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Phase diagram and electrical properties, defect concentration in oxides and sulphides of 3d metals
SOLID STATE IONICS
Solid State Ionics 91 (1996) 315-322
ELSEVIER
Phase diagram and electrical properties, defect concentration oxides and sulphides of 3d metals Andrzej Stoklosa”,
in
Janusz Zajecki
Institute of Chemical Engineering and Physical Chemistry, Cracow University of Technology, ul. Warszawska 24, 31-155 Krako’w, Poland Received
1 April 1996; accepted
19 June 1996
Abstract The analysis of the equilibrium pressure function of oxide and sulphide of 3d metals and their phases was carried out in order to determine the correlation between them and their properties. The extent of the chemical potential changes of the oxidant in the phase stability range was determined and these changes were correlated with the deviations from stoichiometry and electrical properties of the metal oxides and sulphides. It was shown that the changes in the chemical potential of the oxidant in the phase stability range for the compounds exhibiting semiconducting properties could be quantitatively correlated with the changes in the Fermi level which might provide new information on the changes in electronic structure. Keywords: Phase diagram;
Defect concentration;
3d metal oxides; 3d metal sulphides:
1. Introduction The stability range of an oxide or a sulphide, more precisely of a given phase, is determined by the range of temperature and the pressure of the oxidant with which the phase under consideration exists in equilibrium. A phase diagram can be drawn provided the standard Gibbs free energies for the equilibrium between a metal and its oxide (or sulphide) phase or between the given phases in equilibrium are known. The minimum pressure at which a given oxide or sulphide is formed at a given temperature, as well as the equilibrium pressure for coexisting phases can be determined by employing these diagrams. The analysis of phase diagrams shows that some compounds
*Corresponding
author.
0167-2738/96/$15.00 Copyright PII SOl67-2738(96)00454-7
01996
Thermodynamics
have very narrow stability ranges, whereas other ones cover several orders of the magnitude of pressure. The magnitude as well as the range of the oxidant pressures, at which the given phase exists, undoubtedly depend on the metal and the oxidant forming a compound. Simple oxides and sulphides and even phases of oxides or sulphides of the same metal differ in the magnitude of deviations from stoichiometry, in electrical properties, hence in the type and concentration of ionic and electronic defects. As yet there is no quantitative correlation between the thermodynamic functions and the above mentioned phase properties known. In the present work, an attempt to determine correlations between the phase stability range and its electrical properties and the deviation from stoichiometry is presented on the example of oxides and sulphides of 3d metals.
Elsevier Science B.V. All rights reserved
316
A. Stokbsa, J. Zujgcki 1 Solid State tonics 91 (1996) 315-322
2. The stability range and the properties phase
of the 10
In order to determine correlations between phase diagrams and electrical properties, thermodynamic data for oxides and sulphides of metals were collected and analysed in terms of their properties. Equilibrium pressures for oxides and sulphides in equilibrium with metals are presented in Fig. 1 and Fig. 2 [l-4]. As can be seen, in the case of oxides, the sequence of equilibrium curves follows the order of the number of d electrons in the metal. Exceptions are zinc, which is located between manganese and iron, and scandium which does not form the MO type oxide like chromium. It appears interesting that the curves extrapolated into high temperature regions (beyond melting points) cross roughly at one point at the pressures of several atmospheres. Small deviations observed may probably result from limited accuracy of the determination of equilibrium equations. A similar image is obtained for the sulphides. As can be seen, in the sulphide system, the situation is more complex. At the equilibrium with the metal compounds of various composition exist. Particularly significant is high equilibrium pressure in the system Cr/CrS. In the case of sulphides the effect of
zoo0
-100
0
0.0005
r 500
IWO
0.0015
0.001
l/T Fig. 1. Dependence oxide/metal [ 1,2].
of equilibrium
T lK1 I
I
Ix1
0.002
400
0.0025
IlKI
pressure on 1 IT in the system
-80
0
2000
1004
0.0005
0.001
500
0.0015
0.002
l/T Fig. 2. Dependence of equilibrium sulphide/metal [1,4].
400
pressure on l/T
0.0025
lIK1 in the system
intersection of the equilibrium curves at one common point is not so evidently observed. It can generally be stated that equilibrium pressures for M/MS are significantly higher than for the corresponding oxides. The described minimum equilibrium pressure at which a given phase exists, more precisely the chemical potential of the oxidant indicates its lattice energy or generally the interaction force between atoms. According to the physical sense of the chemical potential it is the work required to insert or remove of an oxidant atom or mole from the compound. The curves presented in Fig. 1 and Fig. 2, as already mentioned, determine the minimum pressure of the oxidant at which a given oxide or sulphide exists. When the equilibrium pressure of the oxidant rises, so the chemical potential of oxygen or sulphur in its compounds also increases. This rise proceeds until the next equilibrium pressure is achieved at which the next phase of a given oxide (sulphide) can exist. This fact is presented in Fig. 3 exemplified by Co-O. As a result of the oxygen pressure rise above the pressure pb, the oxygen activity in Co0 also increases. The increase in oxygen activity, as showed in many studies [5], results from the formation of cationic vacancies in the compound, at the same
A. Stoklosa, J. Zajgcki I Solid State lonics 91 (1996) 315-322
0
\ -20 g 0" -lo ,a 9 -60
-80
0.001
0.0015
0.002
UT Fig. 3. Dependence of the equilibrium in the system Co-O [I].
