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Point defects and diffusion in NiO
.L Phys. Chem Solids Vol. 46, No. 1, pp. 43-52, 1985 Printed in the U.S.A.
0022-3697/85 $3.00 + .00 Pergamon Press Ltd.
POINT DEFECTS AND DIFFUSION IN NiOt N. L. PETERSON a n d C. L. WILEY Materials Science and Technology Division, Argonne National Laboratory, Argonne, IL 60439, U.S.A.
(Received 5 March 1984; accepted 26 April 1984) Abstract--A defect model for NiO is developed and is fit to the electrical-conductivity data [26], the deviation-from-stoichiometry data [7], and the cation-self-diffusiondata [14, 17]. This model involves neutral, singly charged, and doubly charged nickel vacancies and charge-compensating electron holes. Both singly and doubly charged cation vacancies are required to explain the data; neutral cation vacancies (if present) are not required by the present data. However, the jump frequencies of the two types of charged cation vacancies are generally not equal; the doubly charged cation vacancy moves with the smaller activation enthalpy. The defect data are quantitatively consistent with the chemical-diffusiondata [26] and with a correlation factor f~ = 0.75.
1. INTRODUCTION Nickel oxide, NiO, is known to be a metal-deficient p-type semiconductor with cation vacancies and electron holes as the primary defects [1]. NiO exists as a sodium-chloride-structured oxide from its melting point to room temperature, thus permitting the growth of rather pure, high-quality single crystals. The deviation from stoichiometry is larger than the impurity content in available crystals (thus allowing the determination of intrinsic defect properties), but is small enough so that theories based on dilute, noninteracting defects are probably appropriate. For these reasons, NiO has been a prototypic transition-metal oxide, and numerous studies have been made of its defect properties [1]. Knowledge about the defect structure of NiO has prompted detailed investigations and interpretations of the growth of oxide scales on nickel metal during high-temperature oxidation [2]. Studies of the deviation from stoichiometry in NiO [3-9] show a deficiency of nickel ions relative to the oxygen ions. The rapid cation self-diffusion [10-19] relative to the anion self-diffusion [20, 21] (D~i/DI~ 10 7 at 1200°C) strongly suggests that the excess oxygen ions are accommodated by the formation of cation vacancies (and electron holes) rather than by the formation of interstitial anions. The electrical conductivity in NiO has been measured over a wide range of temperature and oxygen partial pressure [3, 6, 7, 14, 22-26]. The recent measurements of Farhi and Petot-Ervas [26] are quite extensive and provide a detailed study of the oxygenpartial-pressure dependence of the electrical conductivity. The value of the activation enthalpy for defect formation deduced from the electrical-conductivity measurements appears to be a function of both temperature and oxygen partial pressure [26] and is not in agreement with the values deduced from thermogravimetric measurements. In NiO, the oxygen-partial-pressure dependencies of the electrical f Work supported by the U.S. Department of Energy. 43
conductivity [26] and the cation self-diffusion [14, 18] require at least two types of charged atomic defects, e.g. singly and doubly charged cation vacancies. Extensive measurements of the chemical-diffusion coefficient in NiO, performed using both electricalconductivity and thermogravimetric techniques, have been reported [9, 26-30]. Although considerable disagreement exists between the results of the various investigations, both the activation enthalpy and the value of the chemical-diffusion coefficient are in good agreement in the more recent studies [26, 28, 29]. Although several investigators have recognized the fact that at least two types of charged atomic defects on the Ni sublattice in NiO are important [7, 9, 14, 18, 26, 28, 30], in only two studies of NiO [7, 9] have attempts been made to analyze, in depth, their results in terms of the simultaneous existence of more than one type of charged atomic defect. Osburn and Vest [7] considered a model in which neutral, singly charged, and doubly charged cation vacancies are the important atomic defects in NiO; and using computed ionization energies of the vacancies as necessary input, they fit their thermogravimetric measurements of the deviation from stoichiometry to this model. However, the values of the defect parameters from this model are not in good agreement with more recent measurements of electrical conductivity in NiO. Koei and Gellings [9] have analyzed their own (unpublished) measurements of both the deviation from stoichiometry and the electrical conductivity in NiO. Koel and Gellings deduced the concentrations of singly and doubly charged cation vacancies as functions of oxygen partial pressure and temperature and concluded that the neutral cation vacancy is a minority defect. Using this defect-concentration information, Koel and Gellings deduced defect diffusivities based on their chemical-diffusion results and the assumption that the diffusivities of the singly charged and the doubly charged cation vacancies are equal.
44
N. L. PETERSONand C. L. WILEY
In the present study, we develop a defect model and obtain a self-consistent fit of this model to measurements of the electrical conductivity a, the deviation from stoichiometry 6, the cation self-diffusion coefficient D ' i , and the chemical-diffusion coefficient /)o (as determined by measurements of electrical conductivity) in Ni~_~O. In Section 2, our defect model for Ni~_60 is developed; this model includes neutral cation vacancies V~, singly charged cation vacancies V~i, and doubly charged cation vacancies V~, as well as charge-compensating electron holes h'. In Section 3, this model is least-squares fit to the extensive measurements of electrical conductivity [26] in order to deduce the relative importance of V~i and V~. In Section 4, the model is least-squares fit to the measurements of the deviation from stoichiometry [7] in order to demonstrate that neutral cation vacancies are a minority atomic defect and to evaluate the electron-hole mobility Uh, thus allowing a determination of the absolute values of the atomicdefect concentrations [V~] and [V~i], expressed in number of defects per cation-sublattice site. In Section 5, the atomic-defect diffusivities are determined by incorporating the various defect concentrations in a least-squares fit of the cation-self-diffusion data of Volpe and coworkers [14, 15] and of Atkinson and Taylor [17]. In Section 6, the previously determined defect concentrations and diffusivities are shown to be consistent with the chemical-diffusion measurements of Farhi and Petot-Ervas [26] and with a correlation factor for tracer diffusion by a vacancy mechanism f~ = 0.75. 2.
