High temperature nonstoichiometric rutile TiO2−x

Prog. Solid St. Chem. Vol. 17, pp. 263-293, 1987 Printed in Great Britain. All rights reserved.

0079~786/87 $0.00 + .50 Copyright (~ 1987 Pergamon Journals Ltd.

HIGH TEMPERATURE NONSTOICHIOMETRIC RUTILE TiO2-x F. Millot,* M-G. Blanchin,? R. T6tot,* J-F. Marucco,* B. Poumellec,* C. Picard* and B. Touzelin* *Laboratoire des Compos6s non stoechiometriques, Brit. 415, Universit6 Paris-Sud, 91405 Orsay Cedex, France and ? Departement de Physique des Mat6riaux, Universit6 Claude Bernard, Lyon I, 43, Bd. du 11 Novembre 1918, 69622 Villeurbanne Cedex, France

I. INTRODUCTION

The description of the defect structure in nonstoichiometric oxides at high temperature,during the past thirty years,has been a main subject for a number of laboratories involved in high temperature chemistry and physics. Such oxides the

an interest

arose from

were different

the facts that numerous properties of nonstoichiometric

from those

understanding of

the defect

of the

corresponding

stoichiometric oxides

and that

structure would be the key to predict many properties of

these compounds• In the course of time,however,it became clear that the concept of defect structure covered

various aspects

range

interaction energies,charge state,etc.)each aspect contributing in a specific way to

of a same reality (e.g.: morphology,formation energy,short and long

the properties of the material• The

present paper

describes the

specific contributions of

various experimental and

theoretical procedures involved in high temperature studies of solids. We

shall focus

our attention

on the

results obtained

oxides, TiO2_ x, because there still exists some

doubt

for

concerning

the

nonstoichiometric

their defect structure,

in spite of the considerable attention these oxides have received• For purposes of simplicity,we shall divide the present report in five sections,respectively dealing with: • Thermodynamical properties • Transport properties • Spectroscopies . Structural studies • Theoretical calculations The particular phase Fig.l. It chiometry

of the Ti-O system in which we are interested,TiO2_ x is shown on

consists of a slightly defective phase for which the maximum deviation from stoiat 1273 K does

not exceed

x =

10 -2 . It

is bounded

on the

left side

by the

Anderson phases containing regularly spaced shear planes which extend all over the crystal.

II. THERMODYNAMICAL PROPERTIES.

Let us start with the thermodynamical quantities which allow the

determination of the

stability range of TiO2_ x depending on temperature,T,and oxygen partial pressure, PO2.

JPSSC 17/4 - A

263

264

F. Millot et al.

r(K) 2000-

--

1500

TiO 2_x + Ti n 0 2 n - 1 1000

1.99

1.98

Fig.1

%,

2.0

Temperature-Composition phase diagram of nonstoichiometric rutile TiO2_ x.

These quantities are: • The partial molar free enthalpy of mixing of oxygen in the oxide: ~G(O 2) = RTLnPo2 • The partial molar enthalpy of mixing of oxygen in the oxide:

~H(O2)

• The partial molar entropy of mixing of oxygen in the oxide: AS(O 2) = ( ~ ( O 2) - dG(O2)) / T Among

these quantities,

~G(O 2) is most generally considered to predict which kind of

defect is responsible for departure from stoichiometry.Different theories have been developed to relate thermodynamical quantities to defect properties.The basic concepts were first introduced

by KrSger I to

describe behaviour

of small defect populations, i.e. 10 -5- 10 -6

molar concentrations.These involved Schottky or Frenckel equilibria in stoichiometric crystals.lt

was assumed

that formation of such defects could be explained in terms of virtual

reactions which obey the mass action law. From

there,a relationship

could be

deduced between defect concentration and a measurable

quantity,such as the chemical potential of one of the constituents of the solid.As an exemple,consider

the formation

of doubly

ionized oxygen

vacancies,V~',in TiO2_x.The virtual

reaction of formation is:

o~g) + v ~ - + 2 e'

=

oXo

(l)

K(T) = 1 / p1/2 02 IVY'] [e' ]2

(2)

and application of the mass action law leads to:

gq.(2)

establishes a

trons,e',and

relation between concentration of oxygen vacancies,V~',and of

the equilibrium

pressure of

oxygen,which can

elec-

he easily measured at a given

temperature T. In the particular case where it is assumed that V~'is the majority defect in

High Temperature NonstoichiometricRutile TiO2_x

265

this system,a supplementary relation exists between defect concentrations:

.: Ivy.}

= [e']

/ 2

(3)

Combining Eq.(1) and Eq.(3),it follows that:

x E This to

p-i/6 02

(4)

is a very simple relation between x and P02, which are both measurable quantities.Due that simplicity,such

analyses have been extensively used by scientists involved in the

prediction of defect structures in solids.Subsequently,they have been applied to molar concentrations much larger than those initially considered by Kr6ger (typically up to I0-2). Lidiard 2, first, pointed out that application of lids of

electrically charged defects is increased. This author s suggested that the Debye-H~ckel

(DH) approximation an

the mass action law to imperfect so-

is bounded by the existence of long range coulombic interactions as the concentration

estimate of

(generally used to describe behaviour of liquid electrolytes) may permit

deviation from the ideal behaviour assumed on the basis of the mass action

law. This theory

simple approach

of long

was extended

range interactions

by Allnatt and Cohen 3 who developed a statistical

in ionic crystals. In the case of sodium chloride, for

instance, it was shown that the DH approximation is valid for temperature higher than 800"C and defect concentrations as large as 10 -5 which is, in fact, much larger than expected. Wagner 4 first Extending the

reported that

Lidiard's previous

cases of

long range interactions could not be ignored in oxides.

ideas, he

showed that application of the mass action law to

CU2_xO and COl_ x O was likely not valid and, consequently, defect structures

could be hardly deduced from the dependence of x on PO2. Dieckmann 5 calculated involved

V~

and

the dependence

h'(instead

of x on Po2in COl_xO using a defect model which

of V~o and h" as generally deduced from Krbger's theory) and

the existence of long range interactions. It was shown that experimental data were perfectly

fitted by

the results

of calculations but Dieckmann concluded that such a perfect fit

could result from mere chance ! We have now to examine the influence of long range interactions in the case of TiO2_ x. Let

us consider

the simple defect structure previously assumed and electrons localized on

titanium sites, we have :

K(T)= 1 /

o5[ o i

p1/2[v..]

(5)

A mean activity coefficient 7± for V~' and e' has been introduced which can be expressed from the DH approximation as :

7_+

-

-

exp

2~kT(l + ra)

Iz+z_l

(6)

111

(7)

with r =

-

-

6kT

ZiNiZ~

266

F. Millot et al.

Where e o is the elementary charge,k the Boltzmann constant and a the smallest distance between charged defects

which is taken here as the Ti-O distance. Z i is the charge of spe-

cies i, Niits concentrations per cm3and F is the "dipole potentiel". From Eq.(3),it follows :

x ~ p-l/6 02 7~1 _ The

(8)

e q u a t i o n s can be n u m e r i c a l l y s o l v e d p r o v i d e d t h a t t h e s t a t i c

dielectric

constant,

6 , o f TiO 2 i s known. ~ h a s been measured by P a r k e r 6 up t o 1060 K. R e s u l t s show an a n i s o t r o p y of ~ d e p e n d i n g on t h e c r y s t a l l i n e

orientation.

~#c(300 K) = 170

~#c(lO00K) = 90

~#a(300 K) =

~#a(lO00 K) = 60

86

Extrapolation of these results to higher temperature gives :

~#c ~ ~#a ~ 50 at 1500 K

,

~

dlogPo~ dlog x

100

~ =

~

//

/

~= 20

10 ,

I

-6 Fig 2.

_4

,ogx

Example of deviation from the ideal mass action law applied to doubly ionized

oxygen vacancies on applying the Debye-HQckel approximation.