0.0025
0.003
0.0035
11Kl pressure of oxygen on 1 lT
time, in the oxide, the concentration of electronic holes rises. So, together with the oxygen pressure increase the rise in concentration of ionic and electronic defects occurs. In cobaltous oxide, the maximum concentration of ionic and electronic defects is achieved at the equilibrium pressure p& corresponding to the CoO/Co,O, equilibrium. It is more correct to use the term chemical potential instead of the activity of a component or point defects. So, the change of the chemical potential of the oxidant in the phase stability range is proportional to the difference of the chemical potential of components, the difference of the chemical potential of ionic and electronic defects, which can be written as follows:
The change in the potential of electronic defects is approximately equal to the change in the Fermi level of electrons. By analogy, the chemical potential of ionic defects can be termed the Fermi level of ions [6]. So, the change of the Fermi level of ions in the phase stability range can also be discussed. A question arises, why a phase transition follows the rise in the oxidant pressure and how it is related to the Fermi level of both the ions and electrons.
317
According to the above discussion, the rise in the oxidant pressure forces a determined change in the chemical potential of the oxidant in the compound, which is caused by the formation of ionic defects of a determined type. They could be cationic or anionic vacancies as well as cations or anions in interstitial sites. For the majority of compounds at equilibrium with a metal, the dominating defects are cationic vacancies, so the increase in the oxidant pressure will bring about the rise in the concentration of the vacancies. For example, in zinc oxide the dominating defects are ions in interstitial sites, so their concentration would be the highest at the lowest pressure of the oxidant and it would decrease with the increase of the pressure. In the case of anionic sublattice defects, the maximum concentration of vacancies would occur at the lowest pressure of the oxidant. It would decrease with the increase in the oxidant pressure and the concentration of anions in interstitial sites would rise with the increase of the pressure. Thus the type of the dominating defects and their determined concentration corresponding to a given equilibrium state (equilibrium pressure at a given temperature) would depend on the type of the compound (metal or non-metal) and its crystallographic structure. The formation of defects can occur only in a determined range. If the concentration of the defects increases beyond a determined limit, so the interaction between atoms would be weakened, a phase transition will occur, related to a new order of atoms, symmetry and point group changes etc. As mentioned above, in a crystal, the given type of ionic defects must be accompanied by electronic defects. The compound must have such an electronic structure which, in case the changes in the concentration of ionic defects occur, could accept electronic defects. It can be concluded that compounds with metallic properties, characterized by a continuous state density function and their high density will be able to exhibit larger deviation from stoichiometry. On the other hand compounds having a semiconductor structure, unless the transition into the metallic state occurs during the formation of defects, would show small deviation from stoichiometry. Thus the extent to which (in the phase stability range) the defects are formed depends on the bonding energy and its possible changes caused by the changes in the concentration of ionic defects, as well
A. Stoklosa, J. Zajgcki I Solid State Ionics 91 (1996) 315-322
318
as on the electronic structure and its ability to modification caused by changes in metal-metal distances (as a result of the lattice parameter changes). As results from Eq. (1) the variation in the oxidant equilibrium pressures in the phase stability range can be related to the variation in the chemical potential of the ionic and electronic defects, hence with the variation in the Fermi level. In the case of oxides or sulphides with simple defect structure a quantitative correlation between the above given values can be obtained. For example, in the described case of cobaltous oxide, in which single ionized cationic vacancies can be assumed the dominating defects, the defect reaction can be written as follows:
Applying Eq. (5), the change of the potential of oxygen can be expressed as: = RT(ln APO2
chemical
pi2 - In pb2) = RTAln po,
= 4RTAln[V~,]
= 4RTAln[h’] = 4AEr.
In the case under consideration,
(6)
the changes of the
(2) The equilibrium
constant
can be expressed
as:
K = [V;,][h~]/&‘. Applying
(3)
the electroneutrality
condition:
fh.1 = [V;,l
(4)
I
-soo’
we obtain
700
“0 1100
1500
1900
2300
[Kl
[h‘] = [V&J = K&4.
(5)
320 -a
0
6CoCHO,=ZCo,O,
0
-440
400
600
800
T
1000
1200
120
1100
2300
lK1
WI
4. Temperature dependence of the free enthalpy of formation of cobalt oxides and equilibrium CoO/Co,O, (left axis) [l] and the changes of the chemical potential of oxygen within the stability range of the oxide Co,_,0 (right axis).