A simple defect model of dilute, noninteracting defects is developed for NiO; impurity-induced extrinsic defects are ignored in this model. A fit of this model to data obtained at high temperatures in highpurity material indicates that the model is quantitatively consistent with a wide range of experimental observations. The measurements of 6 indicate that the predominant atomic defect in Ni~_nO is vacancies on the Ni sublattice. The formation of these vacant sites can be described by the reactions
½02(g) ~ ~ o + v ~
(1)
V§i ~ V~i + h"
(2)
V~i ~ V~ + h',
(3)
where (Yo, V~i, V~i, V~i, and h" denote a neutral anion on an anion site, a neutral cation vacancy, a singly charged cation vacancy, a doubly charged cation vacancy, and an electron hole, respectively. Application of the law of mass action to these reactions yields
[ V~i] = Kvg,~pg 2
~'02~1/2
[V~] :
KVN~ [ V~qi] _ K V ' N i K ~ , p ~ '2
[h'] Kv.~,
[h']
[ v~d[h'] - -
(7) (8)
and [ V~i] - Kv'~,[ V~qi] Kv'N~Kr~Kv~p~)/2 2 [h'~ [h'] 2 ,
(9)
where the square brackets denote concentrations in number of defects per cation-sublattice site, and K~ denotes the equilibrium constants for the reaction to form the atomic defect indicated by the subscript ~. We assume that the primary defects in NiO are V~, V~, V~, and h'; impurity atoms, defects on the oxygen sublattice, and free electrons are assumed to be minority defects and are ignored. U n d e r these conditions, electroneutrality requires [h'] : [V~] + 2[V~d.
(10)
From eqns (7), (9), and (10), one obtains the cubic equation ~/2 = O. [h'] 3 - ([h'] + 2 K v.N~)KwN~Kv~Po2
(11)
3. ELECTRICAL CONDUCTIVITY
= (uhe)(NiO)[h'],
(12a)
where uh is the mobility of an electron hole (~h is assumed to be independent of [h']), e is the magnitude of the charge on an electron, and ( N i O ) is the absolute concentration of NiO (expressed in molecules per cm3). For convenience we introduce the "reduced" mobility of an electron hole ah -= #h(NiO) so that we can write a ~- (~he)[h'].
(12b)
The factor ( N i O ) in eqn (12a) permits the use of [h'] expressed in number of electron holes per cationsublattice site. A "reduced" equilibrium constant k~ for the formation of defect ~ can be defined by the relation
k~ =- ([~he)K~.
(l 3)
(4)
Since K~ represents an equilibrium constant, it has an Arrhenius-type temperature dependence. It will be shown in the next section that the temperature dependence of uh is of such a form that one can write
(5)
kr, = (kr,)oe -"~mr,
[~ol[V~] _ [V~l Pg~
(6)
[V~l
The electrical conductivity a can be expressed by the equation
DEFECT MODEL FOR Nil-~O
Kv~i -
[ V~il[h'] Kv~, - - -
(14)
Point defects and diffusion in NiO T (*C1
where H~ is the activation enthalpy for the reaction to form the defect ~. Using eqns (12b) and (13), one can transform eqn (1 l) into the form (73 -- ((7 +
2kv')OkvN~kv~p~ = 0.
1600
I
(15)
I
"~
1400
1200
I I I 902 (arm) 1.oo -~_
I
1000
I
I
0.z09 ~ L , ~
~oo
Of the numerous studies of the electrical conductivity in NiO, the recent measurements of Farhi and Petot-Ervas [26] provide the most extensive observations as a function of oxygen partial pressure at numerous temperatures. Such extensive measurements are necessary in order to be able to separate the contributions from the various types of defects. We have least-squares fit the electrical-conductivity data of Farhi and Petot-Ervas to eqn (15), where Po2 is in atmospheres and (7 is in (~2 cm)-L Eighty-nine percent of the data points are within 3% of the fits to eqn (15), indicating that the model gives an accurate description of the data. (The graphical results from Ref. [26] were kindly provided by G. PetotErvas [31].) The fitting procedure consists of using a nonlinear least-squares procedure on each isotherm of a data in order to fit the observed Po2 dependence of (7 to the positive real root of eqn (15) in which the two fitting parameters are kv-~, and the product k~N,kvf~i. The results of this fitting procedure are shown in Fig. l; the two components (TvN, and av-~, of the electron-hole conductivity which results from the formation of V~i and V~i, respectively, at 1000°C are also shown in the figure. The two fitting parameters kv-N~and the product kv'N~kv~ were subsequently leastsquares fit to an Arrhenius-type equation, yielding
45
N, "x.",,. \ " x
IO-I
10-2
.
i
I 0.6
~
i
I 0.7
i
I\ 0.8
(x")
Fig. 2. Electrical conductivity (r in NiO as a function of temperature. The heavy curves show the fit of the experimental data to our defect model; the light curves show the contributions to the electrical conductivity for .Oo2 = 1.89 × 10-4 atm by holes introduced through the presence of singly and doubly charged cation vacancies. The data are from the work of Farhi and Petot-Ervas [26].
I (kv-n~)0 = (1.07 ± 0.43) × 105,
10°
kcal Hkv;, = (43.0 ± 1.2) mole ' (kvN,k~)o = (4.44 ± 0.27) × 105,
"7E .g b
kcai HkvN,k~, = (42.15 ± 0 . 1 7 ) - mole "
10-1
O'V~I i(100C
10-2
o'v~ ~ (ioo(
-6
I
-4
-2
log Pozl*tm)
0
Fig. 1. Electrical conductivity tr in NiO as a function of oxygen partial pressure; the heavy curves indicate the fits of the data to eqn (15) of our defect model. The light curves show the contributions to the electrical conductivity at 1000°C by holes introduced through the presence of singly and doubly charged cation vacancies. The data are from the work of Farhi and Petot-Ervas [26].
The fit of the model to the temperature dependence of a is shown in Fig. 2; the two components arN, and av-N, of the electron-hole conductivity which results from the formation of VNi and V~, respectively, for Po2 = 1.89 × 10 -4 atm are also shown in the figure. Although the ratio [V'~d/[V'~i] may be determined from eqns (7), (9), (11), (13) and (14), the determination of the absolute values of these defect concentrations requires a knowledge of ~h. In Section 4, absolute values of ~he and of the individual defect concentrations are deduced from measurements of the deviation from stoichiometry. 4. DEFECT CONCENTRATIONS AND THE DEVIATION FROM STOICHIOMETRY The deviation from stoichiometry 6, in number of atomic defects per cation-sublattice site, can be written in the form
46
N. L. PETERSON a n d C. L. WILEY
6 = [ V ~ j + [V~i] + [V~J.