Fig. 2 shows

variation of (-~InPo2/~inx) T versus defect concentration for different values

of ~ at the temperature of 1273 K. Dotted lines represent sections corresponding to definitive values of I/U ; I/U = lOa corresponds to the highest value of the defect concentration for which the DH approximation is regarded as quantitatively valid ; in the region where 1 a < - < 1 0 a , t h e c u r v e s s h o u l d be r e g a r d e d as i n d i c a t i v e o n l y . F We o b s e r v e t h a t TiO2_ x seems t o be a r e l a t i v e l y f a v o r a b l e c a s e w i t h r e s p e c t t o t h e

application varies

of

from 5 . 6

t h e i d e a l mass a c t i o n law . Taking f o r i n s t a n c e f o r x = 10 -3 t o 5 . 4 f o r

a l l x i f t h e i d e a l mass a c t i o n law s t r i c t l y

6 = 50,

,/- 8 1 n P o 2 / ~ l n x )T .

x = 1 0 - 2 . T h e c o r r e s p o n d i n g v a l u e s h o u l d be 6 f o r held.Such slight

departures

from i d e a l v a l u e of

High Temperature Nonstoichiometric Rutile TiO2_x

6

nevertheless prevent

nonstoichiometry first

us from

concluding definitely

in TiO2_ x. Four t y p e s

step since

Kr6ger's theory is a

of d e f e c t s

267

t h e n a t u r e of d e f e c t s accommodating

V ~ ' , V ~ , T i ~ ' , T i ~ ' m a y be c o n s i d e r e d i n a

good a p p r o x i m a t i o n

f o r TiO2_ x as d i s c u s s e d above.

following equations can be written: ~ o~g) + V~. + 2 e, ~ OoX

K1 - 1 / p1/2IV~" ] O 2 [e']2

(9)

if W~'is the majority defect. and x ~ p-I/6 02 O~ g) + V~ + e . = O xo

K 2 = 1 / pI/2IV~I,e'] 02

(i0)

K 3 = 1 / PO2 [Ti~'l|e']4

(11)

K 4 = 1 / PO2 ITi~'][e']3

(12)

if V~ is the majority defect. and x ~ p-i/4 02 O~ g) + Ti~" + 4 e' = Ti~i + 2 0 ox if Ti~'is the majority defect. and x ~ p-i/5 02 O~ g) + Ti~" + 3 e' = Ti~i + 2 0 xo

and x ~ p-I/4 02 if Ti~'is the majority defect. In Table I, there is a summary of the Po2dependence on x and on other quantities considered

in next

paragraphs. All the values listed in Table I were obtained using the mass

action law approximation.

Table

I

:

Expected slopes from the mass action law applied to virtual reactions of formation of defects in TiO2_ x.

majority defect

slope

moving defect

%,

5

4

5

- (~InPo2/~InDTi)

4

2.5

- (~inPo2/~inD;i)

0

5 10

-

-

Ti~"

4

- (~inPo2/~inx)

T i i4

Ti~" or V~

(~InPo2/~in~ e)

Vo •

-

(~InPo2/~InD ;)

0

vo

-

(~inPo2/~InD ~)

4

3.33

268

F, Millot et al.

II-1

- Thermodvn-m~eal

DroDertie. ~

the departure from sto~eh~ometrv .x. in terms of the

euuilihrium ~artlal ~ressure of oxv,@~, PO2. The departure from stoichiometry, x, in terms of PO2 has been determined with two different experimental techniques : Method of ture,

such as,

the physico-chemical

equilibrium between the oxide and a redox gas mix-

CO 2- CO, H20 - H 2 or H 2- O2.The

composition of the solid is determined by

thermogravimetry. This technique has been used by Kofstad 7, Forland 8, Picard Rosa I0 and Marucco et al.ll'12.

and Gerdanian 9, Dirstine and

Method of electrochemical cells with a solid electrolyte : the Po2Value is determined from the e.m.f, of the cell. Samples of constant composition are prepared by coulometry or by reaction with a redox gas mixture. Their composition is also determined by thermogravimetry.This technique has been used by Blumenthal and Whitmorel3'14,Alcock and Zador 15'16, Dirstine and Rosa 10. Finally, a particular method used by Zador and Alcock 17 consisted of equilibrating the composition

of the

oxide in an H20-H 2 atmosphere during a reasonable time (20 to 48 hours

at temperatures between 1600 and 1900 K).The PO2 value is determined with lid

state cell.

a

(ZrO2-Y203) so-

When the equilibrium is obtained, the sample is quenched and its composi-

tion is determined by gravimetry. The

results

obtained

at

1273 K

are

reported

on

Fig.3.

~...~

-4-

~."

Logx

./

...'~ °...''"/

-3'

~.,'~

.,..."~/ij / K o f s t a d (7) Alcock & Zador (15,161 • -----Dirstine& Rosa (10) ........... Marucco& al (11,12) o Blumenthal &Whitmore(13,14 •

.° .~',eil

..,~'~t~/II ~"~o I I 4

-2

-15 Fig.3 .

In the

so far

as

-1~0

Log Po2 (atrn)

Composition - oxygen pressure relationship at 1273 K in TiO2_ x.

the slope (~InPo2/~Inx) T can give some information about the nature of

defect structure,

we have

plotted

the straight line deduced by

from their experimental points between 10 -12 - 10 -18 atm :

Dirstine and Rosa I0

High Temperature Nonstoichiometric Rutile TiO2-x

logx = - O.218 logPo2

269

6.005

and also the straight line corresponding to the general expression proposed

- ~ RTInPo2 = 3RTInx + 131,000

for 10-17<

PO 2

Finally,

< 10-13atm at 1273K. we have

plotted the

straight line

The results which consist of the various slopes, are summarized

discrepancy

in Table

between authors,

stoichiometry vacancy V O

Kofstad 7 :

29.8T (cal.mol. -I)

corresponding to 1068 K. This author has proposed a law of the kind x ~ p-I/6 02 .

tures

by

II. It a slope

(x < 10 -3 ) which

Forland's results 8 at

(~InPo2/@InX)T,for different tempera-

is observed that, with the acknowledgement of some of six is obtained for the smallest departures from

is easily

interpreted in terms of a doubly ionized oxygen

in this range (see Table I and Fig.2).

Table II :

ref.

Slopes - (~inPo2/@inx)

in TiO2_ x from various authors.

I0

ii

9

T ('C)

800

i000

1200

995

1194

977

1227

800

900

i000

Ii00

1050

x < 10 -3

4.25

4.6

4.9

6.4

6.0

2.0

6.0

6.0

6.0

6.0

6.0

6.0

x > 10 -3

4.25

4.6

4.9

6.0

6.0

5.0

5.0

4.3

The than

Such

larger departures from stoichiometry (x > 10 -3 ) are characterized by slopes lower

6, the

slope

effect being

has generally an idea

TiO2_x 12

6.0

more pronounced

been accounted

for in

when the temperature is raised. This change of terms of one or two new major defect species.

appears correct because slopes of nearly 4 are observed for the most reduced

which can hardly be explained with long range interactions as we have noticed be-

fore. However, it is certainly incorrect from these results only to state the nature of the new majority defect(s), which could be Ti~',

V~, or Ti~" as well.

II-2 - Limits of the homogeneity range of Ti02_ x

The

results which concern the maximum departure from stoichiometry , x m, limiting the

homogeneity to

domain of

a straight

TiO2_ x are reported on Fig.4. The plot of lOgXm= f(I/T) corresponds

line between

900 K and 1800 K. This very simple behaviour has not received

any explanation at the present time. Fig. 5 shows the values of dG(O 2) / T = RlnPo2 corresponding to

x m in terms ~f ) / T.

The points have been rationalized with a straight line :

~G(O 2) / T = - 7.555xi05 / T + 242.8

(J K -I molo~)

270

F. Millot et al.

15b0

1200

1000

860

760

T(oc)

Logx

-2.5

-2

~r Alc°ck'& Zad°r!15', 16) "~Mar.ucco&al(11,12) , , o Blumenthal&Whitmore(13) OPicard &Gerdanian (9) -1.5

~

Fig.4 .

8

~)

10yT (K-')

1'0

Maximum deviation from stoichiometry of TiO2. x at 1273 K.

-500"

AG(Oi)/I" J.mo~K t

- 400

- 300

-200'

.j,J'°

IX/0

• "~ o O "k x

Alcock & Zador (15,16) Marucco & al. (11,12) Blumenthal & Whitmore Picard & Gerdanian (9) Saumard &al. (18) Bogdanova 8 al. (19) §

Fig.5 . Oxygen chemical potential of the TinO2n_l- TiO2_xm phase ecuilibrium at various temperatures.