Fig. 5. (a) Temperature dependence of the free enthalpy of the formation of VO and equilibrium between vanadium oxides [3]. (b) Temperature dependence of the changes of the chemical potential of oxygen within the stability range of the oxide for V-O system.
319
A. Stokiosa, J. Zajgcki I Solid State Ionics 91 (1996) 315-322
corresponding to the change in the chemical potential of the oxidant within the Co,_,0 phase range. For example, at the temperature of 1100 K, the change of the chemical potential of oxygen within the Co0 stability range equals 270 kJ, which in the assumed case of single ionized cationic vacancies brings about a change in the Fermi level by 0.7 eV. Analogous calculations under the assumption of the domination of double ionized vacancies yield the value of 0.35 eV.
chemical potential of ionic and electronic defects, i.e. the magnitude of the changes of their Fermi level, can be exactly correlated with the equilibrium pressures of the oxidant (within the phase stability range). The left-hand axis of Fig. 4 represents the change of the free enthalpy of the formation of cobalt oxides expressed in terms of one mole of oxygen, i.e. the temperature dependence of ,L+,*.The right-hand axis of Fig. 4 shows the difference of those two functions,
400
600
800
1000
1200
1400
T
-loo0
1600
1
400
800
[Kl
400
1200
T
800
1200
T
1600
2
[Kl
1600
[Kl
Fig. 6. (a)(b) Temperature dependence of the standard free enthalpy of formation for selected oxides and equilibrium between the oxides [1,2]. (c) Temperature dependence of the changes of the chemical potential of oxygen within the stability range of the oxide for selected oxides.
320
A. Stokiosa,
J. Zajgcki
I Solid State Ionics 91 (1996) 315-322
Generally, a change in the structure of defects occurs with the increase in the oxidant pressure in the most systems, many compounds though exhibit metallic properties, thus such rigorous calculations cannot be made for these compounds. For many systems though, within certain ranges or even within the entire stability range, the dependence of the deviation from stoichiometry as well as the electrical conductivity on the oxidant pressure can be described to a good approximation with the following equations:
E,, the variation range of the experimentally determined values of the exponent l/m in Eq. (7) [5,7-111 are summarized, as well as maximum deviations from stoichiometry (the variations of X/ Me) for the discussed oxides and sulphides in the range of high temperatures. Comparing values summarized in Table 1 and calculated changes Ak2 presented in Fig. 5b, Fig. 6c and Fig. 7b, it can be
lny=llnlnpx2, (7) Therefore, the changes in the Fermi level of ions and electrons can be determined at the assumption of the types of defects or the knowledge of the n and m coefficients in the Eq. (7) by calculating the changes in the chemical potential of the oxidant from thermodynamic data (according to the analogous Eq. (6)). It results from the above considerations that there should exist a correlation between the electrical properties and a change in the potential of the oxidant within the stability range of a given phase. In the case of semiconductors, according to the band model, the variation of the Fermi level with temperature, and hence the variation in the chemical potential of the oxidant within a phase stability range should be relatively significant and increase with temperature, since the maximum concentration of defects increases with temperature in general. In semiconductors, the Fermi level should shift within the energy gap depending on the concentration of electronic defects. In turn, in metallic compounds, the variation of the Fermi level should be small and rather independent of temperature. In order to confirm the above general considerations, thermodynamic data were used and the standard free enthalpies AGO for the several equilibria between the phases in 3d metal oxides and sulphides were calculated, which is depicted in Fig. 5, Fig. 6a,b and Fig. 7a. In turn, in Fig. 5b, Fig. 6c and Fig. 7b, the variations of the changes of the chemical potential of oxidant in the stability range of given oxide and sulphide phases are presented. In Table 1 the metallic (M) and semiconducting (SC) properties, the magnitude of the forbidden gap
’
-300 400
lkl
T
., .. .,,/
co,s8
CG4
1200
loo0
800
600
CG3.X
O600
1
700
800
900
1000
T
1100
1200
1300
1x1
Fig. 7. (a) Temperature dependence of the standard free enthalpy of formation for selected sulphides and equilibrium between the oxides [ 1.41. (b) Temperature dependence of the changes of the chemical potential of sulphur within the stability range of the sulphide for selected sulphides.
A. Stokiosa, J. Zajgcki I Solid State Ionics 91 (1996) 315-322 Table 1 Electrical sulphides
properties,
Compound
TiO Ti,O, Ti,O TiO &-’ vo2 VA V”L, VO* VA VA MnO Mn,O, MnA Fe0 Fe@, coo Co@, NiO cu,o cue ZnO M. =Metallic
forbidden
Electrical properties M. M. M. SC. M. M. M. M. M. SC. SC. SC S.C. SC. S.C. S.C. SC SC. S.C. S.C. S.C. properties;
gap E,, coefficient
E, (eV)
I/m
1 lm (Eq. (7)) and the deviations
O/Me
Compound
0.69-1.33
3.05
-l/2-115
TiS VS CrS MnS FeS
0.98-l 0.71-1.29 1.50-1.52 1.995-2.002 2.165-2.175
2.5 3.5
1.8 2.7 4.0 2.0 2.2 3.2
1/2-l/6
l/2-1/14 l/30 I/4-1/5 l/4-116 118 -114
SC. = semiconducting
I .052- 1.127 Small Small 0.85-0.95 1.334- 1.333 1.02-l.ooo Small l.OOl-l.ooo 0.502-0.500 Small 0.999-l IHI0 properties.
