 2 for u n d o p e d BaTiO 3 at 1270 K. 2-4't4-16

decreasing temperature. 9 According to Sharma et al. 9 the solubility of TiO 2 in BaTiO 3 is below 0-1%. BaTiO3 exhibits different crystallographic structures such as hexagonal (>1730K), regular (3931730K), tetragonal (273-393K), rhombohedral (173-273K) and monoclinic (<173K). The Curie temperature (To), corresponding to the ferroelectric/ paraelectric transition, for undoped BaTiO 3 is 384 K (111°C). The T¢ for high purity BaTiO 3 is slightly higher (403 K)} ° The T¢ also depends on the grain size and porosity} mE Undoped BaTiO3 is a good insulator, with its resistivity varying between 109 and 10Z2~cm at room temperature. Haayman e t al. 13 have shown that doping with aliovalent ions results in a variation in electronic properties of BaTiO 3 within a substantial range. Doping with 0.1 at.% of La leads to a decrease in resistivity down to l0 (2 cm. The colour of undoped BaTiO 3 depends on its nonstoichiometry. Oxidized BaTiO 3 is white while a reduced specimen assumes a dark colour. The effect is well reproducible. Predominant defects in undoped BaTiO3 are oxygen vacancies within a wide range of oxygen partial pressure in the n-type regime (low p(O2) ). BaTiO3 becomes p-type after annealing at high p(O2)2 -4,14- 16 (Fig. 2). Then the Schottky-type disorder becomes predominant. 7 Low mobility of cation vacancies results in severe difficulties in the equilibration of BaTiO3 with the gas phase and, therefore, in the experimental determination of their concentration.

[

CUB.BaTi0 3 •

30 50 TiO 2 CONCENTRATION [ MOL. % ]

Fig.

~"

0 1BoTi205

:~O/IU3 • * o

~L /

~ ,.s

..;,,.

a

o,. =" o. x

3 S P E C I A L P R O P E R T I E S OF BaTiO3 I

70

Phase d i a g r a m of the T i O 2 - B a O system, a

Besides the PTC effect, BaTiOa exhibits several other interesting properties. One of them involves the

Positive

temperature

Dy-DOPED

BaTi

Sb-DOPED

BaTi03

Gd-DOPED

BaTiO,

I I I I , I I 0.2

Fig. 3.

229

coefJicient of resistivity for BaTiO,

0.4

0.6

0.8

Log R as a function of composition for donor-doped BaTiO, involving Dy, ” Sb18 and Gd19 additions.

effect of doping on resistivity. For most nonstoichiometric oxides doping with aliovalent ions results in a monotonous change in electrical properties as a function of dopant concentration within the solubility range. In contrast, the effect of doping of BaTiO, involves an unmonotonous change of electrical conductivity. As seen in Figs 3 and 4,17-20 donor doping, up to about 0*2at.%, results in a sudden decrease in room temperature electrical conductivity, while above this concentration the reverse effect is observed, resulting in restoration of the conductivity to the initial value or close to it. One should emphasize that this effect corresponds to the solubility range of the dopants in BaTiO,. This effect is characteristic for ceramic materials and may be considered in terms of inhomogeneous distributions of the dopants within the BaTiO, grains. Experimental data for single crystals would be very helpful in developing an understanding of this phenomenon. 1 La-DOPED

At high temperatures (> 1000 K) and high ~(0,) BaTiO, assumes a p-type semiconductivity. After quenching or cooling, the p-type material never exhibits p-type conductivity at room temperature, even if the original material is doped with acceptors. Instead it assumes insulating properties. This feature of BaTiO, has a strong impact on its technological application as a component of dielectric devices.21 -25 The PTC effect is certainly one of the most important and interesting features of BaTiO, and BaTiO,-based materials. This effect is closely related to the ferroelectric/paraelectric transition which is accompanied by the tetragonal/cubic phase transition. The PTC effect can be displayed only by donordoped BaTiO, exhibiting the n-type conductivity. Furthermore, the PTC effect is a specific property of polycrystalline ceramic materials. It may be expected to be enhanced by an increase in the concentration of grain boundaries. There is, however, a certain critical limit of grain size below which the PTC effect is not displayed. This effect will be considered below in more detail. Before considering different models and theoretical approaches to explain the PTC effect we will briefly discuss several characteristic features of the effect itself. 4 EFFECT OF MATERIALS ON THE PTC EFFECT

According to available literature the PTC effect in BaTiO, involves the following general features: (1)

1

BaTI

(2)

01 05 La CONCENTRATION Fig. 4.

Resistivity

IO [AT%1

as a function of composition BaTiO,.”

PROPERTIES

for La-doped

(3)

The effect is a property of polycrystalline material. As seen from Fig. 5,26 single crystals do not exhibit the PTC effect at a11.27.28 Consequently it appears that the presence of grain boundaries is a condition for a display of the PTC effect. The PTC effect has been observed only for ntype BaTiO, containing a donor addition ranging between O-1 and 0.5 at.%. Although it has been observed that a small amount of acceptor, in addition to the donor, may result in an increase in the PTC effect,29-31 the bulk phase of the specimen must have an n-type conductivity. It has been observed that Mn, which forms the deepest acceptor centers within the grain boundary structure of BaTiO,, exhibits the strongest influence on the PTC effect.32 The PTC effect is very sensitive to preparative

230

J. N o w o t n y , 1. -O.5 AT. % Sm (POLYCRYSTALINE) 2. -O.S AT. % Sm (SINGLE CRYSTAL)

1o6! E

105

(J X

10 4 >I'--

_> (/I 01 ILl IZ

10 3

(4)

10 2

101 '

2

1

I

I

300

400

500

TEMPERATURE [Ki Fig. 5.

Resistivity as a function of temperature for Sin-doped BaTiO3 .26 6

[ f ~

(5)

conditions involving high temperature treatment and subsequent quenching or cooling procedures. The effect of the temperature of annealing on the PTC effect is illustrated in Fig. 6. 33 The main parameters which have an effect on the composition of grain boundaries and resulting PTC effect involve the oxygen partial pressure and the temperature of the equilibrium state as well as the rate of cooling. The PTC effect is closely related to the Curie temperature. The Tc can be shifted depending on composition. As seen from Fig. 7 additions of Sr and Pb result in the shift of both the Tc and the PTC effect towards lower and higher temperature ranges, respectively. 34 The resistivity versus temperature characteristics within the PTC effect are very sensitive to composition. Figures 8(a) and (b) illustrate the effect of FezO 3 and CuO, respectively, on the resistance versus tem-

1623K

10 ? . 0 ~

-

5

///-"X2

4 t--

Fe203-DOP.ED

0.015 %

- ;373K

E u

M. Rekas

B°TIO3

rr

--

tE

3

fie

O

0~4 O

2

-o005%j 300 Fig. 6.