(13)

10~T (K-')

High Temperature Nonstoichiometric Rutile TiO2_x

271

we then find an enthalpy variation for the reaction:

TinO2n-Z+ 02

-

equal

to - 180,5 kcal

mol -I . (TinO2n_ 1

=

tz -

n/xm~

is the first

-"m

Anderson phase in equilibrium with

TiO2-xm)This value is in reasonable agreement with ~ ( O 2) = - 195 Kcal mol -I directly determined by microcalorimetry 9 .

II-3 - Partial molar enthalp~ L~H(O2) of mixin~ in TIO2_ x

Two methods have been used in order to determine bH(O 2) in TiO2_ x and domain TiO2_xm - TinO2n_ 1

in the diphasic

:

• This quantity can be deduced as the slope

(~(dG(O 2) / T) / ~(i / T))x-

This method has been used by a number of authors 7,8,10,13,14,16,20 • dH(O 2)

can be measured directly

by

high temperature microcalorimetry.

This method

has been used by Picard and Gerdanian 9 . The Gerdanian

results are shown on Fig.6 . Corrected results of the original data of Picard and are also represented as dotted lines. They are based on the previous analysis of

the limits of the homogeneity domain of TiO2_ x which predicts a value of TIO1.9915 at 1323K {see Fig.4) instead of the observed value in calorimetry viz 1.9900• The discrepancy probably

comes from

a heterogeneity in the oxygen absorption during a calorimetric experiment.

The corrections were done by equating areas a and b on Fig.6.

-AH(O~ [71

Kcal.mol

250 T in 02n-I + T iOn,_Xm .,."f

19) (16) (14)

(6) (16)

200

(13)

1(9)

•

I

T i O2-~

•

1,99 Fig.6 .

Oxygen mixing enthalpy in TiO2_ x at 1323 K.

O/Ti

,

I0

272

F. Millot et al.

The

whole data

suggest that

a positive discontinuity appears at the boundary of the

homogeneity range. This agrees with the shape of the coexistence line between phases 21. dH(O 2) in TiO2_ x appears as a flat curve. This property may give information about the relative

energies of

formation H i of defects. It has been shown by Tetot and

Gerdanian ZZ

that : ~8i - h (g) ~H(O 2) = 2 ZiH i ~--02

(13)

Where 8 i is proportional to the concentration of the defect i, Zi8 i = x ,and H i is the corresponding enthalpy of formation, and h(g)is the enthalpy of the gas phase. 02 Assuming that H i does not vary with x, we may conclude that the coexistence of two important defects in TiO2_ x implies a definite ratio between their H i . For instance, considering a mixture of V~" and Ti E" we should have HV~" ~ ~ HTi4"'i

II-4 - Conclusion of thermodynamical properties

At

first sight,

thermodynamical properties of TiO2_ x are smooth functions of the

parture from stoichiometry. It is observed that x ~

p-I/6 02

de-

for x < 10-3and x ~ p-I/4 02 at the

limit of the homogeneity domain (x ~ 10 -2 at 1273K). This variation of the slope is not likely

to be explained with a unique majority defect in the whole range of

nonstoichiometry

even if long range interactions are considered. Thus, within a point defect model,it may he said that the majority defect for x < 10 -3 is a doubly ionized oxygen vacancy and that larger

departures from

stoichiometry are

accomodated by

a new majority defect which may be

Ti~', Ti~'or V;. It

has become

analysis 12).

customary among authors to deduce defect structure from a mathematical

of the variation of inPo2 with inx using a combination of all or part of Eqs.( 9.

This procedure

is not, however, correct because it supposes that long range interac-

tions can be omitted. Taking into account of this last effect may, in principle, extend the domain

of predominance of Y~" in Ti02_ x.

In any case, it questions the

relevance of such

methods in deducing a defect structure involving various defects.

I I I - TRANSPORT PROPERTIES IN TiO2_ x III.l - Electrical properties.

III.l.1 - Conductivity measurements,~. vestigated important

The neighborhood of stoichiometry has been in-

by Rudolph23,Yahia 24 and Greener et a125. There,metallic impurities can play an role in

the electrical

behaviour.Nevertheless a unique value of the gap energy

has been reported by these authors,viz,3 eV. Measurements in a CO-CO 2 reducing atmosphere allow the observation of the influence of structural defects on the electrical properties. Tannhauser 26 ,Blumenthal et a127,Baumard 18'28 and Marucco et a111 have reported results which,apart isotherms

from some disagreements between them,indicate a variation of the slopes of the (- ~InPo2/~In~) T from

6 to 4

when PO2 decreases and

T increases.

High Temperature Nonstoichiometric Rutile TiO2 ~

273

These values are collected in table III.The interpretations of these slopes are similar and subjected

to the

same criticisms

as those

relative to

the departure from stoichiometry

reported above.

Table I I I :

Slopes - (~InPo2/~in~ e) in TiO2_ x from various authors.

ref.

26

1500

987

1480

1100

5

5.6

4.8

4.96

4.7

6

5

4.2

4.8

4.96

4.7

5

1115

1316

1467

x < 10 -3

5.3

5

4

4.6

III.l.2• along with ~

Thermoelectric power

11

i000

T ('C)

10-2> x > 10 -3

18

27

measurementsfS.

They

have generally been carried out

measurements 24'28 at constant temperature and for various PO2 in order to ap-

preciate the mobility of the charge carriers• Bogomolov

et a129 have determined ~ and S

crystallographic

directions "a"

and "c".They

for single crystals oriented along the two have observed that the important anisotropy

for conductivity (~c/~a ~ 2) is not observed for the thermoelectric power (Sc ~ Sa). III•l.3•

Mobility of charge carriers.

Various methods have been used to deduce this

quantity: • The comparison of ~ and S. • The comparison

of ~ and the Hall effect• This method has been used by Breckenhridge

and Husler 30 and Frederikse et a131'32• • The comparison of ~ and the departure from stoichiometry, x, 32'II within the context of a point defect model• • The analysis

of the

behaviour of ~

over a wide range of temperatures• This method

has been recently applied by Poumellec et a133 to the results of Iguchi et a134. All these methods indicate that the at 300 K and 0.I cm2/V sec

III•2.

mobility of the charge carrier is about

! cm 2 / V sec

at 1200 K along the "c" crystallographic direction of TiO2_ x.

Diffusion.

III.2.1. Diffusion of titanium.

$tolchlom®tric TiO2: by

Ventaku and

Self diffusion coefficients DTi in TiO 2 have been reported

Poteat35,Lundy and Coghlan 36 and Akse and Whitehurst37.The three groups of

results show the same activation energy for DTiRc,Vlz,respectively,

- 61.4 ,- 59.9 and - 57

Kcal

mol -I •Moreover,Akse and Whitehurst 37 reported that DT~ was not changed on increasing

the

oxygen pressure in equilibrium with the crystal by a factor of five.Absolute values of

DTi

are about

the same

for Akse

and Whitehurst

and for

Ventaku and

poteat. Lnndy and

Coghlan results appear twice as large• These

results indicate that the

diffusional behaviour of Ti in TiO 2 is controlled by

metallic impurities of the crystal (mainly AI) which explain the invariancy of DTi with PO2 and

the differences

between authors.Observed activation enerq~es of DTi can then be asso-

ciated with the migration energy, Z~Hm,of Ti.

274

F. Millot et al.

16'00

DIc

1,foo

lfOO

10'00

860 T°C

Dj.~

I

"

• D~., In TIO, (36) O ~ in TiO,_, (4"/) 0 D~ in TlO, (381

~ t

0.5-

Fig.7 .

Anisotropy for the diffusion of oxygen and titanium in TiO 2 and TiO2_ x.

Lundy and Coghlan 36 reported measurements of DTi along the two crystallographic directions of TiO 2 ("c" and "a"). They observed a significant anisotropy represented

on Fig.7.

(The solid line was deduced from the Arrhenius plots of

DTi#c and DTilc ) indicating

ferential the r

I

c

diffusion of

axis

of

Ti

a pre-

perpendicular

TiO 2 .They

to

interpreted this

result as indicating that titanium ions do not have

any significant

interstitial site resi-

dence in stoichiometric TiO 2. They based their I

I

argumentation

on the

of

small foreign

Ni

which all

anisotropy

diffuse

by an interstitial me-

chanism. This "Chimney effect" Fig.8 .

to

the

the rutile structure.

T~O 2 represented tentatively as an end v3ew on

circles are oxygen and small

Fig.8

circles titanium.

nels along "c".