T (°C)
(16) 1400
From a knowledge of 6, the quantity (/~e) can be determined from the equation
~-
I
1200
I
I
1000
[
Ip0z(01m)
i
kv~,[l
kwN,)]p~
+ - k - ~ (or +
(~he) =
'zx.~
"k-,.
w t.) b--
Osburn and Vest [7] have reported the most extensive measurements of 6 as a function of oxygen partial pressure ( 10-4-10- ~ atm) and of temperature (900-1400°C). Their results are shown in Fig. 3 in the form of an Arrhenius plot. The results of Koel [32] cover an extensive range of Po2 (10-7-1 atm) over a limited temperature range (1200-1350°C) and are in excellent agreement with the results of Osburn and Vest [7]. The Arrhenius plots for a are rather straight and have nearly the same effective activation enthalpy at all Po~ (see Fig. 2); the slight curvatures in the isobars arise from the coexistence of V~ and V~. However, the Arrhenius plots for /~ show considerably more curvature, with smaller effective activation enthalpies at lower Po~ (see Fig. 3). Although V~ will contribute to 6 but not to ~r, contributions from V~ would be more important at higher Po~. Unfortunately, this behavior would not explain the differences in the temperature dependencies of a and (5; nor would it explain the differences in the Po~ dependencies. However, the discrepancies in the temperature and Po~ dependencies of a and ~ could be explained if the reference point for the absolute value of /~ were a little high, or if a constant vacancy concentration, independent of Po, and T, were introduced by impurities. If a defect concentration of 12 parts per million (ppm) is subtracted from each value of shown in Fig. 3, one obtains the points shown in Fig. 4. The uniform subtraction of 14 ppm defects from the data of Fig. 3 tends to cause the curvature in the Arrhenius plots to be concave downward (particularly T (°C} iooo
1200
I
I--03
I
I
I
I
I
Po2(Otm)
I-
cx 1.0' I0 -I o 1.01.10 "3 1.1.10 -4
A ,5 0
10-4
0
0 []
0 O
0
0
0 0
0
z~
v
[3
0 [3
W
Io
I 0.6
,
I 0.7
0
1.1-10 -4
(17)
6
1400
I
~ 10 • I0-I o 1.01.10-3
,
I 0.8
,
Fig. 3. Reported (uncorrected) values o f the deviation from s t o i c h i o m e t r y t5 in N i O as a function of t e m p e r a t u r e . The d a t a are from the work of O s b u r n a n d Vest [7].
10-4
-
.,43.
z
o
10-5 ¢,o t:3
=~
~¢m(I.1.10 -4 otto)
,-,
\
Io-~
( 0.6
,
i 0.7 103
\
< 0.8
J
7- (K-ql Fig. 4. Temperature dependence of the corrected deviation from stoichiometry (5 in NiO; the measured values have been corrected by the uniform subtraction of 12 ppm from each datum point reported by Osburn and Vest [7]. The dashed lines show the Arrhenius behavior of the data. The heavy solid curves show the values of ~5 predicted by our defect model under the assumption that the only atomic defects present are singly and doubly charged cation vacancies. The light curves show the contributions to ~ by singly and doubly charged cation vacancies for po~ = 1.1 × 10-4 atm. at the lower Po2), while the uniform subtraction of 10 ppm defects does not entirely remove the curvature that is apparent in Fig. 3. The correction of 12 ppm defects to the observed values of ~ yields nearly linear isobars for the Arrhenius plots for /~, with nearly the same effective activation enthalpies at all Po2 (see Fig. 4)--as was expected, based on the electrical-conductivity data. The dashed lines in Fig. 4 represent the temperature dependencies of the corrected/~ data; the heavy solid lines show the values of t5 calculated from the "reduced" equilibrium constants determined from the experimental values of a. These calculations of 6 are based on the assumptions that the only atomic defects present are V~i and V~i and that ~h is temperature independent. If neutral cation vacancies V~ are present, they will contribute to ~ but not to a, and their contribution to t3 will be larger at higher values of Po2. Hence, if V~ are present, the vertical separation between the dashed line for high Po~ and the dashed line for low Po2 must be greater than the vertical separation between the corresponding heavy solid lines. This is contrary to the observations in Fig. 4; therefore we must conclude that V ~ q i is at best a minority defect in NiO and need not be considered further in the present analysis. The near parallelism of the dashed and heavy solid lines for a given Po2 in Fig. 4 requires ~h to be
Point defects and diffusion in NiO independent of temperature. Recent analyses of electrical-conductivity data, performed for constant thermoelectric coefficient in CoO by Chen et al. [33], indicate that /~h is independent of temperature in CoO over the range 1000-1200°C; preliminary measurements on NiO lead to the same conclusion [34]. Using the values of kw~ and the product kv~k~.N~, deduced from the electrical-conductivity data, and the values of 5 shown in Fig. 4, we can use eqn (17) to calculate values of/~he. The mean of these values is (hne) = 2115 __+ 53 (~2 cm)-L If ~ e is allowed to be temperature dependent with an Arrhenius-type behavior, the fitting of the values of ~ne calculated from eqn (17) gives ~ne an effective activation enthalpy of 0.62 ___ 0.81 kcal/mole. This result is statistically consistent with the assumption that ~ is temperature independent. Since ~ is found to be temperature independent, the activation enthalpies which describe the temperature dependencies of Kv-~, and the product K~'N,Kv~, are given by the previously determined quantities Hkv',.,, and H~;,~,~,. Once V~i is excluded from consideration, eqns (1) and (2) can be combined to give 1
= O2(g) ~ ~ o + V~i + h'. z
(18)
Application of the law of mass action to eqn (18) yields gv,,
[ VNi][h'] p~
(19)
/~ ~1/2 V'niPO2 - - ,
(20)
and [ Vtqi]
[h']
47 11500
I
1400
I
I
_
I
r {*C) 1200
I
I000
I
I
I
[v.J
~__ 10-3
i
io-4
_
\
,.'5, Q.
~
10-5
8
10-6
1
0.5
i
1
0.6
i
I
0.7
i
I
0.8
Fig. 6. Temperature dependence of the defect concentrations in NiO, as predicted from our defect model for Po~ = 0.209 atm. which include all three atomic defects V~i , VNi , and V~ can be restated for the restricted case which excludes V§~ by (a) setting kv~, = 0 when it appears alone and not as a product with other k~; and (b) setting kv~ = 1 when it occurs in a product with another k~. Arrhenius-type expressions for /~V'Ni a n d Kv-~, can be determined from the values of the fitting parameters kw,~,, [~he, and the product krN,kvg,; /('v~, = (9.92 ___0.78) × 10-2
i . e . / ( ~ = KrN,Kv~,. The remaining general equations 10-3
t
I
I
I
I
I
I
I.-
kcal / -1 × exp -(42.15 _ 0.17)-~-~ole/RTJ Kv'N~= (50.0 + 20.0) X e x p [ - ( 4 3 . 0 +_ 1.2) k c a l / R T 1 mole/ d '
J
g
10-4
a.
g
8 bt~
t/t~
:,
~0-~
-6
-4
-2
0
log poz(otm) Fig. 5. Oxygen-partial-pressure dependence of the defect concentrations in NiO, as predicted from our defect model for a temperature of 1400°C.