400

I

I

I

500

600

350

I

400

450

TEMPERATURE (a)

TEMPERATURE [K] The effect of annealing temperature of BaTiO3 on the extent of the PTC effect.33

!O-1"o~ C u O -

[K]

DOPED

8

B o T i O3

~ .c

70% SrTiO 3

8 -

"--- 0.015 %

o.o

.c_ ntw 4

~o 4 -

II/

Tc

y .pO; 03

2 I

I

[

200

300

400

0%

t'f

__y Tc

/

-J

I 500

T E M P E R A T U R E {K] Fig. 7. Effect of Sr and Pb on the resistivity versus temperature characteristics for n-type BaTiO3 (0'3 at.% of donor).34

0.003 % j ] / _

,

350

,

, 400

Cu-DOPED ,

,

, 450

TEMPERATURE [ K ]

(b) Fig. 8. Effect of acceptor additions on the log R versus temperature characteristics for Sb-doped BaTiO3: (a) Fe203; (b) CHO.29

Positive temperature coefficient of resistivity for BaTiO 3

231

perature relationship for Sb-doped BaTiO3 .29 It can be seen that above certain critical concentrations both CuO and Fe20 3 may even result in the disappearance of the PTC effect. 29

dimensional layer along the grain boundaries of BaTiO3, and this layer exhibits different electrical properties from those of the bulk phase. According to Heywang 38 the BaTiO 3 grain is composed of the bulk and the grain boundary region. Assuming both donor centers and electrons as predominant defects in the bulk phase, the following lattice electroneutrality condition can be considered:

5 THEORETICAL MODELS

Several models have been developed to explain the PTC effect at Tc. The construction of most of the models is based on an assumption that grain boundaries exhibit special properties which differ substantially from those of the bulk material. It seems that the difference is caused by both adsorption and segregation (adsorption from the solid phase). It is not clear which of these two processes has a major contribution into the PTC effect. An early model proposed by Saburi 3s was based on an assumption that the conductivity mechanism changes at Tc. Assuming that the conductivity involves hopping of electrons between'Ti 3+ and Ti 4+, Saburi considered the PTC effect in terms of different electron mobilities in the ferroelectric state on the one hand and in the paraelectric state on the other hand. He argued that hopping in the ferroelectric state is favored by the local electric field. He assumed that in the paraelectric state this field is much weaker, and therefore the retarding effect caused by the lattice oxygen ions predominates. This model, however, does not explain the absence of the PTC effect in single crystals or its disappearance in polycrystalline BaTiO 3 after annealing in a reducing atmosphere. The model proposed by Brahmecha & Sinha 36 has similar disadvantages. This model is based on an assumption that the electron scattering changes at Tc due to special polarization fluctuations. Peria et al. 3v considered the PTC effect in terms of changes in the contact resistance between the grains as a result of strains generated during the phase transition at T¢. Again this model, which does not explain the strong effect of thermal treatment, seems to be unsatisfactory. 5.1 Model of Heywang The model proposed by Heywang 3a with its further modifications has found the most general acceptance in the literature. This model is based on the formation of a potential barrier at grain boundaries. The model of Heywang has been further developed by Jonker 2° and then termed the Heywang-Jonker model. The model assumes the formation of a bi-

[e']b = [D'] = [DLo,]

(1)

where [e']b denotes the concentration of electrons in the bulk and [D'] and [Dtot] denote the concentrations of ionized donors and the total concentration of donors, respectively. Heywang postulates that surface states present at grain boundaries can be considered as electron traps, resulting in the depletion of the grain boundary layer in electrons. The model of Heywang 38 can be derived in terms of material data such as thickness of the grain boundary region, temperature and bulk resistance. The relation between the negative charge located at grain boundaries and the grain boundary thickness can be written: 2b[e']b = [e']s

(2)

where h is the effective thickness of the grain boundary region on both sides of the grain. Subscript 's' corresponds to the grain boundary region. The bulk and the grain boundary concentration terms ([e']b and [e']~) are expressed in cm -a and c m -2, respectively. On the other hand, from Poisson's equation we can write e[e']~ Wo - 8eeo[e,]b

(3)

where Wo is the potential barrier height, e is the elementary charge, and e and eo denote the electric permittivity of the material and of vacuum, respectively. Accordingly ~o - eb2 [e']~ _ eb 2 [D']

2eeo

(4)

2eeo

Both [e']~ and e in eqns (2)-(4) are functions of temperature. Based on the Fermi-Dirac statistics one may determine the parameter [e']~: [e']~ =

Ns 1 + exp [(E v + eW o - E j / k T]

(5)

where EF = k T In { Ne/[e']b }

(6)

N s and Are denote the density of surface states and conduction electrons, respectively, and E s is the

232

J.

Nowotny, M. Rekas

i..._k.2___l GRAIN BOUNDARY I I"

I I I

Bali03

TEHPERATURE [K] m

Fig. 9. The grain boundary band model after Heywang.38 energy of the surface state in relation to the bottom o f the conduction band (Fig. 9). On the other hand, the temperature dependence of e may be determined from the Curie-Weiss law: C e= - T-To

(7)

Assuming that

Fig. I0. Calculated potential barrier as a function of temperature according to eqn (4) at different numbers of surface states

(N~)(after Heywangaa). R above T~and R b is the bulk value of R. For BaTiO 3 this ratio is equal to 104.38 In modern materials the PTC effect may exhibit a change in resistivity by a factor of 10 v or even more. Assuming that z = 1 0 2 cm -1 and b = 10 -5 cm, we derive from eqn (9) that E F -- 0.25 eV

(11)

E s = 0.9 eV

(12)

Therefore

C = 1.2 x 105 K

To = 3 8 0 K

Ne = [T/Ti] = 1"56 x 10 -3

F r o m eqns (2) and (3) we find that

the resistance of 1 cm 2 o f the grain boundary layer is

b R - e[e,]s ~

kT x etp~ exp

(e~o ~ \kT ]

(8)

where p is the mobility of electrons. Then the effective resistance o f the ceramic specimen assumes the form e~Oo where R b is the resistance o f the bulk phase and z is the number of grains along 1 cm of the specimen. At the ferroelectric state (below To) the barrier height is low because of a very high value of the electric permittivity. Then the resistivity o f the grain boundary layer is comparable with the resistivity of the bulk phase. Above Tc the parameter Refr sharply increases as a result o f a substantial decrease in e and a consequent increase in ~bo. The distance between the level of surface states and the Fermi energy level (EF) is close to kT and ~o assumes its m a x i m u m value: e~Oo -~ Es -- EF

(10)

The analysis of eqns (5) and (7)-(9) may be performed based on the experimental ratio o f Rm.x/Rb, where Rm.x denotes the m a x i m u m value of

N s = 1014 cm -2

(13)

Figure 10 illustrates the functional dependence of etp/kTon T f o r several values o f N s determined from eqn (5).