Nonstoi~hiometrie TiO2_ x

is related

End view along the c axzs of Large

particular

(D#c ~ D±c)

ions like Li,B,Cr,Fe,Co and

has been

reported by

T t 0 2 . x.

Self diffusion

and which

crystallographic structure of

shows the interstitial chan-

of Ti a l o n g "e" i n n o n s t o i c h i o m e t r i c

Akse and Whitehurst 37 between 1273 and 1373 K and for O / Ti

ratios ranging from 1.999 to lo99.These authors observed a slope (- ~inFo2/OlnD;i) T varying from

4.64 at 1273 K to 4.86 at 1373 K that they interpreted as indicating the predominance

of the moving Ti~" ion with some influence of Ti~" as the temperature is lowered.

High Temperature Nonstoichiometric Rutile TiOz_x

275

The majority of their data are at 1331 K. A significant decrease of D~i_ is observed on this

isotherm

for

PO2 < 10 -16 arm,

that

(Po21i m = 10 -16"9 arm, see Fig.5).They

is

inside

the

homogeneity

range of

TiO2_ x

have interpreted these results as indicating a pre-

transition drop associated with the formation of extended defects for the most reduced part of

the TiO2_ x phase.They have not,however,explained why they failed to observe this pheno-

menon at nearly the same temperatures of 1273 and 1373 K. Another on

strange result

measuring DTi

stoichiometric

concerned the very large differences

with different

TiO2,this

single crystals.In

result is

(a factor of 3) observed

contradistinction to measurements on

difficult to explain since diffusion in TiO2_ x may,in

principle,depend upon defects accommodating nonstoichiometry. Finally,they

have observed an activation energy of D~i(PO2 = lO-16atm) of *

mol -I that they have compared with the activation energy (DTi) x

=

-

- 57 Kcal mol -I

112.3 Kcal observed

on stoichiometric TiO2.They deduced from the relation:

(14) x

the be

PO 2

"T

partial enthalpy of mixing of oxygen in TiO2_ x, dH(O 2) = - 276 Kcal mo1-1 which should compared with

the results

shown on Fig.6 ~ ~ ( O 2) ~ - 237 Kcal mol -I may be used as a

reference value ]. III.2.2. Diffusion of oxygen. Self

diffusion

coefficients

have

Available data concern stoichiometric TiO 2 exclusively. been

determined

by

Haul

and Dumbgen38,Doskocil and

Pospici139,Derry et al40,Gruenwald and Gordon 41 and Harita et a142. Haul TiO 2

of

760

and Dumbgen 38 have reported an activation energy of migration of O in powders of -

60 Kcal mol-l.No variation of D O was detected in the oxygen pressure range 10 -3-

mm Hg

and it was concluded that the oxygen vacancy concentration is controlled by me-

tallic

impurities

(Al).The comparison of the self diffusion coefficient for two crystallo-

graphic orientations indicated that D~c / D l c = 0.62 at 1620 K (see Fig.7). The

results of

Doskocil and Pospici139 and Derry et a140 confirmed the previous fin-

dings of Haul and Dumbgen. Derry et al found an activation energy of - 66 Kcal mol -I. Gruenwald and Gordon41determined the anisotropy of O diffusion at 1079 K They obtained D#c / Dlc = 0.19 (see Fig.7). Finally, an

Harita et a142 reported an activation energy along "c" of - 60 cal mol -I and

anisotropy of diffusion D#c / Dic = 0.66 at 1353 K (represented on Fig.7 with its error

bar).They

also reported

a significant

increase of D O in TiO 2 doped with 0.08 mol % Cr203

that they qualitatively explained by the formation of new oxygen vacancies on intentionally doping. All

the preceeding

authors reported comparable values of D O which appear to he about

thirty times lower than the corresponding DTi whatever the temperatures studied. III.2.3. ported by

Chemical diffusion in TiO2_ x.

Chemical diffusion measurements have been re-

Barbanel and Bogomolov 43, Iguchi and Yajima 44, Baumard 45, Picard and Gerdanian 9,

Ait-Younes et a146 and Millot 47. The most cerns

two first

authors 43'44 have determined chemical diffusion coefficients D for al-

stoichiometric TiO 2 the influence

scope of this paper.

crystals.The interpretation

of impurities

is difficult because it mainly con-

on the transport behaviour of TiO 2 which is out of the

276

F. Millot et al.

Data

obtained by

ceramics

or as

other authors 45,9,46,47 concern nonstoichiometric TiO2_ x either as

oriented single crystals• Their comparison with the self diffusion coeffi-

*

cient DTi by standard methods 4 indicates a good self consistency between these various data , (except,however,for the decrease of DTi observed at 1331 K by Akse and Whitehurst 37 for reduced TiO2_x). Millot 47 have reported an anisotropy D#c / D±c = 0.3 independent of the departure from stoichiometry at 1323 K (see Fig.7). III.2.4. E19ctric charge carried by the moving defect in TiO2_ x. Ait-Younes et a148 and later Millot 47 have reported experimental determinations of the effective

charge number,

Z*,of

the moving defect in TiO2_ x. A unique value of Z* = 3 has

been

obtained for

from

TIO1.9985 to TiOi.9925.These authors have interpreted their results as the moving de-

ceramics and

oriented single

crystals at 1323 K for compounds ranging

fect of Ti~" in TiO2_ x at 1323 K. III.2.5. port

ionic transport

Singheiser and Auer 49 have determined ionic trans-

numbers, t i , by Tubandt's method in the range 1125-1255 K and

2.5xi0 -14 atm. They crease

PO2 from 1.3xlO -9 to

observed a weight increase of the cathode equivalent to the weight de-

of the anode which indicated that cations move in TiO2_x.The exact shape of the va-

riations In

number.

of tiwith Po2is difficult to interpret as well as the observed activation energy.

effect,their

(x ~ 10 -4 )

investigations

where the

correspond

influence of

important.Nevertheless,this

to

rather

metallic impurities

experiment is

a simple

low departures from stoichiometry (50 ppm

proof of

iron in their sample) is

the importance

of titanium

diffusion in rutile.

III.3.

Conclusion of transport properties.

Electrical

properties of high temperature TiO2_ x are well documented.The variation of

the electronic conductivity with

is mainly similar to that of x with PO 2 PO 2 • Electrical conduction can be classed in the category of large polaron conduction. Ionic transport data are much more scarce than electrical data.

Oxygen self diffusion data are titanium

only

available

self diffusion, chemical diffusion and ionic

TiO2_x• They all

point toward

for stoichiometric Ti02• In contrast, transport

number

are available in

a moving titanium defect in these compounds. The electrical

polarization method indicates that its charge is +3.The analysis of the variations of these data

with PO2

indicate that

x = 10-2• However,the exact

V~" cannot nature of

be the

the new

majority defect

majority defect

in the range x = 10-3to is not settled ( Ti~" for

ref•37 and Ti~ • for ref.46)• Finally,oxygen and titanium diffusion show similarities in TiO2: • same anisotropy (Fig•7) • same activation energies• This

singular hehaviour

and the fact that titanium migrates faster perpendicular than pa-

rallel to the "c" direction of TiO2_ x call for new questions: • Why is the "chimney effect" (preferential migration along "c") not operative for interstitial titanium ions? • Is there any relationship between transport properties of titanium and oxygen? •

.

A study of the self diffusion of oxygen,Do,ln nonstolchiometric TiO2_ x crystals may provide an answer to this last question.

High Temperature Nonstoichiometric Rutile TiO2 x

277

IV - SPECTROSCOPIES IN TiO2_ x.

In

contradistinction to

previous guish

paragraphs, the

the thermodynamical

and transport properties treated in the

spectroscopic experimental

techniques cannot, a priori, distin-

between the effects induced by the defects and by the atoms occupying a normal posi-

tion in the crystal. It is, then, necessary to observe modifications of the properties with the departure from stoichiometry to assess the influence of constitutional defects. It

supposes, obviously, that experiments are carried out under well defined thermody-

namical

conditions which, however, is almost never the case in practice, and thus strongly

limits the usefulness of the data. Many

spectroscopic methods are very sensitive. Active centers can be detected at con-

centrations (optical

of the

order of

one p.p.m.

