PHYSICAL REVIEW B 75, 094107 共2007兲
Vacancies on the Ti sublattice in titanium monoxide TiOy studied using positron
annihilation techniques
A. A. Valeeva,1,2 A. A. Rempel,2 W. Sprengel,1,3 and H.-E. Schaefer1
1Institute
of Theoretical and Applied Physics, University of Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart, Germany
2
Institute of Solid State Chemistry, Ural Division of the Russian Academy of Sciences, Pervomayskaya 91,
GSP-145, Ekaterinburg 620041, Russia
3Institute of Materials Physics, Graz University of Technology, Petersgasse 16, A-8010 Graz, Austria
共Received 31 July 2006; revised manuscript received 13 December 2006; published 12 March 2007兲
The local atomic environment of lattice vacancies in titanium monoxide has been specifically studied by
positron annihilation techniques in the composition range from TiO0.74 to TiO1.26. For nonstoichiometric
titanium monoxide, a high concentration of vacancies has been suggested on both the titanium and the oxygen
sublattice. From the analysis of the core electron momentum distribution of the atoms surrounding the vacancies in disordered as well as in ordered titanium monoxide, oxygen atoms are identified to form the local
vacancy environment, indicating that positrons are trapped by titanium and not by oxygen vacancies. Positron
lifetime measurements have shown that the decrease of the oxygen content, y, and the process of disordering
in nonstoichiometric titanium monoxide TiOy, are accompanied by an increase of the valence electron density
on the titanium vacancy.
DOI: 10.1103/PhysRevB.75.094107
PACS number共s兲: 78.70.Bj
INTRODUCTION
From x-ray-diffraction and density measurements, it is
suggested that in transition-metal carbides, structural vacancies are located on the nonmetal sublattice, and in some
transition-metal monoxides, TiOy, VOy, and Nb0.75O0.75 or
nitrides, NbNy, structural vacancies are located as well on
metal sublattice.1 Assuming that there are no antisite or interstitial atoms in TiOy, a high amount of structural vacancies up to 10– 15 at. % on both the Ti and the O sublattices
can be calculated from the x-ray and density
measurements.2–4 According to the phase diagram at elevated
temperatures, titanium monoxide is disordered. At temperatures below 1250 K, titanium monoxide undergoes atomic
ordering. Structural vacancies strongly influence not only the
structure but also the magnetic and semiconducting properties of titanium monoxide.5,6
The formation of the ordered Ti5O5 phase leads to a redistribution of the titanium and oxygen vacancies. As a result
of ordering, the nearest-neighbor environment of the vacancies is assumed to change. According to the model of fully
ordered stoichiometric titanium monoxide,7 TiO1.00, the first
coordination shell of the vacancy is formed by six atoms
exclusively, i.e., without vacancies 关see Fig. 1共a兲兴. The second coordination shell consists of 12 atoms and also is free
from vacancies. On the third coordination shell, four vacancies are located. Depending on the oxygen content y and
structural state, the arrangement of oxygen atoms and vacancies in the first coordination shell of a titanium vacancy 共䊏兲
in titanium monoxide TiOy is changing.
According to the model of the disordered phase, i.e., a
nearly random distribution of atoms and vacancies in
nonstoichiometric titanium monoxide TiOy, at least six different local arrangements of oxygen vacancies in the first
coordination shell around the titanium vacancy are probable
关see Figs. 1共a兲–1共f兲兴.7 The same local arrangement is valid
for an oxygen vacancy.7
1098-0121/2007/75共9兲/094107共6兲
Regarding the oxygen content in disordered titanium
monoxide, each titanium vacancy can be surrounded by a
number of oxygen atoms in the first coordination shell varying from three to six. As the Doppler broadening of the
positron-electron annihilation radiation is sensitive to the
type of atoms surrounding the positron annihilation site and
to the number of these atoms, this technique can be applied
to investigate the order-disorder phase transformation in titanium monoxide.