5.2 Model of Jonker Jonker 2° assumed that the electric permittivity in the ferroelectric state is smaller than that assumed by Heywang, as a result of strong electric fields which are formed at grain boundaries. He argued that the resultant real value of e is too low to produce an effective lowering of the surface potential (below To) to negligible values as predicted by the model of Heywang. According to Jonker lowering of the surface potential barrier is explained by assuming compensation o f the surface charge along different directions o f the ferroelectric domains (Weiss domains) at both sides of the grain boundary (Fig. 11). The surface charge f o r m e d at the grain boundary layer is compensated (within 50% of domains as dictated by the nature of the ferroelectric material) by the charge related to the ferroelectric component. The grain boundary polarization vector (PN) normal to the grain boundary is illustrated in Fig. 11. The difference between the two vectors yields

233

Positive temperature coefficient o f resistivity for BaTiO 3 I

i

i

i

Tc ,,-..BaTi03

i0 e

g

10:

", "%

:% •

"10

~

UJ I0~ (.9 Z 10 '~ (D 03 W 10 ~ ,,y

\

%

ee

]

%

•o

%o

o

\ o

o

o2

I ie I

÷ 102

¢'YO

•eel

I

I0 100

Fig. 11.

Ferroelectric domains at grain b o u n d a r y Jonker. 2°

APN = e[e']s

Fig. 12.

(14)

These domains may be considered as a short-cut path resulting in the disappearance of the grain boundary resistance. Of course the resistance across the other 50% of the domains is even higher than before polarization; however, the resistance in the ferroelectric state is determined by the domains forming the short-cuts. According to Jonker 2° the P, component at Tc is equal to 1 8 # C c m -2, which corresponds to 1.3 x 10 ~4 electrons per cm 2. This value is of the same order of magnitude as N~ (corresponding to surface potential) in Heywang's model. 3s The PTC model proposed by Heywang and Jonker has been verified experimentally in several studies. Gersthsen & Hfirdt139 have shown that an oil suspension of TiO 2 ceramic material is mainly deposited on the BaTiO3 surface along the grain boundary regions of high resistivity when an electric field is applied across the specimen. Studies based on the impedance spectroscopy, of Heywang 3s and Peria et al., 37 have also confirmed this model. More direct proof of the formation of the potential barrier across grain boundaries has come from scanning electron microscope and cathode luminescence studies.4O-46 Furthermore, experimental data on the temperature slope of resistance and of the voltagecurrent characteristics also have confirmed t h e model of Heywang and Jonker. Sinclair and West 47'4s have studied both La- and Mn-doped BaTiO3 by the impedance spectroscopy method. Their data are reported in the form of the bulk and grain boundary resistance around the T¢ (Fig. 12). Based on these data they argue that the PTC effect is a bulk property. This statement is in

I 300

I 400

TEMPERATURE [°C]

after

the component responsible for the compensation of the surface charge:

I 200

Resistance as a function of temperature for BaTiO3: (1) grain boundary; (2) bulk phase. 4~'4s

obvious conflict with the model of Heywang and Jonker. The authors provided no explanation of this conflicting view. It appears that the 'bulk resistance' term in the paper of Sinclair and West involves the grain boundary layer as well. 6 V E R I F I C A T I O N OF THE HEYWANG-JONKER MODEL 6.1 Extent of the PTC effect

Despite the fact that several sets of experimental data have confirmed qualitatively the HeywangJonker model there have been many experimental observations which are not consistent with the model. The model predicts that at Tc the resistance changes by four orders of magnitude. In contrast, Heywang & Brauer 49 have observed that the resistance of Sb-doped BaTiO3 changes by seven orders of magnitude. Matsuoka et al. 3° have even reported that for Nb- and Mn-doped BaTiO 3 the PTC effect involves eight orders of magnitude. Miller 5° has verified the Heywang-Jonker model by assuming that the distribution of donors within the grain boundary layer is inhomogeneous, involving several surface states of different E S. These verifications were still not sufficient to explain the experimental data. Further modifications of the model, introduced by Hoffmann e t a / . 5~'52 and Brauer, 53 involve the formation of a grain boundary phase enriched in Ti. This postulation, however, was not confirmed by electron microscopy studies. 42 Generally the extent of the PTC effect depends on the chemical composition of the grain boundary region. Also the formation of segregation-induced low dimensional grain boundary structures seems to play an important role.

234

J. Nowotny, M. Rekas

6.2 Surface states and corresponding electrical potential barrier

I

La- DOPED BaTiO3 (O/+AT%)

J IWORK FUNCTION 50~ HEATINGS • " 2.WORK FUNCTION

It has been assumed that the surface states are formed as a result of broken crystalline periodicity. These states, termed Tamm-Shockley states, have been initially considered for an ideal crystal. One should expect, however, that for a real crystal of nonstoichiometric compounds such as BaTiO 3 the nature of the surface states is more complicated. The first complication involves the effect of segregation-induced concentration gradients across the grain boundary layer. The gradient involves nonstoichiometry and related concentration of intrinsic defects such as oxygen and cation vacancies. Seuter 2 and Daniels & Wernicke 54 argue that the grain boundary layer is enriched in cation vacancies which are formed as a result of interactions between oxygen and the oxide crystal. At moderate and lower temperatures their inward diffusion is impossible because of the kinetic factor (a very low j u m p frequency, resulting in negligible diffusive transport). As shown in several studies, segregation of extrinsic defects also has a strong effect on the grain boundary properties of solids. 55-6° Extrinsic defects involve both intentionally and unintentionally introduced elements. One should expect a close relationship between segregation of lattice defects and the temperature characteristics of resistance around T¢. Unfortunately little is known about this. Regular studies on segregation in BaTiO 3 are required for a better understanding of the electrical properties of the grain boundary region. Chemisorption of gases, especially of oxygen, may be considered as an additional complication resulting in changes in chemical composition along grain boundaries. It has been confirmed experimentally that the PTC effect depends strongly on the oxygen partial pressure. The dependence can be considered assuming the formation of different adsorption acceptor-type states at grain boundaries. 2°'26'61-63 The effect of the oxygen potential gradient resulting in the formation of acceptor states such as oxygen chemisorbed species and cation vacancies within the grain boundary region has been considered in many s t u d i e s , t 7 ' 2 ° ' 2 7 " 6 2 - 6 6 Also chemisorption of F - ions has been reported. 6~ Surface potential studies of Holt, 67 performed by using the Kelvin probe, have shown that the work function versus temperature characteristics are independent of composition and surface pretreatment (Figs 13 and 14). Moreover, cooling and