. On

the other

hand, very different properties

transitions between different states, magnetic properties, ~ or O spins) can cha-

racterize

different defects,

of comparable concentrations, which may be implicated in the

properties of the material. Unfortunately, defect

the sensitivity

as well

as the resolution are strongly decreased for

concentrations higher than I00 ppm or at high temperature, the domain of investiga-

tion of nonstoichiometric crystals. Under such circumstances, no information can be obtained on defects in their thermodynamical

equilibrium state (an important exception concerns the optical spectroscopy of ab-

sorption).

Spectroscopic properties of defects can, however, give some structural, energe-

tical or electronic information. We are

shall, in the following, discuss the results of various techniques which, in fact,

not always spectroscopies but have been grouped together because they generally appear

as complementary techniques within the broad range of experimental physics. They are : Electron paramagnetic resonance (EPR) Absorption spectroscopies UV, visible (VAS),!R Photoluminescence (PL) Thermoluminescence (TL) Photostimulated conductivity (PSC) Thermostimulated conductivity (TSC) Internal friction Dielectric Loss Channeling of protons In

so far as VAS is a technique which can be used at high temperature, it shall cons-

titute the starting point of the following presentation.

IY.l. VASrPLrTSrPSCrTSC

The

results of

Yon Hippel et a150 by

VAS were obtained between room temperature and

1273 K. Three electronic transitions in the 3eV gap of TiO2_ x have been detected.Their characteristic features are summarized in Table IV. Stoichiometric TiO 2 did not show any of the observed effects. Thus, it should be deduced

that absorption

Bogomolov but

in the

gap does

not result

from electron

transport as proposed by

and Mirlin 51 because stoichiometric TiO2at 1273K shows appreciable conductivity

a featureless absorption spectra. They may be attributed to the presence of defects of

constitution. These results have been confirmed by other authors52"53~

278

F. Millot et al.

Table IV :

Visible absorption spectroscopy (VAS) results.

Energy (eV)

Comments

0.4

Predominant for low departures from stoichiometry (DFS)

0.7

Exist for low DFS; intensity proportional to DFS and

0.8

energy slightly displaced toward 0.8 for high DFS.

1.5

Predominant at high DFS whatever the temperature and for low DFS at low temperature.

Other methods have also detected these defects (for instance PL,TL,PSC,TSC) other

defects (see

concentration

table V).

However, these

along with

techniques which allow the detection of low

defects (of the order of one p.p.m.) do not give any information about their

concentration or their structural characteristics.

Table V : Energies of defects observed with PSC,PL,TSC and TL (from Hillhouse54).

0.075

Energy

0.13

0.24

(eV)

PL

Experimental method

0.36-

0.50-

0.58-

0.64-

0.37

0.51

0.61

0.69

TSC

TSC

TSC

TSC

TSC

PSC

TL

TL

TL

TL

TL

TSC

0.74

TL

0.81-

].15-

0.86

1.21

PSC

PSC

TL

TL

IV.2. Channelin~ of protons

The has

channeling of

very

elegant way

comments. cooled cross then

protons in the interstitial channels (001) of TiO2_ x

been reported 55 as indicating of detecting

Channeling was

under unreported the coexistence

in the

the

appearance of

the experimental conditions merit some

room temperature on a high temperature, reduced sample,

conditions. A

glance at

Fig.4 indicates

that the sample had to

line between TIO2_ x and the Anderson phase at about 900 K and it is

questionable whether

see

interstitial ions,

done at

(x ~ 5x10 -3)

the existence of interstitial titanium. In spite of the

next chapter

the cool

product corresponded effectively to TiO2_ x . We shall

that it probably did not because conventional quenching leads to

shear planes in the cool specimen for such reduced rutiles. This detail

i s of p r i m a r y i m p o r t a n c e f o r t h e i n t e r p r e t a t i o n

of t h e s e d a t a b e c a u s e c h a n n e l i n g of p r o t o n s

does n o t d i f f e r e n t i a t e

and a s h e a r p l a n e .

between an i n t e r s i t i t i a l

High Temperature Nonstoichiometric Rutile Ti02_~

279

IV.3. ~PR and UV

EPR

and UV have been widely used. These two techniques give characteristic structural

features of materials. The results are shown in Table VI.

Table VI: EPR and UV results.

Center Energy 57

Spin

Characteristic features

Interpretation

L/2 57

x < 1.7x10 -5

Ti~'(MOM) 56

(eV) A

56

Disappears for x = 0 at low temperatures 58

Ti~'(0~0) 58,60 correlated with H + 61

B

Electron trap 5 7

0.22

B12

0.12

1 57

present with A1 59

Ti#i(000)- AI~'(O~O)-

x < 10 -5, fourfold multiplicity.

Al#i(OlO)

x < 10 -5

T~i(000)- Ti~i(MMM)-

59

also called C or D 57

substitutional.

A1)'(O~O)- AI+i(0!0)59 Fe~i - Vo 62

C

1/2

Electron trap,substitutional

57

Maximum concentration at x = 3.3x10 "4. 1/2 57

EH

GH

Electron trap,substitutional

57

Electron trap,interstitial or perturbed

0.12

Ti~" moving ?56 V~ T~'- M÷i 63

Ti - A] - A1 57

substitutiona157.Depends upon Cr,Al or Fe

HH

Depends upon Cr,Al or Fe.

0.12

L

1/2

Fourfold multiplicity. 64

W

1/2 56

x > 5x!0 -4

56:fourfold multiplicity.

Ti - A1 - A1 57

I oo> A1iloool6'

~i~O00~-r~'~O0~ delocalized electron

X

1 56

AH

0.37

x > 5xlO -4

56

Hole trap 57 .Substitutional symetry. Correlated with Fe 3+

JPSSC

17/4

-

B

280

F. Millot et al.

X large

and W

centers. Following Hasiguti 56 , active magnetic centers for relatively

departures from

stoichiometry

are

X and

W• They appear for x > 5x10 -4 increasing

with x and stabilizing for x = 3x10 -3. The

W center

has been

interpreted as

Ti~i(O00 ) - Ti~ • (~00) where the electron is

delocalized on two titanium sites (half spin). The X center has a spin of unity and then concerns two electrons. A possible interpretation have

would involve been proposed

two neighboring by

Carnahan

interstitial titaniums.

Such defect configurations

et a165 and Wachtman et a166 to interpret their internal

friction experiments. These authors have observed that internal friction is anisotropic in posed

Ti02_ x and pro-

that this is the result of double interstitial titanium ions (~00) - (0N)• The con-

centration of these defects varies as that of the W and X centers• The

relatively clear picture of the origin of the X and W centers is, however, obscu-

red by the large difference between the departure from stoichiometry

(30 to 100xl018 cm -3)

and the maximum number of spins deduced from the integration of the X and W peaks 56 , viz 6 to

7xl017cm -3. This observation, which means in practice that, in an EPR experiment, the X

and

W centers

represent a small fraction of the total defect population, can be explained

by the formation of shear planes on cooling the EPR samples previously reduced at high temperature. In effect, shear planes are either, not magnetic and then do not give any visible center or, may give a broad band in EPR. C ween

2 and

Center. The C center is the predominant center for defect concentrations bet5xlO -4. Its spin is not completely established. In 1972, Hasiguti 56 proposed a

value of ~ hut more recently Hodgkiss 57 has preferred a value of I. Up to now, interpretations concern the half spin center : Hasiguti 56

• on

account of

the large

width of

the EPR line proposes a

moving

Ti~"

(a polaron). Nowick 63 relates the

C center

to a Ti~" coupled with an impurity. This interpretation,

however, does not explain the width of the EPR line. • Other interpretations, The

maximum

involving oxygen vacancies have been proposed 59

concentration

of spin

of this

center is 1.4x1018cm-3

for a deviation

from stoichiometry of 1019 cm -3 (x = 3.3xi0-4). This

last result

explained

by the

suggests that the majority of defects are not magnetic. This can be

formation of shear planes on cooling the sample or by the existence of a

majority of doubly ionized or neutral oxygen vacancies. A

centers. The A center exists for departures from stoichiometry

x ( l.Sx10 -5,

the same time the C center disappears 56. There have been many discussions about this center which is characterized by a half spin and an activation energy of 0.37 eV (close to the 0.4 eV VAS band). It

has been interpreted by titanium interstitial ions or by oxygen vacancies associa-

ted with an impurity : Si~i ,H+ 62, trivalent impurity 67 , It

Co 2+

68

is worth noting that dielectric relaxation data 69 have been interpreted with simi-

lar defects :

g

.

V "(0.194 0.194 ~) - Me Ti(0 0 i)

which however is not magnetic and

Ti~i(O 0 O) !