In addition to the change of the number of the surrounding vacancies, the ordering results in a change of the volume
of the vacancies. The volume of the vacancies approximated
3
/ 8 decreases due to atomic and vacancy ordering
as ⍀ = aB1
to a maximum of 1%. Although the positron-electron annihilation technique is sensitive to the size of the vacancy where
the positron is localized, such a small change in vacancy
volume can hardly be detected by positrons.
The aim of the underlying study is the characterization of
the local vacancy environment in the ordered and disordered
states of TiOy by positron annihilation techniques and
thereby testing the structural models outlined above.
FIG. 1. Most probable arrangement of oxygen 共䊊兲 atoms and
vacancies 共䊐兲 in the first coordination shell of a titanium vacancy
共䊏兲 in the disordered titanium monoxide TiOy with different compositions 关共a兲–共f兲兴. Typical configuration for ordered titanium monoxide is shown in 共a兲.
094107-1
©2007 The American Physical Society
PHYSICAL REVIEW B 75, 094107 共2007兲
VALEEVA et al.
FIG. 2. Dependence of the titanium VTi 共䊏兲 and oxygen VO 共䊐兲
vacancy concentrations in disordered titanium monoxide on the
oxygen concentration y. The vacancy content in each of the two
sublattices was determined using the results from lattice constant
measurements, pycnometric density, and thermogravimetric analysis.
EXPERIMENT
The temperature of the synthesis of the TiOy specimens
was chosen according to the Ti-O phase diagram.8 Nonstoichiometric titanium monoxide TiOy was synthesized by
high-temperature vacuum solid-phase sintering from pure titanium dioxide TiO2 and Ti powders. The synthesis of the
titanium monoxide specimens was performed in a vacuum of
P = 1.3⫻ 10−3 Pa at a temperature of 1770 K for 70 h with
intermediate grinding after every 20 h. After the synthesis,
the composition of the titanium monoxide TiOy specimens
was determined by chemical analysis. Most TiOy specimens
synthesized by this method contained both phases, the ordered and the disordered phases, simultaneously in the assynthesized state.
To obtain fully disordered titanium monoxide with a random distribution of atoms and vacancies on their sublattices,
the TiOy specimens in the as-prepared state were heated in
FIG. 3. Experimentally determined positron lifetime ␶free in the
free, delocalized state 共black circles兲 and trapped at vacancies 共in
metal sublattices, black square; in nonmetal sublattices, open
square兲 as a function of the electron density ␳ for selected elements
and compounds with covalent bonding. The data and references are
listed in Table II. For details on the calculation of the values for the
electron density, ␳, see Ref. 1. According to these curves, a positron
lifetime in the delocalized state for titanium monoxide with a valence electron density of 538 e− / nm3 is expected of about 140 ps.
The positron lifetime of about 170 ps is expected according to this
relation in the case of positrons trapped by oxygen vacancies in
titanium monoxide. Lines are given to guide the eye.
evacuated quartz tubes at 1330 K for 3 h and then quenched
in water with a rate of 200 K / s. For the preparation of fully
ordered specimens with structural vacancies, the synthesized
TiOy specimens were also annealed in an evacuated quartz
tube at 1330 K for 3 h but then slowly cooled to 300 K with
a rate of 10 K / h.
The titanium and oxygen content in the specimens were
determined from weight gain resulting from oxidation to
TiO2. The oxidation was run in air to constant weight during
TABLE I. Lifetimes of positrons in the delocalized state, in the vacancy on the metal sublattice and in the
vacancy on the nonmetal sublattice for titanium monoxide 共this work兲 and selected transition-metal compounds taken from the literature. Values derived from theoretical calculations are marked by *.
Substance
TiO0.74-TiO1.26
TiO0.81-TiO1.26
Ti
Crystal
structure
B1
C2 / m
hcp
TiO2
TiC
tetr.