Sources:
concluded that semiconducting properties are exhibited by oxides for which Apo2 is higher than about 100 kJ/mol, and the greater the forbidden gap the higher the values of the changes tof the chemical potential of oxygen. It should be stressed that for the compounds exhibiting semiconducting properties, such as TiO,, NiO, MnS, which are not in equilibrium with their higher oxides, the change in the chemical potential of the oxidant with changing the pressure from the equilibrium value to 1 atm is much bigger than 200 kJ which is depicted in Fig. 8 and Fig. 9. In order to calculate the change in the chemical potential of electronic defects, the magnitude of which would characterize a given compound, the value of the parameter m should be known over the entire phase stability range. For the compounds with semiconducting properties this value is generally small and ranges from 4 to 8. It results both from the defect theory and the experimental studies [5,7]. In the case of metallic compounds the magnitude of the changes of the chemical potential reaches 250 W. The parameter m, as results from numerous measure-
Refs. [5,7-l
from stoichiometry
Electrical properties M. M. S.C. SC. M. M. M. M. M. M. S.C. I M. M. S.C.
Co& Co%& cos Ni,S, NiS cu,s cus zns
321
X/M
E,
for selected
oxides and
l/f?!
S/Me
(eV)
3.2
l/5-1/6 1120
l/100
2.3
l/200 l/3-1/6
3.1
0.98-1.10 0.85-1.15 1.12-1.48 1.00-1.000 1.12-1.00 0.660-0.818 1.16-1.0 0.56-0.79 1.15-1.0 0.53-0.50 Small
11.
ments of electrical conductivity, takes significant values ranging from 20 to over 100. Thus the changes in the Fermi level obtained by dividing Ah2 by the parameter m will be insignificant.
T Fig. 8. Temperature dependence for selected oxides [1,2].
of the free enthalpy
[Kl of formation
322
A. Stoklosa, J. Zajecki
I Solid State Ionics 91 (1996)
within the stability range of a given phase are correlated with the maximum value of the deviation from stoichiometry and its electrical properties. The full correlation can be found when the character of the dependence of the concentration of ionic and electronic defects on the oxidant pressure is known.
100
z E
0
2 2
315-322
-100
B E 2
-200
6 E
-300
NiS MnS ,..“.....
Acknowledgments ‘...._,_
,:’
& ”
The research has been carried out under contract with the Polish Science Research Committee (KBN).
-IO0
% -500 400
600
800
1000
I200
T Fig. 9. Temperature dependence for selected sulphides [ 1.41.
of the free enthalpy
1400
[Kl
References
of formation
From the data summarized in Table 1 results also that the compounds with metallic properties exhibit substantial deviation from stoichiometry. As results from the conducted calculations, the changes in the chemical potential of oxidant in the stability range of a given phase could be applied for the semiquantitative, and in some cases even quantitative calculation of the changes in the Fermi level of electrons or the ‘Fermi level’ of ionic defects.
3. Conclusions Analysis of phase diagrams of a selection of oxides and sulphides of 3d metals demonstrated that the changes in the chemical potential of the oxidant
[I] J. Barin and 0. Knacke, Thermochemical Properties of Inorganic Substances (Springer-Verlag, Berlin, 1973, Supplement 1977). [2] L.B. Pankratz, Thermodynamic Properties of Elements and Oxides, Bulletin 672 (United States, Bureau of Mines, Washington, 1982). [3] H. Oppermann, Vanadiumoxide (Akademie-Verlag, Berlin, 1983). [4] S.R. Shatynski, Oxid Metals 11 (1977) 307. [5] P Kofstad, Nonstoichiometry, Diffusion and Electrical Conductivity in Binary Metal Oxide (Wiley, New York, 1972). [6] J.M. Blakely, in: Electrode Process in Solid State Ionics, eds. M. Kleitz and J. Dupuy (Reidel, Dordrecht, 1976) p. 83. [7] S. Mrowec, Defect and Diffusion in Solids (PWN-Elsevier, Warszawa-Amsterdam, 1980). [8] S. Mrowec and K. Przybylski, High Temp. Mater. Process. 6 (1984) 1. [9] J.B. Goodenough, in: Progress in Solid State Chem. Vol. 5, ed. H. Reiss (Pergamon Press, Oxford, 1971) p. 145. [lo] J.B. Goodenough, in: Solid State Chemistry, ed. C.N.R. Rao (Marcel Dekker, New York, 1974) p. 215. [ll] F. Jellinek, React. Solids 5 (1988) 323.