~ V ~ Ti~i(~ ~ ~)

- MeTi

where the unpaired electron is delocalized between the vacancy and the two titanium•

H i g h T e m p e r a t u r e N o n s t o i c h i o m e t r i c Rutile T i O 2 x

Finally cies.

it should

281

be noticed that this center cannot he explained with simple vacan-

A recent calculation 70 of

the change of the electron density of states induced by a

non-distorted oxygen vacancy shows that no level appears in the gap. Other centers. tures

from stoichiometry.

Other

centers shown

All these

in Table VI correspond to very low depar-

centers depend on the presence of impurities and are

beyond the scope of this paper.

~V.4. Conclusion

As

we mentioned

in the

introduction, spectroscopic

studies were carried out at low

temperature on TiO2_ x samples initially prepared at high temperature. It shear

shall be

seen in

the next paragraph that, at least for the most reduced samples,

planes appear on cooling and that we may expect to preserve the high temperature de-

fect population of TiO2_ x only under a very rapid cooling of samples. Unfortunately, people involved in spectroscopic properties of TiO2_ x were not aware of these effects and consequently the data concerning the range TIO1.999 - TIO1.99 do not permit any quantitative determination of the occurence of high temperature defects. At

the present

time the various data reported on spectroscopies suggest that for de-

partures from stoichiometry higher than x = 5xl0-4,there exist various kinds of di-interstitial titanium ions along with simple interstitials. The

nonstoichiometric range x ~ 5xlO -4 is not as well understood. Oxyqen and titanium

defects have been proposed to account for the same observed centers.

Y - STRUCTURAL STUDIES

V.I. Transmission electron microscopy.

As cal

reported above, various techniques are available to study the dependence of physi-

and chemical properties on the departure from stoichiometry in futile. Models for ele-

mentary or complex structural defects are then proposed to interpret results of such measurements, but defect structures deduced so far can always be regarded as more or less speculative.

Thus, question

arises as

to whether it could be possible to visualize structural

defects in nonstoichiometric crystals. In

the seventies, transmission electron microscopy (T E M) appeared as an appropriate

technique

to observe

stoichiometric

extended defects

like crystallographic shear planes (C S P) in non-

oxides. Extensive T E M observations of C S P in TinO2n_ 1 ordered compounds

were achieved by Bursill and Hyde 71 , who also reported existence of C S P for compositions very close to TiO 2, namely TIO1.998571 This slight

result gave rise to a great deal of argument about the suggestion that even very

departures from

defects,

stoichiometry in rutile could be entirely accomodated by

extended

i.e. C S P, and not by point defects. In fact, Blanchin et a177 showed later that

TIO1.9985 crystals very rapidly cooled from reduction temperature did not reveal any C S P. It

was thus

highlighted that

enough attention had not been paid, in previous studies, to

the cooling procedure for the preparation of specimens. From that point, new T E M structural

studies by Bursill, Blanchin et al were started, the aim of which was to determine the

nature

and the

especially

structure of the defects accomodating nonstoichiometr¥ in futile. This was

in regards

to any

transition from

non extended defects, traditionally called

point defects,to extended defects,mainly planes (C S P),at increasing degrees of reduction.

282

F. Millot et al.

These recent T E M studies are now summarized. Clearly, it would have been interesting to observe by T E H rutile crystals reduced in situ

(in the

electron microscope)

under thermodynamic

equilibrium conditions at various

temperatures• This is difficult to achieve experimentally since equilibrium at the compositions

of interest

requires very

low partial pressures of oxygen. Therefore,treatments in

the microscope of rutile crystals reduced ex situ at 1323 K were limited to in situ heating and

cooling experiments

tions

up to

1273 K under 10 -5 torr vacuum. Despite the obvious limita-

of such experiments, important results were obtained concerning dissolution and pre-

cipitation

mechanisms of

C S p73,74

However, most of the T E M studies were done at room

temperature on foils thinned from crystals reduced ex situ at 1323 K under equilibrium conditions and then cooled to room temperature at controlled cooling rates 72'75-77 . These results of

proved to

be consistent with those of in situ studies• On the other hand, the range

well-determined compositions and cooling rates experimentally available allowed the de-

termination

(in a large extent) of the influence of the cooling rate on the microstructure

observed at room temperature• Some solid conclusions could thus be drawn about the structure of nonstoichiometric defects present at reduction temperature• The investigations Fig•l

which was

were carried out

with respect to

the phase diagram

depicted

in

constructed using various equilibrium measurements. The range of nonstoi-

chiometry investigated extended from TIO1•9994 to TIO1•9915. Concerning, first, the composition range TIO1.9994- TIO1•9950 it was shown that

:

• C S P are not present,even in miniature, in samples cooled rapidly from 1323 K. Such crystals exhibited only {i00} platelet defects showing a precipitate-type contrast, as well as spot contrast or patchy change in the background contrast suggesting clustering of small defects 72 . High resolution electron microscopy (H R E M) images of crystals containing no C S P revealed

spot or

line contrasts

which were

interpreted as

aggregates of

ten or

less small

defects 78 • The

interstitial-versus-vacancy

due

to disturbing surface contrasts though calculations of H R E H images allowed determi-

nation be

character

of such defects could not be recognized, mainly

of the values of crystal thickness and lens defocus in which that distinction would

theoretically possible 78. • The C S P

of

mean orientation close to {132)

appear only in specimens cooled more

slowly through the diphasic region of Fig.l. In crystals C S P exhibit

cooled at

intermediate rates the

both lateral and longitudinal disorder 75. The degree of disorder within both

individual C S P and in lamellae of C S P increases as cooling rate increases 76 Such results were confirmed by in situ T E M studies 73"74 : C S P and platelet defects formed

in crystals reduced ex situ were seen to dissolve during in situ heating and to re-

precipitate

upon cooling ; the temperature at which C S P dissolved was found to depend on

the departure from stoichiometry in a way consistent with Fig. i. In the composition range TIO1.994 - TIO1.9915, even crystals quenched most rapidly (in oil), proved to contain C S P at room temperature, though according to Fig•l, such compositions correspond to region of solid solution of small defects at the temperature of 1323 K. However, rates, less the lable

comparison of as illustrated

microstructures obtained in Fig.9

a and

at room

temperature for various cooling

b for TIO1.9925, shows that C S P obviously become

dense, extended, and ordered as the cooling rate increases. This suggests that due to proximity of the phase limit for such compositions, cooling rates experimentally avaiare not

fast enough to preserve the microstructure existing in TiO2_ x at a tempera-

ture of 1323 K and hence the diagram of Fig.l is still regarded as valid.

High Temperature Nonstoichiometric Rutile TiOz_×

283

0.2 ,m

Fig.9a.

Fig.9b.

Fig.9

: T E M bright-field micrographs the same composition

taken from crystals reduced at 1323 K to

Ti01.9925 but cooled to room temperature

rent rates.

In a. the crystal was cooled

density of

C S P (viewed inclined,

observed.Most

at diffe-

more slowly in air:

a

large

like in A, or edge on like in B) is

of the C S P are fully extended through the specimen.

Ordering process has started to give rise to lamellae of C S P, as seen in C. In b. the crystal was cooled more rapidly in water:C S P are less extended and less ordered than in a.: defects of small size and B) are observed

throughout

the specimen.

( l~ke in A

F. Millot et al.

284

A

~

~

-

---

-

-----

-

-

- New

structural

models

for interstitial defects :

-'~:'7.,'

a) no defect; b) traditional interstitial

defect,i.e.ad-

ditional

Ti 3+

c)

pairs

two

cations; of

drally-coordinated tions and

sharing f)

sions tk

show

octaheTi 3+ ca-

faces; d),e) linear exten-

of c) to produce more

widely-separated

pairs

of

face-shared octahedra. ,~

~

c

d

e

f

-D B

-

Arrows indicate sugges-

ted diffusion mechanisms for movement of Ti 3+ interstitial r

1

i

'

defects parallel to a) [001]

_&

b) [010]

and c) [I/2 0 ,/z]

Dotted arrows

A

indicate more

,-~-~

complex concerted atom move-

N

ments required for bulk dif°I

fusion of M 3+ (e.g. Cr 3+ or "

Fe 3+) interstitial defects. ~ 1 '? 1!

b 1253!

g

(132)

(1.13)

C

- Aggregation

depicted

W

formation

in A

of defects leads to the

of: a)

a pair of

(121)crystallographic

shear

planes

(C S P); b) a pair of

(253)

C S

P : c) a pair of

(132) C S P and d) a pair of (143)

Fig.10 : Bounded projection structure.