COOLING ? . ~ 3. RESISTIVITY

8

E c .-.R

I.L

300

400

500

TEMPERATURE [K]

Fig. 13. Work function and log R for La-doped BaTiO3 as a

function of temperature.67 heating result in hysteresis of these characteristics (Fig. 13). Holt argues that the increase of the work function during heating corresponds to the formation of a potential barrier in the model of Heywang. This barrier is considered in terms of strong oxygen chemisorption within the ferroelectric state (below To) and a sudden desorption above Tc as a result of the decrease in the electric permittivity. However, the decrease in work function above T¢ cannot be explained within Heywang's model. According to Holt 67 several properties of the external surface, related to their effect on the work function, are entirely different from properties of the grain boundary region. Holt argues that the difference involves the nature of the barrier, which for grain boundaries is related to segregation of lattice defects and which for external surfaces is caused by chemisorption of gases. Seuter 2 has reported that the mobility of electron carriers is 0.2 c m 2 V-1 s-1. This value is higher by two to three orders of magnitude from that assumed in the model of Heywang-Jonker (0.01-0"001 cm 2 V -1 s-l). Thus the concentration of electrons is lower. This indicates that the donors are predomi5~)

TAL

2 /,.t, 3OO

tOO

5OO

TEMPERATURE [ K]

Fig. 14. Work function as a function o f temperature for BaTiO 3 single crystal. 67

235

Positive temperature coefficient of resistivity for BaTiO 3

resulting potential barrier, but results in the disappearance of anion vacancies according to the following reaction:

BaTi03

~ G 15

1/202 + 2e' + Vo" ~-~ Oo

Instead interfacial oxidation of n-type BaTiO3 results in the formation of both acceptor-type adsorbed centers along grain boundaries

= 10

02 ~ 02 + h" ~ 2 0 - + 2h" ~ 2 0 2 - + 4h"

5

350

t~00

I+50

TEMPERATURE

Fig. 15.

500 [K]

nantly compensated by the acceptors such as Ba and Ti vacancies. The concentration of these vacancies within the grain boundary layer is much higher than that in the bulk phase. As deterrrtined from the activation energy of the ionization obtained by Seuter the parameter E S is equal to 2.25 eV. Assuming also that Ne =

2 x 1021 c m - 3

The model of Seuter was later developed by Daniels & Wernicke, 54 who assumed that the potential barrier related to the PTC effect is generated during the cooling of the specimen and the accompanying formation of defects along the grain boundaries. The presence of the surface states and their possible effect on the PTC effect was ignored in their model, which is based on the following defect reactions: 3 zero ~

(15)

instead of 1022 considered previously, Seuter has proposed revised parameters describing the PTC effect such as ~,, Reff and b. From Fig. 15 the increase in [e']~ results in an increase of the PTC effect with the maximum shifted towards lower temperatures. At high density of grain boundaries (103 cm- ~) the change in resistivity is ten orders of magnitude. Thus we have 1"4V

(17)

and cation vacancies within the grain boundary region 3/202 ~__ 3 O o + V6'a+ V4'['+6h" (18)

Log R as a function of temperaturc for BaTiO3 at diffcrcnt valucs of surfacc-statc dcnsity. 2

~//max =

(16)

b = 0-6 #m

At higher values of the density of states the barrier ¢'o cannot be neglected in the lower temperature range and, therefore, the resistance value increases below Tc. Furthermore, the increase in the density of states (N~) results in an increase in the thickness of the depleted layer (b). For small grains b may be substantially higher than the grain size. Accordingly the PTC effect is not observed for fine-grained (<1 pm) ceramics. These modifications introduced by Seuter result in the verified model of Heywang and Jonker, which explains a large PTC effect. Negative surface charge at grain boundaries and the resulting PTC effect is not displayed either for undoped or acceptor-doped BaTiO3. In the case of undoped BaTiO3 the process of grain boundary oxidation, usually taking place during cooling, does not result in the generation ofacceptor-type centers, required for the formation of electron traps and

Vo + VB.

O o ~-- Vo + 1/202

(19) (20)

For donor-doped BaTiO3 prepared in air the following three regimes of the lattice electroneutrality conditions can be considered: (a) (b)

(c)

[D'] = [e'] T > 1770 K [D'] = 2[ V~'a] + [e'] 1490 < T < 1770 K [-D'] = 2[ V~'a] T < 1490 K

(21) (22) (23)

The electrical properties of BaTiO 3 at room temperature depend on both T and p(O2) corresponding to equilibrium conditions as well as depending upon the quenching procedure. The effect of equilibrium conditions on the resistance of donordoped BaTiO3 at room temperature is shown in Fig. 16. Let us consider the effect of both rapid and slow quenching on properties of BaTiO3 within different electroneutrality regimes (a)-(c). An ideal rapid quenching of the specimen from equilibrium ((b) and (c)) results in low and high resistivity, respectively. The difference essentially results in the concentration of Ba vacancies being lower in the regime (b). On the other hand, when cooling is slow enough then equilibrium conditions may be reached at different stages of cooling. In consequence, changes in electrical properties may follow the dependence in Fig. 16 from equilibrium down to a certain critical

J. Nowotny, M. Rekas

236 poz = i0 Pa

~"

5

.c: ,,.y.