B1
WC
hex.
VC0.87
B1
Positron lifetime ␶ 共ps兲 in
delocalized state
共147± 5兲
132– 153*
共148± 4兲
98– 107*
共124± 10兲
95*
98*
metal vacancy
nonmetal vacancy
Reference
共160± 2兲
124– 131*
共136± 3兲
116*
共157± 2兲
121*
this work
this work
17
18
13
14 and 16
19
15
20
16
19
184– 210
180– 205
206– 228*
160– 161*
共175± 20兲
161*
共172± 1兲
147*
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PHYSICAL REVIEW B 75, 094107 共2007兲
VACANCIES ON THE Ti SUBLATTICE IN TITANIUM…
TABLE II. Experimental and theoretical 共*兲 lifetimes of positrons in the free, delocalized state, ␶free, and trapped at vacancies on the A
sublattice, ␶共VA兲, and on the B sublattice, ␶共VB兲, of TiO 共this work兲 in comparison with selected nonmetallic AnBm compounds. Additional
information necessary to derive the relation between the electron density, ␳e1, and the positron lifetimes as shown in Fig. 4 are given in Ref.
1; here, ne1 is the number of valence electrons per element, N is the number of atoms per unit cell, a , c are the lattice constants, and V is the
unit cell volume.
A nB m
ne1
N
a , c 共nm兲
V 共nm3兲
␳e1 共e− / nm3兲
␶free, expt. 共ps兲
TiO
Al2O3
4, 6
3, 6
a = 0.4204a
a = 0.4760
c = 1.2993
0.0743
0.2550
538
565
⬃共140± 10兲b
共139± 2兲c
共147± 1兲d
MgO
2, 6
4,4
30
共six molecules
of Al2O3兲
8
共Mg-4 , O-4兲
a = 0.4216
0.0749
427
130h
166i
共152± 15兲e
166*j
121– 131*h
4
8
a = 0.3567
0.0454
705
90– 92*l
4, 4
12
共Si-6 , C-6兲
8
a = 0.3081
c = 1.5116
a = 0.5431
0.1242
386
0.1602
200
4, 4
8
共Ga-4 , As-4兲
a = 0.5653
0.1807
177
共107± 1兲k
共130± 10兲m
146m
共144± 2兲k
共219± 1兲m
共218± 1兲k
230q
4
8
a = 0.5658
0.1811
177
230u
226*j
C
共diamond兲
SiC
Si
GaAs
4
Ge
␶free, theor. 共ps兲
141*n
221*o
229*j
␶共VA兲 共ps兲
␶共VB兲 共ps兲
184– 210b
共160± 20兲e
共137± 2兲f
共223± 8兲g
200i
共205± 10兲e
266*j
180h
146*j
⬃共170± 10兲b
共176± 5兲m
192k
254*o
共266± 1兲p
共295± 5兲r
260– 280t
265*j
263*j
共153± 2兲m
160k
295s
268*j
aReference
39.
work.
c
Reference 21.
d
Reference 22.
eReference 23.
fReference 24.
gReference 25.
hReference 26.
iReference 27.
j
Reference 28.
kReference 29.
lReference
bThis
m
heating to 1200 K over a period of 5 – 8 h in a Q-1500D
thermoanalytic system.
The true density of the specimens was measured by pycnometric method. The measurements were made on fineparticle powders 共3 – 5 ␮m兲 in 1 cm3 pycnometers at a controlled temperature of 298 K. The liquids used for analysis
were water and high-purity kerosene with a density of
0.7886 g / cm3. The relative error of pycnometric density
measurements did not exceed 0.8%.
X-ray-diffraction studies of titanium monoxide were carried out with a Siemens D-500 automatic diffractometer in
the Bragg-Brentano geometry 关Cu k␣1,2 radiation, 2␪ in the
range 4 ° – 160°, step-scan mode with a step size of
⌬共2␪兲 = 0.025°, and an exposure time of 13 s at each point兴.