C S P.

(along [010] for -~/4 ~ y ~ J/4) of the rutile

High Temperature Nonstoichiometric Rutile TiO2 ×

285

Regarding the mechanisms for nucleation and growth of C S P, H R E M observations show that

C S P mostly precipitate as pairs

which then split, leading eventually to the forma-

tion of lamellae of aligned C S p76 . Such

a "pair

mechanism" suggested

to Bursill and Blanchin new structural models for

titanium interstitial defects 79, different from the classical picture depicted in Fig.10Aa. The by

basic structure Ti 3+ ions

for the new models consists of two pairs of oxygen octahedra occupied

and sharing a face in common ; the pairs are separated by a cationic vacancy

(Fig. i0 Ab). The new models readily explain

the precipitation of C S P as pairs, as shown

by Figs. 10A and 10C. H R E M images of single C S P terminating inside crystals reduced at 1323

X unambiguously

show that

the single C S P have an extrinsic displacement vector of

about ~ <011>, thus supporting the interstitial model 80. These new interstitial models also explain the structure of the platelet defects having mean orientation {100) 75 , which decorate C S P in samples cooled at intermediate rates, because they develop at temperatures of 650 to 900 K 75. In fact, single C S P also exist in reduced futile ; they are more frequently observed in

samples corresponding

with

a

larger

to slight degrees of reduction 4 . This is regarded as consistent

concentration

of

oxygen

vacancy

defects

in these domains. Bursill and

Blanchin 81 have proposed new models for the structure of vacancy defects, which are depicted of

in Fig. IIA and B.

The "reconstructed" vacancy model (Fig. liB) implies the formation

single pairs of oxygen octahedra sharing a face, which should naturally lead to nuclea-

tion of single C S P. Recent from or

T E M studies by Bursill, Blanchin et al

firmly suggest that small departures

stoichiometry in rutile are accomodated by small defects, i.e. titanium interstitials oxygen vacancies,

extended

at high

temperatures. For such degrees of reduction, C S P or other

defects form only at lower temperatures. T E M observations clearly show that the

cooling

rate from

reduction temperature is the prominent parameter governing micro struc-

ture of reduced crystals observed at room temperature. H R E M images of C S P terminations and pair mechanism for nucleation of C S P suggest that titanium interstitial defects become

the major

closer

species as departure from stoichiometry increases. Formation of single C S P

to the stoichiometric composition seems consistent with the existence of oxygen va-

cancies

for the

obtained

so far

smallest degrees

of reduction. Unfortunately H R E M experimental images

have not permitted distinguishing between an interstitial or vacancy cha-

racter of small defects.

V.2 - X-rays and density measurements.

Density measurements have been reported by Straumanis et a182 and by Barbanel et a]83~ The

two sets

the

majority defect

marked,

of authors

that

in TiO2_ x (titanium interstitial or oxygen vacancy). It should be re-

however, that

considerations

have deduced opposite conclusions from their data with regard to

neither have

of imposed

most reduced

explored the nonstoichiometric domain of futile. From

oxygen pressure

samples were,

and coloration

of samples it can be concluded

in fact, in the very close neighborhood of stoichiometry

and thus that the reported variations of density have to be considered with cautiousness. High lattice

temperature X-rays

have been

used by Touzelin 84 to determine variations of the

parameters with the departure from stoichiometry.The sample was situated in a high

temperature

furnace adapted

to a

continuously flowed. 0.01"8 accuracy

diffractometric set up in which a redox gaseous mixture was obtained on the position of diffraction peaks and

thus a and c quadratic lattice parameters could not be determined to better than T 0.001 A.

286

F. Millot

et al.

~!~.

•

N

• T,Ion,um

a-

• Ti

b -

O Ox

• T,Ion,um

C-

Fig.11.l : a)

~ Ox),gefl

II10l Pro)echon

o Ouyge n

[i~oi Project,on

Oxygen vacancy structure

following simple removal of

(uu0). Note triangular coordination of

nearest

(TiI, Ti 2 and Ti3), which have distorted coordination,

neighbour

square

Ti atoms

pyramidal fivefold

b) Showing location of tetrahedra]

(1/2 0 1/4) [MO4]

interstitial sites adjacent to the oxygen vacancy, tion of octahedral (-1/2 0 0)[MO6] interstitial

oxygen at

c) Showing loca-

sites

adjacent

to

the oxygen vacancy.

',~ k d ~ i I "%il" ,,,~

, '~.....~;

o

b

Fig.ll.B : Showing production of face-shared pair of displacement of Ti I by vector (-1/2 0 0).

[TiO6~-[~O6] octahedra by Ti 3+ charge

compensation

fects may be placed at either (-I/2 0 O) and (-I 0 0) : a) model I, or alternatively (1/2 1/2 I/2) and (I/2 1/2 -1/2) : b) model IT.

High Temperature NonstoichiometricRutile TiO2-x

287

It was observed that a and c vary linearly with temperature. At 1323 K, O-=Ti

1.992 was obtained with a H2-O 2 Gaseous mixture of PO2- 10_16.66 atm9"

The parameters for TIO 2 and TIO1.992 at 1323 K w e r e f o u n d t o he: TiO2in air : a = 4.629 A , c = 2.989 A TIO1.992

: a = 4.627 A , c - 2.988 A

Given the accuracy of these data, it should be concluded that no apparent variation of a and c can be observed in TiO2_ x on varying x.

VI. THEORETICAL CALCULATIONS A

The formation energy of the point defects V~ • and Ti T "_ in TiO2_ x have been reported by Catlow

et a185 and

groups•

It is

Sawatari

et

a186 • The

fundamental approach is the same for the two

based on the approximate method of description of the solid, first proposed

by Mott and Littleton 87 , which consists of considering two regions in the crystal : • an external region II considered as a polarizable dielectric continuum. • an internal region I surrounding the defect, described by the fully ionic mode~, in which the of

ions are

explicitly relaxed• The total potential energy of region I is taken as a sum

pair interaction

terms each

of which

is dependent only on the distance between ions.

These two-body potentials consist of three terms : • the Madelung energy (term I) . the short range repulsive interaction described by

a simple Born Mayer (Sawatari et

al.) or Buckingham function (Catlow et al), Y(r) = A exp(-r/R) - Cr -6 (term 2.). • the

dispersive

Van der Waals

interactions

( dipole-dipole and dipole-quadrupole )

(term 3). Term 3 has been explicitly it

has been

calculated by

Sawatari et al for each pa~r of ions, while

included in the r -6 term of the Buckingham potential, term 2, by Catlow et a]

(the dipole-quadrupole interaction is then neglected). The

formation energy

gion I on code

varying the

and Sawatari

of the defect is obtained by minimizing the tota~ energy of re-

coordinates of

ions• To that end, Catlow et al. use the HADES 88:89

et al use a method previously developed by Dienes et a~ 90 for the study

of point defects in AI203. The

polarization of

previously

ions is

described in

the two works by means of the shel] mode]

proposed by Dick and Overhauser 93 . Each ion consists of two components, a core

of charge X containing the whole mass of the ion, and a shell of charge Y. X and Y are coupled

by a

harmonic spring with a force constant K, so that the polarisability of the ionr

a, is given by : a = y2/K. The by

choice of

experimental The

the various parameters involved in these potentials, A,E,C,Y,K is done

adjusting the calculated values of selected properties of the perfect crystal to their values :

parameters and

cohesion energy, elastic constants, dielectric constants ~o,~,etc.

values of experimental and calculated properties for the two groups of

authors are shown in Tables VII, VIII and IX. In Table

spite of IX, we

groups,

the satisfactory agreement between calculated and experimental values of

observe

noticeable

especially the shell model

differences between

the parameters proposed by the two

parameters of the Ti 4+ ion. These observations set the

problem of the validity of the empirical parametrisation of the inter-ionic potentials. Let us consider the following processes of formation of V~" and T ~ "

o~

=

v b.