Solid State Ionics 91 (1996) 315-322
ELSEVIER
Phase diagram and electrical properties, defect concentration oxides and sulphides of 3d metals Andrzej Stoklosa”,
in
Janusz Zajecki
Institute of Chemical Engineering and Physical Chemistry, Cracow University of Technology, ul. Warszawska 24, 31-155 Krako’w, Poland Received
1 April 1996; accepted
19 June 1996
Abstract The analysis of the equilibrium pressure function of oxide and sulphide of 3d metals and their phases was carried out in order to determine the correlation between them and their properties. The extent of the chemical potential changes of the oxidant in the phase stability range was determined and these changes were correlated with the deviations from stoichiometry and electrical properties of the metal oxides and sulphides. It was shown that the changes in the chemical potential of the oxidant in the phase stability range for the compounds exhibiting semiconducting properties could be quantitatively correlated with the changes in the Fermi level which might provide new information on the changes in electronic structure. Keywords: Phase diagram;
Defect concentration;
3d metal oxides; 3d metal sulphides:
1. Introduction The stability range of an oxide or a sulphide, more precisely of a given phase, is determined by the range of temperature and the pressure of the oxidant with which the phase under consideration exists in equilibrium. A phase diagram can be drawn provided the standard Gibbs free energies for the equilibrium between a metal and its oxide (or sulphide) phase or between the given phases in equilibrium are known. The minimum pressure at which a given oxide or sulphide is formed at a given temperature, as well as the equilibrium pressure for coexisting phases can be determined by employing these diagrams. The analysis of phase diagrams shows that some compounds
*Corresponding
author.
0167-2738/96/$15.00 Copyright PII SOl67-2738(96)00454-7
01996
Thermodynamics
have very narrow stability ranges, whereas other ones cover several orders of the magnitude of pressure. The magnitude as well as the range of the oxidant pressures, at which the given phase exists, undoubtedly depend on the metal and the oxidant forming a compound. Simple oxides and sulphides and even phases of oxides or sulphides of the same metal differ in the magnitude of deviations from stoichiometry, in electrical properties, hence in the type and concentration of ionic and electronic defects. As yet there is no quantitative correlation between the thermodynamic functions and the above mentioned phase properties known. In the present work, an attempt to determine correlations between the phase stability range and its electrical properties and the deviation from stoichiometry is presented on the example of oxides and sulphides of 3d metals.
Elsevier Science B.V. All rights reserved
316
A. Stokbsa, J. Zujgcki 1 Solid State tonics 91 (1996) 315-322
2. The stability range and the properties phase
of the 10
In order to determine correlations between phase diagrams and electrical properties, thermodynamic data for oxides and sulphides of metals were collected and analysed in terms of their properties. Equilibrium pressures for oxides and sulphides in equilibrium with metals are presented in Fig. 1 and Fig. 2 [l-4]. As can be seen, in the case of oxides, the sequence of equilibrium curves follows the order of the number of d electrons in the metal. Exceptions are zinc, which is located between manganese and iron, and scandium which does not form the MO type oxide like chromium. It appears interesting that the curves extrapolated into high temperature regions (beyond melting points) cross roughly at one point at the pressures of several atmospheres. Small deviations observed may probably result from limited accuracy of the determination of equilibrium equations. A similar image is obtained for the sulphides. As can be seen, in the sulphide system, the situation is more complex. At the equilibrium with the metal compounds of various composition exist. Particularly significant is high equilibrium pressure in the system Cr/CrS. In the case of sulphides the effect of
zoo0
-100
0
0.0005
r 500
IWO
0.0015
0.001
l/T Fig. 1. Dependence oxide/metal [ 1,2].
of equilibrium
T lK1 I
I
Ix1
0.002
400
0.0025
IlKI
pressure on 1 IT in the system
-80
0
2000
1004
0.0005
0.001
500
0.0015
0.002
l/T Fig. 2. Dependence of equilibrium sulphide/metal [1,4].
400
pressure on l/T
0.0025
lIK1 in the system
intersection of the equilibrium curves at one common point is not so evidently observed. It can generally be stated that equilibrium pressures for M/MS are significantly higher than for the corresponding oxides. The described minimum equilibrium pressure at which a given phase exists, more precisely the chemical potential of the oxidant indicates its lattice energy or generally the interaction force between atoms. According to the physical sense of the chemical potential it is the work required to insert or remove of an oxidant atom or mole from the compound. The curves presented in Fig. 1 and Fig. 2, as already mentioned, determine the minimum pressure of the oxidant at which a given oxide or sulphide exists. When the equilibrium pressure of the oxidant rises, so the chemical potential of oxygen or sulphur in its compounds also increases. This rise proceeds until the next equilibrium pressure is achieved at which the next phase of a given oxide (sulphide) can exist. This fact is presented in Fig. 3 exemplified by Co-O. As a result of the oxygen pressure rise above the pressure pb, the oxygen activity in Co0 also increases. The increase in oxygen activity, as showed in many studies [5], results from the formation of cationic vacancies in the compound, at the same
A. Stoklosa, J. Zajgcki I Solid State lonics 91 (1996) 315-322
0
\ -20 g 0" -lo ,a 9 -60
-80
0.001
0.0015
0.002
UT Fig. 3. Dependence of the equilibrium in the system Co-O [I].