10

i_l

""

to'l=2lVaol

~=[O']~a

[~'--" "-~

2 [VBal'le'l

Fig. 17. Representation of BaTiO a ceramic material involving local electroneutrality conditions for the bulk and the grain boundary layer.5a

-9° 15 Poz=2l~Paj~

1/,oo

~o

z= 10S~

ldOO

TEHPERATURE OF EQUILIBRATION [K] Fig. 16. Log R (at room temperature) as a function of equilibrium temperature at different oxygen partial pressure in equilibrium for n-type BaTiO3.53

the grain boundary thickness. At very slow cooling rates the thickness may be much larger than the Debye length: ( 2 e q J ) '/2

b = L kT-

(24)

where

eeokT ,~1/2 temperature, below which equilibrium cannot be reached throughout the entire grain because of the kinetic factor. The equilibrium kinetic is rate controlled by the diffusion of Ba vacancies from grain boundaries into the bulk. Assuming that the critical temperature corresponds to 1490K, then below this temperature the bulk defect structure remains within regime (b) and the grain boundary region assumes the equilibrium represented by the condition (c). Accordingly slow cooling results in a low resistivity in the bulk while the grain boundary layer assumes insulating properties. Inhomogeneous material obtained in this way is shown in Fig. 17, 53 along with local electroneutrality conditions for the bulk phase and the grain boundary region. Then the barrier height is determined by the energy level of Ba vacancies (~1 eV). In ferroelectric material the barrier corresponds to very high resistivity. On the other hand, in the ferroelectric state the barrier is compensated by the electrical polarization, and consequently it has no effect on the resistivity as was assumed in the model of Heywang and Jonker. 7 EFFECT OF PREPARATION AND COMPOSITION ON THE POTENTIAL BARRIER

Z 1 Effect of cooling

From the above considerations we see that the thickness of the grain boundary layer involving concentration gradients of defects depends on the rate of cooling from the equilibrium state down to the temperature of measurements. Essentially a decrease in the cooling rate results in an increase of

L = \ e 2[D.]eff]

(2S)

and [O']eff = [ O ' ] - 2[ V~'a]

(26)

Figure 18 illustrates both grain boundary barrier and accompanying space charge within the grain boundary region at different grain boundary layer thicknesses. Figure 18(a) corresponds to Heywang's model. The increase in grain boundary thickness leads to an increase in the resistivity in the ferroelectric state. This is due to the formation of an intergranular layer of highly increased resistivity in both ferroelectric (cold resistance) and paraelectric states. The formation of the intergranular insulating layer results in very high apparent electric permittivity (eelf ~ 105) characteristic for a so-called boundary layer (BL-type) capacitor. 68-71 This material does not exhibit the PTC effect. Accordingly

o) j~-V L
p

b) ~

L=b

~A

-C|/~ x

L>b

.J

x-

p

A

d)/~_~_~ L>>b

V N

x

P

Fig. 18. Illustration of the potential barrier ~bo and the charge density (p) distribution across grain boundary, s3

237

Positive temperature coefficient of resistivity for BaTiO 3

Nb-DOPEO BaTi03

f

BaTi03

.... E j.

/'J -

I

/I

.=_ rv. t~

=. o

1~ OXYGEN .~/ i/ 3

TEHPERATURE [ K] Fig. 19. LogR as a function of temperature for Nb-doped BaTiO3 (0-19at.%) equilibrated under different oxygen partial pressure. Cooling rates: (1) 210 K/h; (2) 90 K/h; (3) 210 K/h; (4) 90 K/h. Grain size: (1) 30/~m;(2) 42 #m; (3) 30/~m;(4) 140Bm.66

successive increases in the grain boundary thickness correspond to a transition from the PTC thermistor (Fig. 18(a)) to the intergranular capacitor (Fig. 18(b)). This model has been confirmed experimentally by Kahn. 66 As seen from Fig. 19, the resistivity in both ferroelectric and paraelectric states substantially increases with a decrease in the rate of cooling. The effect of the grain size involves the parameter z. Its increase leads to an increase in the amount of the insulating material. Z2 Effect of acceptors

As has been argued above, the PTC effect may be displayed for an n-type material. However, besides the intentional doping with donors most of the materials involve a certain concentration of acceptors which have been introduced unintentionally. Based on the principles of defect chemistry one may expect that predominant donor defects will be compensated by acceptors. The compensation may be effective if the segregation driving force of the acceptors is substantially higher than that of the donors. As is well known the segregation driving force for various ions may substantially differ, and consequently the grain boundary region may be enriched with an element which is present in the bulk phase at negligible concentration, if its segregation driving is high enough. Even a low concentration ofacceptors may have a

~

lt,.00

1600

101=03AI% p~ = 2.104Pa T.[As= 300K 1800

TEHPERATURE OF EQUILIBRATION (K]

Fig. 20. Calculated resistivity at 300K as a function of quenching temperature at different concentrations of acceptors.53 significant effect on the PTC effect. 29-31 Heywang & Brauer 49 assume that acceptor-type impurities are mainly localized along grain boundaries. The effects of acceptors have been considered by Daniels & Wernicke, 54 who assumed a homogeneous distribution within the grain. Then the compensation effects between donors and acceptors were considered assuming the effective (noncompensated) concentration of donors. Based on these assumptions they determined the effect of acceptors on the resistivity versus temperature characteristics (Fig. 20). Based on the computer simulation technique, Lewis e t a / . 72'73 determined the difference in the defect energy formation between the surface and the bulk. The difference can be considered as the segregation energy. The calculations indicate that oxygen, Ti and Ba vacancies segregate to the surface of BaTiO 3. Also acceptors such as Mg~i, Ca~i and Mn4~ have a tendency to segregate. 74'75 In contrast, donors such as La~, will be depleted in the surface layer. Unfortunately the amount of experimental material concerning segregation in metal oxides in general and in BaTiO 3 in particular is still far below requirements from the viewpoint of technological application. The scanning transmission electron microscopy studies of Chiang & Peng 76 have shown that grain boundaries of S r Z i O 3 a r e enriched with Ti. It has also been documented that Nb-doped BaTiO3 is enriched with N b . 77 Z3 Effect of nonstoichiometry

The studies of Tsai-Fa Lin et al. vs have shown that the PTC effect is closely related to the oxide nonstoichiometry, which has a strong impact on the microstructure. They observed that nonstoichiometric Ba-rich specimens exhibit a small grain microstructure which displays a low PTC effect. In

J. Nowotny, M. Rekas

238

p-

: i

I I,=AL

TETR/~..~Ig~ STRUCTURE

'