Absolute values of [ V~i], [ V~i], 6, and [h'] can now be calculated from eqns (9), (11), (16) and (20); these results are shown in Fig. 5 as a function of Po2 for 1400°C and in Fig. 6 as a function of temperature for Po: = 0.209 atm (air). The two components ~rN, and ~v;,~ of t5 which arise from V~i and V~qi, respectively, are shown in Fig. 4 as a function of temperature for Po~ = 1.1 X 10-4 atm. The temperature dependence of the ratio [V'~d/[V'Ni] is similar to that reported by Farhi and Petot-Ervas [26]; however, our values of this ratio are consistently about 25% lower than theirs. The slight curvature in the lines representing the Po2 and temperature dependencies of [V~i] and [V~i] results from the coupling between V'Ni and V~i through the electroneutrality condition. The coupling between charged defects can significantly alter the concentration of the minority defect. As an
48
N. L. PETERSONand C. L. WILEY
example, [ V~i] becomes virtually independent of po~ at low temperatures where [V~] ,> [V~i]. If the diffusivity of V~ is much larger than the diffusivity of V~i, then D ~ can become virtually independent of Po~ at low temperatures. Koel and Gellings [9] report Arrhenius-type expressions for Kv'N, and the product /£v~, = Kv~,KvN, deduced from their measurements of and cr as functions ofpo2 and temperature. Unfortunately, neither the experimental values of 6 and used in their calculations nor a discussion of the formalism used to deduce the equilibrium constants was included in the paper by Koel and Gellings. However, the values of the concentrations of the various defects, as calculated from our eqns (7), (9) and (11) and the values of the equilibrium constants given by Koel and Gellings, are in qualitative agreement with the values obtained in the present study. The Po~ dependence at T = 1300°C (the middle of the temperature range considered in Fig. 6) of the values of [ Vhi], [ V~i], 6, and [h'], as determined from the parameters given by Koel and Gellings, are in agreement with the results of the present analysis within 10%. Values of [V~i], [V;~i], 6, and [h'], calculated from the parameters given by Koel and Gellings, are shown in Fig. 7 as functions of temperature at Po~ = 0.209 atm. The values shown in Fig. 7 are in good agreement with the present results (Fig. 6), except for the values o f rt T:~, ~NiJ near the temperature extremes of the figures; in the temperature range actually studied by Koel and Gellings, the agreement between the two studies is good. However, we shall see that the results of studies of the tracer selfdiffusion coefficient D~i (see Section 5) and the
1600
I 10.3
I
T {*C} 1200
1400
I
I
I
I000
1
I
I
i]
g~
,0-'
Iv',,
I---
~ 10_5
.
8
0,5
0.6
~ (K-h
0.7
0.8
Fig. 7. Temperature dependence of the defect concentrations in NiO for Po2 = 0.209 atm, as predicted from the defectreaction equilibrium constants reported by Koel and Gellings [9].
chemical-diffusion coefficient /)o (see Section 6) are more compatible with the defect data shown in Fig. 6 than with that shown in Fig. 7.
5.
CATION SELF-DIFFUSION IN NiO
The defect model can now be applied to the tracer self-diffusion coefficient D*i for NiO. Since V~ and V~ are the only atomic defects of importance in NiO, we may write
D~i = D~iv, + D'iv. = DvN~f~[V~,] + Dv-NJ~[V'~],
(21a) (21b)
where DN~v~ * ~ denotes the contribution = D ~ f ~ [ VNi] to the tracer self-diffusion coefficient by a cation vacancy in the effective charge state indicated by the superscript a, D ~ , denotes the uncorrelated diffusion coefficient for a cation vacancy in the effective charge state indicated by the superscript a, and f~ denotes the correlation factor for tracer diffusion by a vacancy mechanism. Note that .f~ is a function of crystallattice geometry and does not depend upon the charge state of the vacancy when the total vacancy concentration is small, as in NiO; f~ = 0.781 [35] for diffusion on the cation sublattice of the NaCl structure. The quantity D ~ i can be written in the form
D ~ = ~ r2ZWv~i,
(22)
where Z is the number of nearest neighbors, r is the j u m p distance, and W ~ , is the j u m p frequency for V~. Volpe et al. [14, 15] and Atkinson et al. [17, 18] have provided a mutually consistent set of measurements of the cation self-diffusion in NiO, both as a function of temperature for Po2 = 1.0 atm (Fig. 8) and of oxygen partial pressure for T = 1245 and 1380°C (Fig. 9). We have globally least-squares fit the tracer-diffusion data [14, 15, 17, 18] to eqn (21b) for all available T and Po2. Using the values of [V~i] and [V~i] determined from the fitting of a and 6 in Sections 3 and 4, we obtain values of the product D~,f~. If we assume equal diffusivities for V~ and V~, we obtain a satisfactory fit to the temperature dependence of D ~ , but a totally unsatisfactory fit to the Po2 dependence of D*i (dot-dash lines in Fig. 9). Hence, Wv~ and Wv-N~ cannot be equal. This is consistent with a suggestion by Ikeda and Nii [29] that V~i is much more mobile than V~. If we assume ' unequal diffusivities for VN~ and V"Ni, we obtain equally good fits to the tracer-diffusion data, but with two very different sets (e.g. A and B) of parameters for the defect diffusivities. These two sets of parameters correspond to two equivalent chi-squared minima in parameter space. For fit A,
Dvs~f~ = (1.60 + 0.87) × e x p I - ( 4 8 . 5 -- 1.9) ~ o l e / R
cm2/sec;
49
Point defects and diffusion in NiO
T (%1 i600 1400 i200 I000 IIII I I I I I
800 I
I-
10-s
lO-~C
I0-" %=
i0-12
sented by the solid lines in Fig. 9; fit B is represented by the dashed lines. Both fits A and B are rather similar in the temperature range of the experimental data [14] (i.e. for 1245 and 1380°C), but they become more dissimilar at the higher and lower temperatures illustrated by the lines for 1155 and 1545°C. Unfortunately, the scatter in the data of Atkinson et al. [18] for 1460°C (not shown in Fig. 9) prevents selection of a preference between fits A and B. However, accurate measurements of the cation selfdiffusion in NiO as a function of Po2 at greatly different temperatures could help to identify the appropriate values for DrN~ and Dv'n~. The large difference between the activation enthalpies of migration for V~ and V~ in fit A is unexpected. An indication as to which of the two fits A or B is preferred can be obtained from considerations of the chemical-diffusion coefficient discussed in the next section. 6. CHEMICAL DIFFUSION IN NiO
iO-I~,
When one of the parameters T or Po2 is changed abruptly, the point-defect concentration at the gas/ solid interface should rapidly change to correspond to the equilibrium values consistent with the newly
IE DNi
10-14
IO-mL__L
0.5
I 0.6
0.7
0.8
0.9
I
I
I
I/
I
J
1.0
T Fig. 8. Temperature dependence of the cation self-diffusion coefficient in NiO for Po2 = 1.0 atm. The solid circles represent the data of Volpe and Reddy [14]; the open circles represent the data of Volpe, Peterson, and Reddy [15]; and the open triangles represent the data of Atkinson and Taylor [17]. The heavy curve shows the fit of the experimental data to our defect model; the light curves show the contributions to D~i by singly and doubly charged vacancies, as deduced from fit A.