The concentrations of vacancies on the metal and nonmetal
TiOy sublattices were calculated based on the data on pycnometric density, lattice constant, and chemical composition.
X-ray-diffraction studies of these annealed specimens
showed that the specimens exclusively contained the ordered
monoclinic phase Ti5O5. A detailed description of the preparation method of the disordered and the ordered specimens
and the descriptions of the characterization techniques are
given elsewhere.4
For the positron annihilation studies, a 22NaCl positron
source with an activity of about 1 – 2 MBq on a 0.8 ␮m Al
foil stacked between two identical TiOy specimen plates was
used. The positron lifetime spectra measured by means of a
fast-slow ␥␥ spectrometer with a time resolution of 205 ps
关full width at half-maximum 共FWHM兲兴 and a total number of
coincidence counts of 共1 – 2兲 ⫻ 106 were numerically evaluated by multicomponent fits.9
The Doppler broadening experiments were performed by
coincident measurement of the energies E1 and E2 of the two
positron-electron annihilation photons with two Ge detectors
of high-energy resolution 共FWHM= 1.2 keV at 511 keV兲.
From the two-dimensional E1, E2 spectra with about
3 ⫻ 107 total coincident counts, the Doppler broadened spectra were obtained from a cut along the energy conservation
diagonal E1 + E2 = 1022 keV with a 1 keV energy width.10,11
The spectra were obtained with a high signal-to-noise ratio
⬎105 with optimum statistics up to high electron momenta
from 22 to 50⫻ 10−3 m0c, which is characteristic for positron
annihilation with core electrons. In this range, the momen-
30.
Reference 12.
nReference 31.
oReference 32.
pReference 33.
q
Reference 34.
rReference 35.
sReference 36.
tReference 37.
u
Reference 38.
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PHYSICAL REVIEW B 75, 094107 共2007兲
VALEEVA et al.
VO = 1 − z = 1 − xy = 1 −
FIG. 4. Linear behavior of the positron lifetime ␶ in a
single atomic vacancy vs the vacancy volume ⍀ in disordered
titanium monoxides TiOy 共0.74⬍ y ⬍ 1.26兲 共open circles兲, ordered
titanium monoxides TiOy 共0.81⬍ y ⬍ 1.26兲 共closed circles兲,
titanium carbides TiCy 共0.5⬍ y ⬍ 1.0兲, and niobium carbides
NbCy 共0.72⬍ y ⬍ 1.0兲. The measurements of the positron lifetime
were performed on compounds with the same B1 basic structure.
The variation of the nonmetal content y in the compounds leads to
a change in the lattice constant aB1 and the mean atomic volume
⍀ = a3B1 / 8.
tum distribution substantially differs for different elements
and the technique can therefore be employed for an atomicscale chemical analysis of vacancy surroundings.
In order to facilitate the analysis of the Doppler broadening measurements on TiOy, measurements on the pure components Ti and O2 were performed with the Ti measurements
carried out in the sandwich geometry used for the TiOy studies. For the studies of oxygen, a 22NaCl positron source enclosed in a 5 ␮m Mylar foil was immersed into liquid O2
cooled by liquid nitrogen.
RESULTS AND DISCUSSION
For disordered TiOy, the B1 structure has been determined
by x-ray diffraction with the lattice constant aB1 decreasing
from 0.4195 to 0.4166 nm with increasing oxygen content, y,
which is in accordance with earlier data.2,3 In Fig. 2, the
dependence of the concentration on titanium VTi and oxygen
VO vacancies on the oxygen content y for quenched, disordered titanium monoxide TiOy as derived from the x-ray and
density measurements is shown. An increase of the oxygen
content y leads to an increase of the titanium vacancy concentration and to a decrease of the oxygen vacancy concentration.