Fig. 21. The tetragonality gradient across a BaTiO3 grain. 79

contrast, the Ti-rich samples have a large grain microstructure with much better PTC characteristics. The dependence of the PTC effect on the grain size may be considered involving the structural gradient across the BaTiO 3 grain, v9 It has been reported that below Tc the tetragonal structure is stable only in the crystalline bulk phase while the outer part of the crystal forms a low dimensional layer of the cubic structure, resulting in the formation of the tetragonality gradient between the two structures (Fig. 21). Concordantly, the PTC effect cannot be displayed below a certain critical grain size which corresponds to the thickness of the surface cubic structure. 8 CONCLUSIONS There is still much disagreement as to a model for the PTC effect. The disagreement involves composition of the grain boundary layer and the interpretation of the nature of the potential barrier which is formed across this layer. The main difficulty is caused by the fact that the purity of the BaTiO 3 material exhibiting the PTC effect is not well defined. There is general agreement that the PTC effect appears only for a donor-doped polycrystalline BaTiO 3. After quenching or cooling, the material still remains n-type in the bulk of grains, which are covered with an insulating shell. The difference in electrical properties between the bulk and the grain boundary layer is produced by chemical composition and involves enrichment in acceptor-type defects. Oxygen, Ba and Ti vacancies as well as

DISTANCE FROM THE GRAIN BOUNDARY

Fig. 22.

Schematic representation of the donor-doped BaTiO3 grain within the potential barrier O.

surface states, acceptor-type impurities and adsorbed oxygen species have been considered here. Above the Curie temperature a potential barrier develops between the bulk and the insulating grain boundary layer (Fig. 22). The barrier results in a high resistivity for charge transfer across the grain boundary. Below Tc the barrier significantly decreases as a result of either a high permittivity (Heywang) or the compensation effect produced by electrical polarization (Jonker). The extent of the PTC effect (Fig. 23) depends on chemical composition and resulting properties of the bulk phase and the grain boundary shell. The bulk phase is relatively well defined in terms of the defect chemistry models corresponding to the thermodynamic equilibrium. On the other hand, there are many conflicting reports on the properties of the grain boundary shell and its effect on the charge transport behavior between the grains. The chemical HOT RESISTIVITY (LOW e,HIGH ~O )

o_ E u --

e-

tv- 4 tY

o

-

2 -

-J

Z

[

-,,J.....,,/iCOLDRESISTIVITY (HI(3H ¢,LOW ~10) I 1 I I 350 Tc 450 550

TEMPERATURE IK } Fig. 23.

Schematic representation of the PTC effect.

Positive temperature coefficient o f resistivity f o r B a T i O 3

239

composition of this shell depends on many uncontrolled processes such as segregation of bulk defects and interaction between the gas phase (mainly oxygen) and grains during cooling. Because of the kinetic reason the gas/solid interactions are limited to the grain boundary shell. The outstanding variety of models considered in the literature correspond mainly with the properties of this shell. Unfortunately there is a lack of experimental data about properties about grain boundaries and of the grain boundary region of BaTiO3. Accumulation of such data is an urgent necessity from the point of view of preparation of materials with enhanced properties. Da & Umeya 8° have recently reported the methodology of the determination oflocal dielectric properties of the grain boundary region while the PTC effect is determined. Their model enables analysis of the grain boundary resistance and capacitance data. As seen from these considerations, an understanding of the properties of grain boundaries is of fundamental importance for correct interpretation of the PTC effect. Further grain boundary studies of BaTiO3-based materials are therefore very necessary.

barium metatitanate. Sci. Bull. Acad. Mining Metall., No. 1094, Krakow (1986) 1-149. 6. CHOI, G. M. & TULLER, H. L., Defect structure and electrical properties of single-crystal Bao.03Sro.97TiO 3. J. Am. Ceram. Soc., 71(4) (1988) 201-5. 7. NOWOTNY, J. & REKAS, M., Defect chemistry of BaTiO 3. Proc. Int. Syrup. on Advanced Materials. Elsevier, Amsterdam (in press). 8. RASE, D. E. & ROY, R., Phase equilibria in the system BaO-TiO2. J. Am. Ceram. Soc., 38(3) (1955) 102-13. 9. SHARMA, R. K., CHAN, N. H. & SMYTH, D. M., Solubility of TiO 2 in BaTiO 3. J. Am. Ceram. Soc., 64(8) (1981) 448-51. 10. REMEIKA, J. P., A method for growing barium titanate single crystals. J. Am. Chem. Soc., 76 (1954) 940-1. 11. BRAJER, E. J., Effect of some ceramic techniques on the piezoelectric properties of barium titanate. Am. Ceram. Soc. Bull., 36 (1957) 333-6. 12. BELL, A. J. & MOULSON, A. J., The effect of grain size on dielectric properties of barium titanate ceramics. Br. Ceram. Proc., 36 (1985) 57-66. 13. HAAYMAN, P. W., DAM, R. W. & KLASENS, H. A., German Patent 929,350 1955-06-23; Netherlands Patent 84,015 1957-02-15. 14. LONG, S. A. & BLUMENTHAL, R. N., Ti-rich nonstoichiometric BaTiO 3. I: High temperature electrical conductivity measurements. J. Am. Ceram. Soc., 54(10) (1971) 515-19. 15. PRISEDSKII, V. V. & TRETYAKOV, Y. D., Chemistry of point defects in an oxide ferroelectric. Izvestia Akad. Nauk SSSR, Niorg. Materially, 18(12) (1982) 1926-38. 16. EROR, N. G. & SMYTH, D. M., Nonstoichiometric disorder in single-crystalline BaTiO 3 at elevated temperatures. J. Solid State Chem., 24(3-4) (1978) 235-44. 17. ASHIDA, T. & TOYODA, H., The effect of additives and of ambient atmosphere during heating on the electrical resistivity of semiconducting BaTiO 3. Jap. J. Appl. Phys., 5(4) (1966) 269-74. 18. HEYWANG, W., Resistivity anomaly in doped barium titanate. J. Am. Ceram. Soc., 47(10) (1964) 484-90. 19. MURAKAMI, T., MIYASHITA, T., NAKAHARA, M. & SEKINE, E., Effect of rare-earth ions on electrical conductivity of BaTiO 3 ceramics. J. Am. Ceram. Soc., 56(6) (1973) 294-7. 20. JONKER, G. H., Some aspects of semi-conducting barium titanate. Solid State Electron., 7(12) (1964) 895-903. 21. WAKU, S., Studies on the boundary layer ceramic capacitor. Rev. Electric. Comm. Lab., 15(9-10) (1967) 689715. 22. BRAUER, H., German Patent 1 646987, 5 March 1970. 23. HENNINGS, D., Barium titanate based ceramic materials for dielectric use. Int. J. High Technol. Ceram., 3 (1987) 91111. 24. ICHINOSE, N., Recent R& D trends of electronic ceramic materials. J. Jpn Int. Ed., 97 (1989) R31-R38. 25. YAMAMOTO, H., OGASAWARA, T., NAKAMURA, T., WATANABE, Y. & FUJIWARA, S., Material development of high dielectric constant (Sr,Ca,Pb)TiO 3 series ceramics. J. Ceram. Soc. Jpn Int. Ed., 97 (1989) 693-6. 26. GOODMAN, G., Electrical conduction anomaly in samarium-doped barium titanate. J. Am. Ceram. Soc., 46(1) (1963) 48-54. 27. MAcCHESNEY, J. B. & POTTER, J. F., Factors and mechanisms affecting the positive temperature coefficient of resistivity of barium titanate. J. Am. Ceram. Soc., 48(2) (1965) 81-8. 28. BROWN, F. & TAYLOR, C. E., Electrical conductivity of