IO-S
T,.c;..j io-IO
Dv-n,f~ = (1.18 ± 0.43) X 10 -4
,38o / . -
[ k c a l / R T ] cm2/sec. X exp - ( 1 3 . 6 + 1.0) m o l e / / For fit B,
iO-u 1245 / ~ ~ / / / /" 1155/~ /
DrN,fo = (4.0 ___ 1.6) X 10 -2 X exp [ - ( 3 6 . 9 _ 1.0) ~kcal l I R T I ] cm2/sec;
~
D v k ~ = (2.5 +_ 1.7) X 10 -2
io-~Z kcal/RT] X e x p I - ( 3 0 . 4 _+ 2.2) m~ole/ J cm2/sec" The values of x 2 and the systematics of the deviations of the individual points from the least-squares fit are virtually identical for fits A and B. All of the calculations converged to either fit A or fit B, even though a wide variety of values for the activation enthalpies and pre-exponential factors was used as starting points in the least-squares fitting procedure. Fit A is repre-
I -6
I
I I -4 Ioq PozlOtm)
[ -2
I
0
Fig. 9. Oxygen-partial-pressure dependence of the cation self-diffusion coefficient in NiO. The data are from the work of Volpe and Reddy [14]. The solid and the dashed curves show the fits A and B, respectively, at four temperatures, including the two temperatures (1245 and 1380°C) for which experimental data are available. The dot-dash curves show the fit to the experimental data under the assumption of equal diffusivities for the singly and doubly charged cation vacancies.
50
N.L. PETERSONand C. L. WILEY
imposed conditions of temperature or oxygen partial pressure. In the interior of the sample, thermodynamic equilibrium will be achieved after some time t by the diffusion of point defects to or from the surface. The rate of achieving thermodynamic equilibrium throughout the sample is determined by the chemicaldiffusion coefficient /5, provided that the oxygen transport across the gas-solid interface is not rate limiting. When the chemical-diffusion coefficient is determined through electrical-conductivity measurements, it is denoted b y / ) , and is defined by an equation of the form of Fick's first law: ]f.[ = e(NiO)/).lV[h'][,
(23)
where ./~ = j~ ~ + j~ ~, denotes the total charge flux generated by the cation-vacancy migration, The charge flux j~f for a given defect is related to the corresponding particle flux ~b~f for that defect through the relationship
jo,.," +-elzo~d4~o,,,,
(24)
=
where zoa is the effective valence of the defect; the + sign is used for defects having a positive effective charge. The particle flux for a given defect can be determined from the equation
4~a~,= ___(NiO)Dd~.f{-Vtdef] (25)
+ Izd~,rl[def](eV~b~l -
\k.Tl)
'
where Doff is the uncorrelated diffusion coefficient for the defect; [def] is the concentration of the defect, expressed in number of the defects per cation-sublattice site; ~7~bis the gradient of the electric potential set up in the oxide by the migration of charged defects; and k~ is the Boltzmann constant. The external + sign is to be used for particle defects, e.g. holes; the external - sign is to be used for nonparticle defects, e.g. vacancies. The internal + sign is to be used for defects having a negative effective charge. Application of the coupled-current condition for NiO yields
+Jw~[.
IJ~'l = IJ~,
(26)
Assuming that Dh" ,> D ~ , one can determine the (eWk~ quantity \ k ~ ! ;
ev~ _
vW] - [h']
ksT
-
-V
ln[h'l.
(27)
This result is the same as that obtained by simply setting ~¢ = 0 . The electroneutrality condition, eqn (10), allows one to calculate the quantity ~7[h']; ~Tlh']
=
V[ V~] + 27[ V~,].
(28)
Then one can write
I/ol e(NiO)lV[h']l
(29a)
= (NiO)IV[ V~] + 2V[ V~,]I
(29b)
2(D, N,[ V~,] + 2D~ ~[ V~,])([ V~,] + 3[ V~]) 2 r I/~t 1~2
(29c) In going from eqn (29b) to (29c) no assumptions are made; one simply performs the algebraic operations indicated in eqn (29b). From eqns (21a, b), one can write (D~,. + 2D*~,..) = (D, 'N+[V~,] + 2Dv'~i[V'~])f..
(30)
Then from eqns (29c) and (30), one obtains L =
(D~i,.. + 2D'i,..)