Assuming that there are no antisite atoms and no titanium
or oxygen atoms on tetrahedral interstitial positions, the titanium and oxygen vacancies content in titanium monoxide
Tix䊐1−xOz䊏1−z are calculated as
VTi = 1 − x = 1 −
3
CTi
daB1
,
NmuATi
3
yCO
daB1
,
NmuAO
where aB1 is the crystal lattice constant from x-ray diffraction, d is the mass density from pycnometric measurements,
N = 4 is the number of sites in the crystal unit cell, mu
= 1.66⫻ 10−27 kg is the atomic mass constant, ATi, AO are the
atomic weights of titanium and oxygen, and CTi, CO are the
mass weights from chemical analysis.
The positron lifetime ␶ of disordered as well as ordered
titanium monoxide was found to increase with increasing
oxygen content. For disordered titanium monoxide, with the
B1 structure, the positron lifetime varies from 184 to 210 ps
in the composition range from TiO0.74 to TiO1.26 共Table I兲.
The positron lifetime is lower in ordered titanium monoxide
by ⬃3 ps for y ⬎ 1.00 and by ⬃5 ps for y ⬍ 1.00.
The positron lifetimes in defect-free, covalently bonded
solids in dependence of the valence electron densities are
given in Fig. 3 as derived according to the data in Table II.
For titanium monoxide with a valence electron density of
538 e− / nm3, a positron lifetime in the delocalized state of
about 140 ps is expected according to this relation. The positron lifetimes measured for TiOy 共see Fig. 4兲 are substantially higher than this value. Therefore, the higher values in
the range from 180 to 210 ps for titanium monoxide of the
present study demonstrate the presence of structural vacancies in TiOy.
In Fig. 4, positron lifetimes ␶ in a monovacancy versus
the mean atomic volume ⍀ for nonstoichiometric, disordered
titanium monoxides 共present work兲, for titanium carbides,16
and niobium carbides14 are shown that follow a linear relationship, however with different slopes for the different materials. A comparison of experimental and calculated positron
lifetimes in the free state-and in the localized states, i.e., the
metal and the nonmetal vacancy states are given in Table I,
additionally.
The increase of the vacancy volume ⍀ in carbides leads to
an increase of the positron lifetime 共Fig. 4兲. In carbides, the
change of the positron lifetime with the mean atomic volume
resembles the behavior of pure metals. Indeed, applying external hydrostatic pressure to a metallic system leads to a
decrease of the vacancy volume ⍀ due to the decrease of the
lattice constant, which enhances the valence electron density
and decreases the positron lifetime due to the increased probability of an electron-positron annihilation. In the case of
carbides, assuming that the valence electron density of
nearest-neighbor metal atoms extends into the carbon vacancy inversely proportional to the mean atomic volume
where the charge of that vacancy should be for a given carbide unchanged with pressure. The different slopes of the
positron lifetimes in dependence of the atomic volume for
titanium carbide and for niobium carbide 共Fig. 4兲 arise from
the different number of valence electrons of metal atoms.
The slope for titanium carbide is higher than the slope for
niobium carbide because titanium has one valence electron
less than niobium.14
In the case of titanium monoxide, the relation between the
positron lifetime and the atomic volume has a negative slope
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PHYSICAL REVIEW B 75, 094107 共2007兲
VACANCIES ON THE Ti SUBLATTICE IN TITANIUM…
FIG. 6. Ratios of Doppler broadened spectra for disordered
TiO0.99 共q兲, TiO1.07 共q兲 共sub- and superstoichiometric兲, and ordered
TiO0.99 共a兲 to the Doppler broadened spectrum of ordered TiO1.07
共a兲. In the range from 10 to 25⫻ 10−3 m0c, the curves differ significantly for ordered and disordered states.