0.0025
0.003
0.0035
11Kl pressure of oxygen on 1 lT
time, in the oxide, the concentration of electronic holes rises. So, together with the oxygen pressure increase the rise in concentration of ionic and electronic defects occurs. In cobaltous oxide, the maximum concentration of ionic and electronic defects is achieved at the equilibrium pressure p& corresponding to the CoO/Co,O, equilibrium. It is more correct to use the term chemical potential instead of the activity of a component or point defects. So, the change of the chemical potential of the oxidant in the phase stability range is proportional to the difference of the chemical potential of components, the difference of the chemical potential of ionic and electronic defects, which can be written as follows:
The change in the potential of electronic defects is approximately equal to the change in the Fermi level of electrons. By analogy, the chemical potential of ionic defects can be termed the Fermi level of ions [6]. So, the change of the Fermi level of ions in the phase stability range can also be discussed. A question arises, why a phase transition follows the rise in the oxidant pressure and how it is related to the Fermi level of both the ions and electrons.
317
According to the above discussion, the rise in the oxidant pressure forces a determined change in the chemical potential of the oxidant in the compound, which is caused by the formation of ionic defects of a determined type. They could be cationic or anionic vacancies as well as cations or anions in interstitial sites. For the majority of compounds at equilibrium with a metal, the dominating defects are cationic vacancies, so the increase in the oxidant pressure will bring about the rise in the concentration of the vacancies. For example, in zinc oxide the dominating defects are ions in interstitial sites, so their concentration would be the highest at the lowest pressure of the oxidant and it would decrease with the increase of the pressure. In the case of anionic sublattice defects, the maximum concentration of vacancies would occur at the lowest pressure of the oxidant. It would decrease with the increase in the oxidant pressure and the concentration of anions in interstitial sites would rise with the increase of the pressure. Thus the type of the dominating defects and their determined concentration corresponding to a given equilibrium state (equilibrium pressure at a given temperature) would depend on the type of the compound (metal or non-metal) and its crystallographic structure. The formation of defects can occur only in a determined range. If the concentration of the defects increases beyond a determined limit, so the interaction between atoms would be weakened, a phase transition will occur, related to a new order of atoms, symmetry and point group changes etc. As mentioned above, in a crystal, the given type of ionic defects must be accompanied by electronic defects. The compound must have such an electronic structure which, in case the changes in the concentration of ionic defects occur, could accept electronic defects. It can be concluded that compounds with metallic properties, characterized by a continuous state density function and their high density will be able to exhibit larger deviation from stoichiometry. On the other hand compounds having a semiconductor structure, unless the transition into the metallic state occurs during the formation of defects, would show small deviation from stoichiometry. Thus the extent to which (in the phase stability range) the defects are formed depends on the bonding energy and its possible changes caused by the changes in the concentration of ionic defects, as well
A. Stoklosa, J. Zajgcki I Solid State Ionics 91 (1996) 315-322
318
as on the electronic structure and its ability to modification caused by changes in metal-metal distances (as a result of the lattice parameter changes). As results from Eq. (1) the variation in the oxidant equilibrium pressures in the phase stability range can be related to the variation in the chemical potential of the ionic and electronic defects, hence with the variation in the Fermi level. In the case of oxides or sulphides with simple defect structure a quantitative correlation between the above given values can be obtained. For example, in the described case of cobaltous oxide, in which single ionized cationic vacancies can be assumed the dominating defects, the defect reaction can be written as follows:
Applying Eq. (5), the change of the potential of oxygen can be expressed as: = RT(ln APO2
chemical
pi2 - In pb2) = RTAln po,
= 4RTAln[V~,]
= 4RTAln[h’] = 4AEr.
In the case under consideration,
(6)
the changes of the
(2) The equilibrium
constant
can be expressed
as:
K = [V;,][h~]/&‘. Applying
(3)
the electroneutrality
condition:
fh.1 = [V;,l
(4)
I
-soo’
we obtain
700
“0 1100
1500
1900
2300
[Kl
[h‘] = [V&J = K&4.
(5)
320 -a
0
6CoCHO,=ZCo,O,
0
-440
400
600
800
T
1000
1200
120
1100
2300
lK1
WI
4. Temperature dependence of the free enthalpy of formation of cobalt oxides and equilibrium CoO/Co,O, (left axis) [l] and the changes of the chemical potential of oxygen within the stability range of the oxide Co,_,0 (right axis).