(D,N,[V'Nd + 2D, ~,[V~,])
= 2(D*w. + 2D~w.)([V~] + 3[V~]) t D,([VNi] + 2[V~i])2
(31a)
(3 l b)
Hence. the correlation factor f. can be determined from measurements of D~, and /). and a knowledge of [V~i] and [VLi]. Among the many measurements of/5. in NiO [9, 26-30], three measurements [26, 28. 29] are in reasonable agreement with one another. Using the extensive measurements o f / 5 . by Farhi and PetotErvas [26], the values of [V~] and [V~,] determined in Section 4, the values of D*.. and D * . . obtained from fits A and B in Section 5, and eqn (31b), we have calculated values off~; the results are tabulated in columns 2 and 3 of Table 1. Since ~ is expected to be independent of temperature, fit A from the present study appears to provide the more consistent set of diffusivities for the singly and doubly charged cation vacancies. We have also used eqn (21b), values of D*, [14, 15, 17, 18], and values of [V~i] and [V~i] computed from the parameters reported by Koel and Gellings [9] to obtain values of the products DrN,f~ and DvN,f~. Again, two equally good, but distinctly different, fits were obtained, similar to the fits A and B reported in Section 5. Using eqn (31b), the results of the A and B fits mentioned earlier in this paragraph, and values of/5~ obtained from the studies by Farhi and Petot-Ervas [26], we have calculated the values of f~ reported in columns 4 and 5 of Table 1. However, the apparent temperature dependence of the values offo deduced from the defect data of Koel and Gellings is inconsistent with the expected temperature independence off~. The calculated values of J~ listed in Table 1 involve the combined interpretation of ~r, D~,, 6, and /9o. The values off~ from all four fits near the center of the temperature range investigated (1300°C) are in
51
Point defects and diffusion in NiO Table 1. The correlation factor f~ for tracer diffusion by a vacancy mechanism at po~ = 0.209, as calculated from measurements of Dg~ and/),, eqn (31b), and defect concentrations obtained either in the present study or from values of the equilibrium constants reported by Koel and Gellings [9] Present study
Study of Koel and Ge11£n~s Fit k Fit B
T (°C)
Fit A
Fit B
1000
0.66
0.40
0.62
0.47
1100
0.70
0.51
0.69
0.58
1200
0.73
0.63
0.77
0.71
1300
0.76
0.76
0.85
0.85
1400
0.77
0.87
0.93
0.99
1500
0.77
0.98
1.02
1.14
1600
0.78
1.06
1.11
1.29
1700
0.78
1.13
1.20
1.44
acceptable agreement with the theoretical value off~ = 0.78. It is the temperature independence of the f~ values obtained from fit A of the present study which suggests that this fit gives the preferred values of defect parameters for NiO. As noted in Section 4, [V~i] becomes virtually independent of Po2 at low temperatures where [ V~i] >> [ V~i]. If the defect parameters deduced from fit A are correct, Dv'N, >> DvN~ at low temperatures such that DNi * v. dominates D * i even though VNi " is a minority defect. Hence, D~i in NiO should be virtually independent of Po, at low temperatures (e.g. 800°C) where studies of the oxidation of nickel metal have been pursued. The effective diffusivities of cation vacancies having different effective charge states are often observed to be equal. As an example, Dieckmann [36] noted that in CoO D~o oc { [ V~o] + [ V~:o] + [ V~o] }, a condition that is not in complete agreement with the Po2 dependence o f / ) ~ in CoO reported by Petot-Ervas et al. [37] who suggest that Dye,, > Dvbo. Similar observations of equal diffusivities for the various vacancies have been reported for M n O [38] and CuaO [39]; these may be rationalized as follows. If the lifetimes of the various charge states of the cation vacancies are small compared to the mean time of stay of a vacancy, a vacancy may change its charge state many tens or hundreds of times between j u m p s so that only one activation energy, e.g. that corresponding to the energetically most favorable vacancy charge state, is required to describe the cation-vacancy exchanges. The actual charge state of the vacancy during the j u m p is unknown. In the present study of NiO, the diffusivities of V~4i and V~qi a r e distinctly different. This different behavior of vacancies in NiO relative to that in, say, CoO may be rationalized as follows. The hole concentration in NiO is about 50 times smaller than that in CoO, and ~h is slightly smaller in NiO. Hence a given vacancy would change its charge state approximately 50 times less often in NiO than in CoO, and j u m p s of V~i and V~i would occur as separable events with different activation energies. A cation
vacancy in NiO might remain in the same charge state for more than one successive jump. This explanation may be only a part of the correct interpretation, depending on the localization of the electron hole during the j u m p process. 7. CONCLUSIONS (1) A defect model for N i O is developed and quantitatively fit to the data for electrical conductivity, deviation from stoichiometry, and cation self-diffusion. (2) Both singly and doubly charged cation vacancies are required to explain the experimental observations; neutral cation vacancies are not required. (3) The diffusivities of the two types of charged cation vacancies are not equal; the doubly charged cation vacancy moves with the smaller activation enthalpy. The electron-hole mobility is temperature independent. (4) The defect data are quantitatively consistent with the chemical-diffusion data and with a (temperature-independent) correlation factor f~ = 0.75. Acknowledgements--The authors thank G. Petot-Ervas and coworker for making available the graphical results of their extensive measurements of a in NiO as functions of temperature and Po2, T. O. Mason for disclosing his preliminary results regarding the temperature independence of uh in NiO, and Janet B. Anderson for many helpful discussions regarding the least-squares fitting procedures. REFERENCES
1. Kofstad P., Non-Stoichiometry, Diffusion, and Electrical Conductivity in Binary Metal Oxides. Wiley Interscience, New York (1972). 2. Gesmundo F. and Viani F., Solid St. lonics 6, 33 (1982). 3. MitoffS. P., J. Chem. Phys. 35, 882 (1961). 4. Sockel H.-G. and Schmalzried H., Ber. Bunsenges Phys. Chem. 72, 745 (1968). 5. Tretyakov Y. D. and Rapp R. A., Trans. A I M E 245, 1235 (1969). 6. Tripp W. C. and Tallan N. M., J. Am. Ceram. Soc. 53, 531 (1970). 7. Osburn C. M. and Vest R. W., J. Phys. Chem. Solids 32, 1331 (1971).
52
N. L. PETERSON and C. L. WILEY
8. Sockel H.-G. and llschner B., Z. Phys. Chem. (N.F.) 74, 284 (1971). 9. Koel G. J. and Gellings P. J., Oxid. Met. 5, 185 (1972). 10. Lindner R. and Akerstr6m A., Z. Phys. Chem. (N.F.) 6, 162 (1956); Disc. Faraday Soc. 23, 133 (1957). 11. Shim M. T. and Moore W. J., J. Chem. Phys. 26, 802 (1957). 12. Choi J. S. and Moore W. J., a( Phys. Chem. 66, 1308 (1962). 13. Klotsman S. M., Timofeyev A. N. and Trakhtenberg I. S., Fiz. Metal. Metalloved 14, 428 (1962); Phys. Metals Metallogrv. 14, 91 (1962). 14. Volpe M. L. and Reddy J., J. Chem. Phys. 53, 1117 (1970). 15. Volpe M. L., Peterson N. L. and Reddy J., Phys. Rev. B3, 1417 (1971). 16. Ikeda Y., Nii K., Beranger G. and Lacombe P., Trans. Japan Inst. Met. 15, 441 (1974). 17. Atkinson A. and Taylor R. I., J. Mater. Sci. 13, 427 (1978); Phil. Mag. A39, 581 (1979). 18. Atkinson A., Hughes A. E. and Hammou A., Phil. Mag. A43, 1071 (1981). 19. Lesage B., Huntz A. M. and Lacombe P., J. Phys. Chem. Solids 42, 705 (1981). 20. Meyer M., Barbezat S., El Houch C. and Talon R., J. Phys. Colloq. (C6) 327 (1980). 21. Dubois C., Monty C. and Philibert J., Phil. Mag. A46, 419 (1982). 22. Uno R., J. Phys. Soc. Japan 22, 1502 (1967). 23. Bransky I. and Tallan N. M., J. Chem. Phys. 49, 1243 (1968). 24. Eror N. G. and Wagner, Jr. J. B., Phys. Status Solidi 35, 641 (1969). 25. Meier G. H. and Rapp R. A., Z. Phys. Chem. (N.F.) 74, 168 (1971). 26. Farhi R. and Petot-Ervas G., J. Phys. Chem. Solids 39, 1169 (1978); 39, 1175 (1978).