A C K N O W L E D G EM ENTS We would like to thank Dr Keith Reeve for proofreading and several comments which were helpful in the preparation of the final version of this paper. We also thank Dr Detlev Hennings for his remarks. We acknowledge the support of the Commonwealth of Australia through the Department of Industry, Technology and Commerce as well as the Commission of the European Communities (grant # BREU-0144-C). REFERENCES 1. LONG, S. A. & BLUMENTHAL, R. N., Ti-rich nonstoichiometric BaTiO3. II: Analysis of defect structure. J. Am. Ceram. Soe., 54(1) (1971) 577-83. 2. SEUTER, A. M. J. H., Defect chemistry and electrical transport properties of barium titanate. Philips Res. Repts, Suppl. No. 3 (1974) 1-84. 3. DANIELS, J. & H~RDTL, K. H., Defect chemistry and electrical conductivity doped barium titanate ceramics. Part I: Electrical conductivity at high temperatures of donor doped barium titanate ceramics. Philips Res. Repts, 31 (1976) 489-504. 4. CHAN, N. H., SHARMA, R. K. & SMYTH, D. M., Nonstoichiometry in undoped BaTiO 3. J. Am. Ceram. Soc., 64(9) (1981) 556-62. 5. REKAS, M., Electrical properties and defect structure of

J. Nowotny, M. Rekas

240

29. 30. 31. 32. 33.

34.

35. 36. 37.

38. 39. 40. 41.

42. 43. 44. 45.

46. 47. 48.

49. 50.

niobium-doped barium titanate. J. Appl. Phys., 35 (1964) 2554-6. BRAUER, H., The problem of the surface states in BaTiO 3 cold conductors. Z. angew. Phys., 23(3) (1967) 373-6. MATSUOKA, T., MATSUO, Y., SASAKI, H. & HAYAKAWA, S., PTCR behaviour of BaTiO3 with Nb205 and MnO2 additives. J. Am. Ceram. Soc., 55(2) (1972) 108. UEOKA, H., The doping effects of transition elements on the PTC anomaly of semiconducting ferroelectric ceramics. Ferroelectrics, 7(1-4) (1974) 351-3. IHRIG, H., PTC effect in BaTiO 3 as a function of doping with 3d elements. J. Am. Ceram. Soc., 64(10) (1981) 617-20. IHRIG, H. & PUSCHERT, W., A systematic experimental and theoretical investigation of the grain-boundary resistivities of n-doped BaTiO 3 ceramics, J. AppL Phys., 48(7) (1977) 308 I-8. HENNINGS, D., Grain size and grain boundary effects in passive electronic components. In Surface and Near-Surface Chemistry of Oxide Materials, ed. J. Nowotny & L.-C. Dufour. Elsevier, Amsterdam, 1988, pp. 479-505. SABURI, O., Properties of semiconductive barium titanates. J. Phys. Soc. Jap., 14(9)(1959) 1159-75. BRAHMECHA, B. G, & SINHA, K. P., Resistivity anomaly in semiconducting BaTiO 3. Jap. J. Appl. Phys., 10(4) (1971) 496-504. PERIA, W. T., BRATSCHUN, W. R. & FENITY, R. D., Possible explanation of positive temperature coefficient in resistivity of semiconducting ferroelectrics. J. Am. Ceram. Sot., 44(5) (1961) 249-50. HEYWANG, W., Barium titanate as a semiconductor with blocking layers. Solid State Electron., 3(1) (1961) 51-8. GERTHSEN, P. & HARDTL, K. H., A method for the direct proof of the homogeneity of conductivity of grain boundaries. Z. NaturJbrsch. A, 18(3) (1963) 423-4. HAANSTRA, H. B. & IHRIG, H., Voltage contrast imaging of PTC-type BaTiO3 ceramics having low and high titanium excess. Phys. Stat. Sol. (a), 39 (1977) K7-K10. IHRIG, H. & KLERK, M., Visualization of the grainboundary potential barriers of PTC-type BaTiO 3 ceramics by cathodoluminescence in an electron probe microanalyser. Appl. Phys. Lett., 35(4) (1979) 307-9. KNAUER, U., Distribution of the iron dopant in barium titanate ceramic, determined by the scanning transmission electron microscope. Phys. Stat, SoL (a), 53 (1979) 207-10. HAANSTRA, H. B. & IHRIG, H., Transmission electron microscopy at grain boundaries of PTC-type BaTiO 3 ceramics. J. Am. Ceram. Soc., 63(5-6) (1980) 288-91. HEYDRICH, H. & KNAUER, U., Grain boundary effects in ,ferroelectric barium titanate. Ferroelectrics, 31 (1981) 151-6. KOSCHENEK, G. & KUBALEK, E., Grain-boundary characteristics and their influence on the electrical resistance of barium titanate ceramics. J. Am. Ceram. Sot., 68(11) (1985) 582-6. REHME, H., Detection of barrier layers in barium titanate by electron microscopy: cold conductor ceramics. Phys. Stat. SoL, 18 (1966) K101-KI02. SINCLAIR, D, C. & WEST, A. R., Bulk PTC effect on doped BaTiO3. J. Mater. Sei. Lett., 7 (1988) 823-4. SINCLAIR, D. C. & WEST, A. R., Variation with processing conditions of bulk and grain boundary PTCR phenomena in doped BaTiO 3. In Surfaces and Interfaces of Ceramic Materials, ed. U-C. Dufour, C. Monty & G. PetotErvas. Kluwer Acad. Publ., Dordrecht, 1988, pp. 535-43. HEYWANG, W. & BRAUER, H., Formation of barrier regions in cold conducting barium titanate. Solid State Electron., 8(2) (1965) 129-35. MILLER, C. A., Potential barriers on semiconducting barium titanate. J. Phys. D: Appl. Phys., 4 (1971) 690-6.