Fig. 5. (a) Temperature dependence of the free enthalpy of the formation of VO and equilibrium between vanadium oxides [3]. (b) Temperature dependence of the changes of the chemical potential of oxygen within the stability range of the oxide for V-O system.
319
A. Stokiosa, J. Zajgcki I Solid State Ionics 91 (1996) 315-322
corresponding to the change in the chemical potential of the oxidant within the Co,_,0 phase range. For example, at the temperature of 1100 K, the change of the chemical potential of oxygen within the Co0 stability range equals 270 kJ, which in the assumed case of single ionized cationic vacancies brings about a change in the Fermi level by 0.7 eV. Analogous calculations under the assumption of the domination of double ionized vacancies yield the value of 0.35 eV.
chemical potential of ionic and electronic defects, i.e. the magnitude of the changes of their Fermi level, can be exactly correlated with the equilibrium pressures of the oxidant (within the phase stability range). The left-hand axis of Fig. 4 represents the change of the free enthalpy of the formation of cobalt oxides expressed in terms of one mole of oxygen, i.e. the temperature dependence of ,L+,*.The right-hand axis of Fig. 4 shows the difference of those two functions,
400
600
800
1000
1200
1400
T
-loo0
1600
1
400
800
[Kl
400
1200
T
800
1200
T
1600
2
[Kl
1600
[Kl
Fig. 6. (a)(b) Temperature dependence of the standard free enthalpy of formation for selected oxides and equilibrium between the oxides [1,2]. (c) Temperature dependence of the changes of the chemical potential of oxygen within the stability range of the oxide for selected oxides.
320
A. Stokiosa,
J. Zajgcki
I Solid State Ionics 91 (1996) 315-322
Generally, a change in the structure of defects occurs with the increase in the oxidant pressure in the most systems, many compounds though exhibit metallic properties, thus such rigorous calculations cannot be made for these compounds. For many systems though, within certain ranges or even within the entire stability range, the dependence of the deviation from stoichiometry as well as the electrical conductivity on the oxidant pressure can be described to a good approximation with the following equations:
E,, the variation range of the experimentally determined values of the exponent l/m in Eq. (7) [5,7-111 are summarized, as well as maximum deviations from stoichiometry (the variations of X/ Me) for the discussed oxides and sulphides in the range of high temperatures. Comparing values summarized in Table 1 and calculated changes Ak2 presented in Fig. 5b, Fig. 6c and Fig. 7b, it can be
lny=llnlnpx2, (7) Therefore, the changes in the Fermi level of ions and electrons can be determined at the assumption of the types of defects or the knowledge of the n and m coefficients in the Eq. (7) by calculating the changes in the chemical potential of the oxidant from thermodynamic data (according to the analogous Eq. (6)). It results from the above considerations that there should exist a correlation between the electrical properties and a change in the potential of the oxidant within the stability range of a given phase. In the case of semiconductors, according to the band model, the variation of the Fermi level with temperature, and hence the variation in the chemical potential of the oxidant within a phase stability range should be relatively significant and increase with temperature, since the maximum concentration of defects increases with temperature in general. In semiconductors, the Fermi level should shift within the energy gap depending on the concentration of electronic defects. In turn, in metallic compounds, the variation of the Fermi level should be small and rather independent of temperature. In order to confirm the above general considerations, thermodynamic data were used and the standard free enthalpies AGO for the several equilibria between the phases in 3d metal oxides and sulphides were calculated, which is depicted in Fig. 5, Fig. 6a,b and Fig. 7a. In turn, in Fig. 5b, Fig. 6c and Fig. 7b, the variations of the changes of the chemical potential of oxidant in the stability range of given oxide and sulphide phases are presented. In Table 1 the metallic (M) and semiconducting (SC) properties, the magnitude of the forbidden gap
’
-300 400
lkl
T
., .. .,,/
co,s8
CG4
1200
loo0
800
600
CG3.X
O600
1
700
800
900
1000
T
1100
1200
1300
1x1
Fig. 7. (a) Temperature dependence of the standard free enthalpy of formation for selected sulphides and equilibrium between the oxides [ 1.41. (b) Temperature dependence of the changes of the chemical potential of sulphur within the stability range of the sulphide for selected sulphides.
A. Stokiosa, J. Zajgcki I Solid State Ionics 91 (1996) 315-322 Table 1 Electrical sulphides
properties,
Compound
TiO Ti,O, Ti,O TiO &-’ vo2 VA V”L, VO* VA VA MnO Mn,O, MnA Fe0 Fe@, coo Co@, NiO cu,o cue ZnO M. =Metallic
forbidden
Electrical properties M. M. M. SC. M. M. M. M. M. SC. SC. SC S.C. SC. S.C. S.C. SC SC. S.C. S.C. S.C. properties;
gap E,, coefficient
E, (eV)
I/m
1 lm (Eq. (7)) and the deviations
O/Me
Compound
0.69-1.33
3.05
-l/2-115
TiS VS CrS MnS FeS
0.98-l 0.71-1.29 1.50-1.52 1.995-2.002 2.165-2.175
2.5 3.5
1.8 2.7 4.0 2.0 2.2 3.2
1/2-l/6
l/2-1/14 l/30 I/4-1/5 l/4-116 118 -114
SC. = semiconducting
I .052- 1.127 Small Small 0.85-0.95 1.334- 1.333 1.02-l.ooo Small l.OOl-l.ooo 0.502-0.500 Small 0.999-l IHI0 properties.