FIG. 5. Area normalized coincident Doppler broadening spectra
of pure titanium Ti and liquid oxygen O2. 共b兲 Ratios of Doppler
broadened spectra for disordered titanium monoxide,TiO0.99 共q兲, titanium, and liquid oxygen to the spectrum of ordered titanium monoxide, TiO0.99共a兲. In the high electron momentum range which is
solely characteristic for the core electrons, the curves for disordered
and ordered TiO0.99 resemble the curve of liquid oxygen and are
quite different from the curve for titanium indicating an oxygendominated environment of the vacancies in TiO0.99.
in contrast to carbides with a positive slope 共see Fig. 4兲.
Because the positron lifetime in a vacancy is nearly inversely
proportional to metal valence electron density on that vacancy in monoxide, the main reason for the negative slope
could be an increase of the nearest-neighbor oxygen atoms in
the vicinity of the titanium vacancy as a result of an increase
of oxygen content in the monoxide. The more oxygen
nearest-neighbor atoms, the more screening of metal valence
electron density on titanium vacancy, the less valence electron density on titanium vacancy, the longer positron lifetime
in monoxide.
The coincident Doppler broadening data 共Fig. 5兲 at high
electron momenta 关共25– 40兲 ⫻ 10−3 m0c兴 are for TiO0.99 共a兲
and for TiO1.07 共a兲 similar to those of oxygen and differ
substantially from that of pure Ti 共steep slope兲. This demonstrates that the positrons are annihilated in vacancies with an
oxygen surrounding, i.e., in vacancies on the Ti sublattice.
The Doppler broadening data in Fig. 6 show that the oxygen surroundings of the vacancies in the two ordered crystals
TiO0.99 共a兲 and TiO1.07 共a兲 with different compositions are
rather similar. However, these oxygen surroundings vary
slightly with composition approaching a more oxygenlike
Doppler broadening spectrum when the oxygen content is
increased.
As sketched in Fig. 1, there are different titanium vacancy
positions in nonstoichiometric titanium monoxide depending
on the oxygen content y in TiOy and degree of order. Considering the ratio curves, it is presumably that ordered monoxides TiO0.99 共a兲 and TiO1.07 共a兲 are arranged by six oxygen
atoms in the neighborhood of the titanium vacancies 关Fig.
1共a兲兴, and in case of disordered titanium monoxide TiO0.99
共q兲, TiO1.07 共q兲 the number of oxygen atoms in the fist coordination shell is lower 关Figs. 1共b兲–1共f兲兴. Titanium monoxide
with lower oxygen content contains more oxygen vacancies,
and comparing the ratio spectra of disordered monoxides
TiO0.99 共q兲 and TiO1.07 共q兲 suggests that the disordered
TiO0.99 共q兲 has more oxygen vacancies around the titanium
vacancy 关see Figs. 1共c兲 and 1共d兲兴 than disordered monoxide
TiO1.07 共q兲.
CONCLUSION
In the present work, the sensitive local probing techniques
of positron lifetime and coincident Doppler broadening spectroscopy were employed for the study of structural vacancies
in disordered and ordered titanium monoxide. This compound has structural vacancies that are possibly located on
both the Ti and O sublattices. The observation that predominantly vacancies on the Ti sublattice are detected in the
present positron annihilation studies indicates that positrons
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PHYSICAL REVIEW B 75, 094107 共2007兲
VALEEVA et al.
are preferentially trapped at vacancies on the cation sublattice as anticipated from their negative electronic charge. This
directly demonstrates that in titanium monoxide, structural
vacancies are available on the Ti sublattice. Positron trapping
at oxygen vacancies is unlikely because of the repulsion
from their positive apparent charge.
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ACKNOWLEDGMENTS
Financial support 共A.A.V.兲 from the INTAS 共Ref. No. 0355-913兲, RFBR 共Ref. No. 07-03-00040 a兲, and President
Grant 共Ref. MK-1054.2007.3兲 are acknowledged. The authors thank E. Partyka for her support with the positron annihilation experiments.
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