27. Price J. P. and Wagner, Jr. J. B., Z. Phys. Chem. (N.F.) 49, 257 (1966). 28. Deren J. and Mrowec S., J. Mater. Sei. 8, 545 (1973). 29. Ikeda Y. and Nii K., Trans. Japan Inst. Met. 17, 419 (1976). 30. Nowotny J. and Sadowski A., J. Am. Ceram Soc. 62, 24 (1979). 31. We report the data from Ref. [26] for only values of /)o2 > 10 4 atm. However, Farhi and Petot-Ervas have made an additional eleven measurements of a for Po2 < 10-4 atm. A fit of all their available data to eqn (15) is slightly worse than that using only the data for Po2 > 1 0 - 4 atm; for the fit to all of the data, only seventyfive percent of the data points are within 3% of the fits to eqn (15). However, both fits are of nearly the same quality and both fits provide consistent and satisfactory agreement with measurements of 6, D*~, and D,. In our reported analysis, we have arbitrarily chosen to consider only the ~ data for Po2 > 10-4 arm. This choice is based, in part, on our own difficulties in obtaining accurate values ofpo2 for Po2 < 10-6 atm. 32. Koel G. J., Ph.D. dissertation, Technical University of Twente (1971). 33. Chen H. C., Gartstein E. and Mason T. 0., J. Phys. Chem. Solids 43, 991 (1982). 34. Mason T. O., private communication (1983). 35. Compaan K. and Haven Y., Trans. Faraday Soc. 52, 786 (1956); 54, 1498 (1958). 36. Dieckmann R., Z. Phys. Chem. (N.F.) 107, 189 (1977). 37. Petot-Ervas G., Radji O., Sossa B. and Ochin P., Rad. Effects 75, 301 (1983). 38. Peterson N. L. and Chen W. K., J. Phys. Chem. Solids 43, 29 (1982). 39. Peterson N. L. and Wiley C. L., J. Phys. Chem. Solids" 45, 281 (1984).
0022-3697/85 $3.00 + .00 Pergamon Press Ltd.
POINT DEFECTS AND DIFFUSION IN NiOt N. L. PETERSON a n d C. L. WILEY Materials Science and Technology Division, Argonne National Laboratory, Argonne, IL 60439, U.S.A.
(Received 5 March 1984; accepted 26 April 1984) Abstract--A defect model for NiO is developed and is fit to the electrical-conductivity data [26], the deviation-from-stoichiometry data [7], and the cation-self-diffusiondata [14, 17]. This model involves neutral, singly charged, and doubly charged nickel vacancies and charge-compensating electron holes. Both singly and doubly charged cation vacancies are required to explain the data; neutral cation vacancies (if present) are not required by the present data. However, the jump frequencies of the two types of charged cation vacancies are generally not equal; the doubly charged cation vacancy moves with the smaller activation enthalpy. The defect data are quantitatively consistent with the chemical-diffusiondata [26] and with a correlation factor f~ = 0.75.
1. INTRODUCTION Nickel oxide, NiO, is known to be a metal-deficient p-type semiconductor with cation vacancies and electron holes as the primary defects [1]. NiO exists as a sodium-chloride-structured oxide from its melting point to room temperature, thus permitting the growth of rather pure, high-quality single crystals. The deviation from stoichiometry is larger than the impurity content in available crystals (thus allowing the determination of intrinsic defect properties), but is small enough so that theories based on dilute, noninteracting defects are probably appropriate. For these reasons, NiO has been a prototypic transition-metal oxide, and numerous studies have been made of its defect properties [1]. Knowledge about the defect structure of NiO has prompted detailed investigations and interpretations of the growth of oxide scales on nickel metal during high-temperature oxidation [2]. Studies of the deviation from stoichiometry in NiO [3-9] show a deficiency of nickel ions relative to the oxygen ions. The rapid cation self-diffusion [10-19] relative to the anion self-diffusion [20, 21] (D~i/DI~ 10 7 at 1200°C) strongly suggests that the excess oxygen ions are accommodated by the formation of cation vacancies (and electron holes) rather than by the formation of interstitial anions. The electrical conductivity in NiO has been measured over a wide range of temperature and oxygen partial pressure [3, 6, 7, 14, 22-26]. The recent measurements of Farhi and Petot-Ervas [26] are quite extensive and provide a detailed study of the oxygenpartial-pressure dependence of the electrical conductivity. The value of the activation enthalpy for defect formation deduced from the electrical-conductivity measurements appears to be a function of both temperature and oxygen partial pressure [26] and is not in agreement with the values deduced from thermogravimetric measurements. In NiO, the oxygen-partial-pressure dependencies of the electrical f Work supported by the U.S. Department of Energy. 43
conductivity [26] and the cation self-diffusion [14, 18] require at least two types of charged atomic defects, e.g. singly and doubly charged cation vacancies. Extensive measurements of the chemical-diffusion coefficient in NiO, performed using both electricalconductivity and thermogravimetric techniques, have been reported [9, 26-30]. Although considerable disagreement exists between the results of the various investigations, both the activation enthalpy and the value of the chemical-diffusion coefficient are in good agreement in the more recent studies [26, 28, 29]. Although several investigators have recognized the fact that at least two types of charged atomic defects on the Ni sublattice in NiO are important [7, 9, 14, 18, 26, 28, 30], in only two studies of NiO [7, 9] have attempts been made to analyze, in depth, their results in terms of the simultaneous existence of more than one type of charged atomic defect. Osburn and Vest [7] considered a model in which neutral, singly charged, and doubly charged cation vacancies are the important atomic defects in NiO; and using computed ionization energies of the vacancies as necessary input, they fit their thermogravimetric measurements of the deviation from stoichiometry to this model. However, the values of the defect parameters from this model are not in good agreement with more recent measurements of electrical conductivity in NiO. Koei and Gellings [9] have analyzed their own (unpublished) measurements of both the deviation from stoichiometry and the electrical conductivity in NiO. Koel and Gellings deduced the concentrations of singly and doubly charged cation vacancies as functions of oxygen partial pressure and temperature and concluded that the neutral cation vacancy is a minority defect. Using this defect-concentration information, Koel and Gellings deduced defect diffusivities based on their chemical-diffusion results and the assumption that the diffusivities of the singly charged and the doubly charged cation vacancies are equal.