+ o~-

~E 1

respectively

:

~ls)

288

F. Millot et al.

TiO 2 ~

Ti 4" + 2 02-

(16)

l~E2

for which the energies dE 1 and ~E 2 may be deduced from the works of : Catlow et al. 84 and Sawatari et al. 85

: A~ 1 = 17.3 eV,

dE 2 = 41.34 eV

: ~E 1 = 32.06 eV,

~E 2 - 54.1 eV

Large discrepancies can be observed between the two sets of results.

Table VII - Parameters of short range potentials.

A

(eV)

Catlow et al

R (A)

Sawatari et al

Catlow

Sawatari

et al

9.557x103

Ti 4+ - Ti4 +

(ev/A6)

C

Catlow

et al

et al

0.185

0

Ti 4+ - 02-

0.656x103

1.164x103

0,404

0.33

02- _ 02-

0.227x105

1.597x105

0.149

0.15

Sawatari et al

0

27.06

Table VIII - Shell model parameters.

K (ev/A 2)

Y (number of electrons)

y2/K (calc.)

Ion Catlow

of

Catlow

Sawatari

e t al

et al

et al

2.38

- 3.67

18.41

94.6

0.307

0.142

- 35.86

- 1.58

65974

61

0.02

0.041

-

may be

2 02- from

Sawatari

e t al

When the calculations defect,

Catlow

et al

02-

Ti 4+

Sawatari

et al

are made at 0 K, the nature of stable defect, then the majority

brought out by comparing 26E I with AE 2, each corresponding to the removing the crystal. As it will be shown below, the differences are large enough to

neglect the entropy contributions even at high temperature. 0~- is an oxygen ion at rest,at infinite separation. So,from V~"

is the

the results of Catlow et al (26E 1 - dE 2 = - 6.74 eV),it can be concluded that majority defect,whereas

the results of Sawatari et a] (2~E 1 - dE 2 = 10.02 eV)

lead to the contrary conclusion. Very

recently,Tetot and Gerdanian 22'92 have derived,on thermodynamical grounds invol-

ving no hypothesis,a relation which allows a direct comparison between calculated formation energies

and the experimental ~H(O2),the partial molar enthalpy of mixing of oxygen in the

oxyde,for the case of small departures from stoichiometry.

High Temperature Nonstoichiometric Rutile TiO2 ×

289

Table IX - Comparison of experimental values of selected properties calculated

values

using

Sawatari et al

Catlow et al

Calc.

Experim.

Calc.

Experim.

Cll = C12 (1011 dynes cm -2)

25.33

27.01

25.29

26.97

C33 (I0 II dynes cm -2)

77.92

48.19

51.36

48.13

21.2

20.93

Bulk modulus

with their

the parameters of Tables VII and VIII.

(I0 II dynes cm -2)

Cohesion energy

109.97

126

124.3

157.32

170

180

94.76

86

124.6

(eV)

I

Let

us

define

the

quantity

H(O) -

B

89.8

where

H(O)

=

(~H(O 2) + h ( g ) ) / 2 02

and

Ta B = 3 RT + -- V(O) in which h(g)is the standard entha!py of 02 gas at temperature T,e and x x 02 are respectively the thermal expansivity and the isothermal compressibility and V(O) is the partial volume of the constituent O in the oxide. It lues

has been shown 92 that H(O) - B must be necessarily located between the extreme va-

of El'S, which are the energies for particular formation processes of the various de-

fects present in a crystal. Here, these processes are as follow :

o~ +

v~'+ 2 Ti#i

=

O xo + 2 T ~ i

EV~ 0

(17)

El< 0

(i8)

5 o®+

-1 Ti~" + 2 T~÷~. :

ox o +

Ti~i

2 and

correspond respectively

to the disappearence of one V$" and a half Ti~" in particular

conditions detailed elsewhere 92. From the work of Catlow et a185 the formation energies can be calculated : -

E V = 1.358 ± 0.35 eV = 31.2 + 8 Kcal mo] -I.

-

E I = 4.728 ± 0.35 eV = 109 i 8 KcaI mol -I.

Sawatari et a186 have not calculated tron

the energy necessary to create a localized elec-

Ti~i , so it is not possible to calculate E v and E I from their work, and therefore to

compare their results with experimental ones.

290

F. Millot et al.

H(O) can be deduced from ~H(O2) measurements of P i c a r d e t a l 9 and B i s r e d u c e d t o 3 RT because

V(O) i s

n e a r z e r o 84.

So, H(O) - B - - 181 Kcal mo1-1 i n TiO2_ x f o r x ( 5xl0 -3 .

This v a l u e i s not l o c a t e d between t h e v a l u e s o f Ev and EI from Catlow e t a l . , conclued

that the

Til" are really

s o , i t can be

f o r m a t i o n e n e r g i e s c a l c u l a t e d by t h e s e a u t h o r s a r e i n c o r r e c t i f V~" and

t h e p r e s e n t d e f e c t s i n TiO2_ x f o r s m a l l d e p a r t u r e s from s t o i c h i o m e t r y .

R e c e n t l y , new s t r u c t u r a l

models have been p r o p o s e d f o r t i t a n i u m i n t e r s t i t i a l

d e f e c t s 80

and f o r oxygen vacancy d e f e c t s 81 and have been d e s c r i b e d above, i n s e c t i o n IV. From t h e r e cent

work o f

Shen 93, i n which t h e p o i n t d e f e c t f o r m a t i o n e n e r g i e s c a l c u l a t e d by Catlow e t

a185 a r e u s e d , we have c a l c u l a t e d fects

t h e f o l l o w i n g f o r m a t i o n e n e r g i e s f o r t h e s e new s m a l l d e -

: -

-

However,

E¢ = 19.98 eV

= 460.7 Kcal mo1-1

E i - 10.963 eV = 252.8 Kcal mo1-1

these values

depend largely on the energy of the process, Ti~+~ Ti~" ,the forma-

tion of a Ti~" interstitial given by Shen (- 29.87 eV). As far as the authors know,this value

does not appear in any paper on TiO2_ x published since 1982. It must therefore be used

with discretion. Excluding this point and considering that the interactions between the different constituents

are sufficiently

small, we

can conclude

that

E¢

and E i are incorrect because

H(O) - B is not included between them. In of

conclusion, the

techniques of simulation of solids do not allow the determination

the nature of the majority defect in TiO2_ x .Several reasons may be advanced to explain

this point. First,the pair interactions are suitable for pure ionic compounds but the omission

of angle-dependent

triction

on the

forces and

many body effects generally results in a severe res-

application of this model to more covalent systems like oxides. Second, a

weakness of the empirical parametrization of the potentials is to sample ionic interactions only

at their values in the perfect lattice, whereas studies of defective lattices require

accurate

potentials for

accurate

theoretical calculations

a greater

range of internuclear separations. Third, reliable and of the

shell model parameters and of the Van der Waals

coefficients would be a considerable improvement of the technique.

VII. CONCLUSION

We have examined the various experimental and theoretical may

draw

information

Ti02_ x • In doing so,

on

the

defect

techniques from

which one

structure of a particular nonstoichiometric oxide,

we have critically surveyed existing data and we have defined the inhe-

rent limitations for the interpretation of a particular experiment or calculation. What a

defect

emerges from this incomplete panorama is the variety of possible descriptions of structure.

(spectroscopies,

Techniques

involved

in

electron microscopy) stress the

the description

of microscopic properties

importance of structural characteristics

of defects : di-interstitial titanium ions, reconstruction of the neighborhood of an oxygen vacancy. In contrast,

macroscopic properties (thermodynamical and transport properties) are

described with point defects, interstitial and/or oxygen vacancies, which in fact are defined hypothetically. Careful

experimental work

with electron microscopy has improved the understanding of

quenching conditions on the modifications between high and room temperature defect structure. Small defect population in high temperature Ti02_ x appears the most probable from these E M studies.

High Temperature Nonstoichiometric Rutile TiO2 x

Titanium

and oxygen

defects coexist in TiO2_ x, the former appearing for most reduced

samples (x > 10-3). This conclusion is reached from macroscopic vity

and

However,

diffusion the exact

291

data)

as

well

nature, electrical

as microscopic

(thermodynamical, conducti-

(HREM and spectroscopies)

properties.

charge and concentration of these defects is still

under discussion. For

this purpose,

titanium and oxygen self diffusion experiments in TiO2_ x should be

significant contributions to these questions as well as spectroscopic data on thermodynamically well-characterized samples, correctly quenched from high temperature. Finally,

calculations of

defect formation energies have still to be defined in order

to allow guantitative predictions of defect structures in oxides.

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Phys.