51. GERTHSEN, P. & HOFFMANN, B., Current-voltage characteristics and capacitance of single grain boundaries in semiconducting BaTiO3 ceramic. Solid State Electron., 16(5) (1973) 617-22. 52. H O F F M A N N , B., A model of the grain boundary resistance in doped BaTiO 3 ceramic. Solid State Electron., 16(5) (1973) 623-8. 53. BRAUER, H., Normal resistance in a barium titanate ceramic semiconductor in the range below Curie temperature. Solid State Electron., 17(10) (1974) 1013-19. 54. DANIELS, J. & WERNICKE, R., Defect chemistry and electrical conductivity of doped barium titanate ceramics. Part V: New aspects of an improved PTC model. Philips Res. Repts, 31(6) (1976) 544-59. 55. PETERSON, N. L., Grain boundary diffusion in metals. Int. Metals Rev,, 28 (1983) 65-91. 56. NOWOTNY, J., Certain aspects of segregation in oxide ceramic materials. Mater Sci. Forum, 29 (1988) 99-126. 57. NOWOTNY, J., Surface and grain boundary segregation in metal oxides. In Surfaces and Interfaces of Ceramic Materials, ed. L.-C. Dufour et al. Kluwer Acad. Publ., Dordrecht, 1989, pp. 205-39. 58. ATKINSON, A. & TAYLOR, R. I., Impurity diffusion of 63Ni along grain boundaries in nickel oxide. Phil. Mag. A, 43 (1981) 979-98. 59. DECHAMPS, M. & BARBIER, E, Interface transport in monoxides. In Non-stoichiometric Compounds, ed. J. Nowotny & W. Weppner. Kluwer Acad. Publ., Dordrecht, 1989, pp. 221-36. 60. WYNBLATT, P. & McCUNE, R. C., Surface segregation in metal oxides. In Surface and Near-Surface Chemistry of Oxide Materials, ed. J. Nowotny & L.-C. Dufour. Elsevier, Amsterdam, 1988, pp. 247-79. 61. JONKER, G. H., Halogen treatment of barium titanate semiconductors. Mater Res. Bull., 2 (1967) 401-7. 62. THOMAS, R. T., TREDGOLD, R. H. & WILLIAMS, R. H., Influence of gaseous adsorption on the metal-BaTiO a contact. Proc. Phys. Soc. London, Solid St. Phys. Set. 2, 1(5) (1968) 1370-5. 63. TIEN, T. Y. & CARLSON, W. G., Influence of oxygen partial pressure on properties of semiconducting barium titanate. J. Am. Ceram. Soc., 46(6) (1963) 297-8. 64. UEDA, I. & IKAGEMI, S., Oxidation phenomena in semiconducting BaTiO 3. J. Phys. Soc. Jap., 20(4) (1965) 546-52. 65. HASEGAWA, A., FUJITSU, S. & YANAGIDA, H., Adsorbed oxygen and PTCR effect of semiconducting barium titanate. Jap. J. Ceram., 97(5) (1989) 549-53. 66. KAHN, M., Effect of heat-treatment on the PTCR anomaly in semiconducting barium titanate. Ceram. Bull., 50(8) (1971) 676-80. 67. HOLT, J., Surface space-charge barriers on semiconducting barium titanate. Solid State Electron., 9 (1966) 813-18. 68. WAKU, S., Studies on the boundary layer ceramic capacitor. J. Inst. Elect. Comm. Engrs Jap., 47(7) (1966) 1285. 69. WAKU, S., NISHIMURA, A., MURAKAMI, T., YAMAJI, A., EDAHIRO, T. & UCHIDATE, M., Classification and dielectric characteristics of the boundary layer ceramic dielectrics (BL dielectrics). Rev. Electr. Comm. Lab., 19(5-6) (1971) 665-79. 70. WAKU, S., Studies of the boundary layer ceramic capacitor, Rev. Electr. Comm. Lab., 15(9-10) (1971) 689-715. 71. Grain boundary phenomena in electronic ceramics----cycle of articles. In Advances in Ceramics, Vol. 1, ed. L.M. Levinson & D.C. Hill, American Ceramic Society, Columbus, OH, 1981. 72. LEWIS, G. V. & CATLOW, C. R. A., Computer modelling of barium titanate. Radiat. Eft., 73 (1983) 307-14. 73. LEWIS, G. V., CATLOW, C. R. A. & CASSELTON, R. E.

Positive temperature coefficient o f resistivity f o r BaTiO 3

74. 75. 76. 77.

W., PTCR effect in BaTiO 3. J. Am. Ceram. Soc., 68(10) (1985) 555 8. HAN, Y. M., APPLEBY, J. B. & SMYTH, D. M., Calcium as an acceptor impurity in BaTiO 3. J. Am. Ceram. Soc., 70(2) (1987) 96-100. H A G E M A N , H. J. & HENNINGS, D., Reversible weight change of acceptor-doped BaTiO3. J. Am. Ceram. Soc., 64(10) (1981) 590-4. CH1ANG, Y. M. & PENG, C. J., Grain boundary nonstoichiometry in spinels and titanates. Adv. Ceram., 23 (1987) 361-77. BERNASIK, A., HIRSCHWALD, W. & STOLZE, F., SIMS studies of undoped and Nb-doped BaTiO 3 (in preparation, to be published).

241 78. TSAI-FA LIN, CHEN-TI HU & I-NAN LIN, Influence of stoichiometry on the microstructure and positive temperature coefficient of resistivity of semiconducting barium titanate ceramics. J. Am. Ceram. Sot., 73(3) (1990) 531-6. 79. NIEPCE, J. C., Some aspects of the influence of particle size on properties and behaviour of a dielectric material: example of barium titanate. In Surfaces and Interfaces of Ceramic Materials', ed. L.-C. Dufour, C. Monty & G. PetotErvas. Kluwer Acad. Publ., Dordrecht, 1989, pp. 521-33. 80. DA YU-WANG & K A Z U M A S A UMEYA, Electrical properties of PTCR barium titanate. J. Am. Ceram. Soc., 73(3) (1990) 669~7.