Sources:
concluded that semiconducting properties are exhibited by oxides for which Apo2 is higher than about 100 kJ/mol, and the greater the forbidden gap the higher the values of the changes tof the chemical potential of oxygen. It should be stressed that for the compounds exhibiting semiconducting properties, such as TiO,, NiO, MnS, which are not in equilibrium with their higher oxides, the change in the chemical potential of the oxidant with changing the pressure from the equilibrium value to 1 atm is much bigger than 200 kJ which is depicted in Fig. 8 and Fig. 9. In order to calculate the change in the chemical potential of electronic defects, the magnitude of which would characterize a given compound, the value of the parameter m should be known over the entire phase stability range. For the compounds with semiconducting properties this value is generally small and ranges from 4 to 8. It results both from the defect theory and the experimental studies [5,7]. In the case of metallic compounds the magnitude of the changes of the chemical potential reaches 250 W. The parameter m, as results from numerous measure-
Refs. [5,7-l
from stoichiometry
Electrical properties M. M. S.C. SC. M. M. M. M. M. M. S.C. I M. M. S.C.
Co& Co%& cos Ni,S, NiS cu,s cus zns
321
X/M
E,
for selected
oxides and
l/f?!
S/Me
(eV)
3.2
l/5-1/6 1120
l/100
2.3
l/200 l/3-1/6
3.1
0.98-1.10 0.85-1.15 1.12-1.48 1.00-1.000 1.12-1.00 0.660-0.818 1.16-1.0 0.56-0.79 1.15-1.0 0.53-0.50 Small
11.
ments of electrical conductivity, takes significant values ranging from 20 to over 100. Thus the changes in the Fermi level obtained by dividing Ah2 by the parameter m will be insignificant.
T Fig. 8. Temperature dependence for selected oxides [1,2].
of the free enthalpy
[Kl of formation
322
A. Stoklosa, J. Zajecki
I Solid State Ionics 91 (1996)
within the stability range of a given phase are correlated with the maximum value of the deviation from stoichiometry and its electrical properties. The full correlation can be found when the character of the dependence of the concentration of ionic and electronic defects on the oxidant pressure is known.
100
z E
0
2 2
315-322
-100
B E 2
-200
6 E
-300
NiS MnS ,..“.....
Acknowledgments ‘...._,_
,:’
& ”
The research has been carried out under contract with the Polish Science Research Committee (KBN).
-IO0
% -500 400
600
800
1000
I200
T Fig. 9. Temperature dependence for selected sulphides [ 1.41.
of the free enthalpy
1400
[Kl
References
of formation
From the data summarized in Table 1 results also that the compounds with metallic properties exhibit substantial deviation from stoichiometry. As results from the conducted calculations, the changes in the chemical potential of oxidant in the stability range of a given phase could be applied for the semiquantitative, and in some cases even quantitative calculation of the changes in the Fermi level of electrons or the ‘Fermi level’ of ionic defects.
3. Conclusions Analysis of phase diagrams of a selection of oxides and sulphides of 3d metals demonstrated that the changes in the chemical potential of the oxidant
[I] J. Barin and 0. Knacke, Thermochemical Properties of Inorganic Substances (Springer-Verlag, Berlin, 1973, Supplement 1977). [2] L.B. Pankratz, Thermodynamic Properties of Elements and Oxides, Bulletin 672 (United States, Bureau of Mines, Washington, 1982). [3] H. Oppermann, Vanadiumoxide (Akademie-Verlag, Berlin, 1983). [4] S.R. Shatynski, Oxid Metals 11 (1977) 307. [5] P Kofstad, Nonstoichiometry, Diffusion and Electrical Conductivity in Binary Metal Oxide (Wiley, New York, 1972). [6] J.M. Blakely, in: Electrode Process in Solid State Ionics, eds. M. Kleitz and J. Dupuy (Reidel, Dordrecht, 1976) p. 83. [7] S. Mrowec, Defect and Diffusion in Solids (PWN-Elsevier, Warszawa-Amsterdam, 1980). [8] S. Mrowec and K. Przybylski, High Temp. Mater. Process. 6 (1984) 1. [9] J.B. Goodenough, in: Progress in Solid State Chem. Vol. 5, ed. H. Reiss (Pergamon Press, Oxford, 1971) p. 145. [lo] J.B. Goodenough, in: Solid State Chemistry, ed. C.N.R. Rao (Marcel Dekker, New York, 1974) p. 215. [ll] F. Jellinek, React. Solids 5 (1988) 323.