44
N. L. PETERSONand C. L. WILEY
In the present study, we develop a defect model and obtain a self-consistent fit of this model to measurements of the electrical conductivity a, the deviation from stoichiometry 6, the cation self-diffusion coefficient D ' i , and the chemical-diffusion coefficient /)o (as determined by measurements of electrical conductivity) in Ni~_~O. In Section 2, our defect model for Ni~_60 is developed; this model includes neutral cation vacancies V~, singly charged cation vacancies V~i, and doubly charged cation vacancies V~, as well as charge-compensating electron holes h'. In Section 3, this model is least-squares fit to the extensive measurements of electrical conductivity [26] in order to deduce the relative importance of V~i and V~. In Section 4, the model is least-squares fit to the measurements of the deviation from stoichiometry [7] in order to demonstrate that neutral cation vacancies are a minority atomic defect and to evaluate the electron-hole mobility Uh, thus allowing a determination of the absolute values of the atomicdefect concentrations [V~] and [V~i], expressed in number of defects per cation-sublattice site. In Section 5, the atomic-defect diffusivities are determined by incorporating the various defect concentrations in a least-squares fit of the cation-self-diffusion data of Volpe and coworkers [14, 15] and of Atkinson and Taylor [17]. In Section 6, the previously determined defect concentrations and diffusivities are shown to be consistent with the chemical-diffusion measurements of Farhi and Petot-Ervas [26] and with a correlation factor for tracer diffusion by a vacancy mechanism f~ = 0.75. 2.
A simple defect model of dilute, noninteracting defects is developed for NiO; impurity-induced extrinsic defects are ignored in this model. A fit of this model to data obtained at high temperatures in highpurity material indicates that the model is quantitatively consistent with a wide range of experimental observations. The measurements of 6 indicate that the predominant atomic defect in Ni~_nO is vacancies on the Ni sublattice. The formation of these vacant sites can be described by the reactions
½02(g) ~ ~ o + v ~
(1)
V§i ~ V~i + h"
(2)
V~i ~ V~ + h',
(3)
where (Yo, V~i, V~i, V~i, and h" denote a neutral anion on an anion site, a neutral cation vacancy, a singly charged cation vacancy, a doubly charged cation vacancy, and an electron hole, respectively. Application of the law of mass action to these reactions yields
[ V~i] = Kvg,~pg 2
~'02~1/2
[V~] :
KVN~ [ V~qi] _ K V ' N i K ~ , p ~ '2
[h'] Kv.~,
[h']
[ v~d[h'] - -
(7) (8)
and [ V~i] - Kv'~,[ V~qi] Kv'N~Kr~Kv~p~)/2 2 [h'~ [h'] 2 ,
(9)
where the square brackets denote concentrations in number of defects per cation-sublattice site, and K~ denotes the equilibrium constants for the reaction to form the atomic defect indicated by the subscript ~. We assume that the primary defects in NiO are V~, V~, V~, and h'; impurity atoms, defects on the oxygen sublattice, and free electrons are assumed to be minority defects and are ignored. U n d e r these conditions, electroneutrality requires [h'] : [V~] + 2[V~d.
(10)
From eqns (7), (9), and (10), one obtains the cubic equation ~/2 = O. [h'] 3 - ([h'] + 2 K v.N~)KwN~Kv~Po2
(11)
3. ELECTRICAL CONDUCTIVITY
= (uhe)(NiO)[h'],
(12a)
where uh is the mobility of an electron hole (~h is assumed to be independent of [h']), e is the magnitude of the charge on an electron, and ( N i O ) is the absolute concentration of NiO (expressed in molecules per cm3). For convenience we introduce the "reduced" mobility of an electron hole ah -= #h(NiO) so that we can write a ~- (~he)[h'].
(12b)
The factor ( N i O ) in eqn (12a) permits the use of [h'] expressed in number of electron holes per cationsublattice site. A "reduced" equilibrium constant k~ for the formation of defect ~ can be defined by the relation
k~ =- ([~he)K~.
(l 3)
(4)
Since K~ represents an equilibrium constant, it has an Arrhenius-type temperature dependence. It will be shown in the next section that the temperature dependence of uh is of such a form that one can write
(5)
kr, = (kr,)oe -"~mr,
[~ol[V~] _ [V~l Pg~
(6)
[V~l
The electrical conductivity a can be expressed by the equation
DEFECT MODEL FOR Nil-~O
Kv~i -
[ V~il[h'] Kv~, - - -
(14)
Point defects and diffusion in NiO T (*C1
where H~ is the activation enthalpy for the reaction to form the defect ~. Using eqns (12b) and (13), one can transform eqn (1 l) into the form (73 -- ((7 +
2kv')OkvN~kv~p~ = 0.
1600
I
(15)
I
"~
1400
1200
I I I 902 (arm) 1.oo -~_
I
1000
I
I
0.z09 ~ L , ~
~oo
Of the numerous studies of the electrical conductivity in NiO, the recent measurements of Farhi and Petot-Ervas [26] provide the most extensive observations as a function of oxygen partial pressure at numerous temperatures. Such extensive measurements are necessary in order to be able to separate the contributions from the various types of defects. We have least-squares fit the electrical-conductivity data of Farhi and Petot-Ervas to eqn (15), where Po2 is in atmospheres and (7 is in (~2 cm)-L Eighty-nine percent of the data points are within 3% of the fits to eqn (15), indicating that the model gives an accurate description of the data. (The graphical results from Ref. [26] were kindly provided by G. PetotErvas [31].) The fitting procedure consists of using a nonlinear least-squares procedure on each isotherm of a data in order to fit the observed Po2 dependence of (7 to the positive real root of eqn (15) in which the two fitting parameters are kv-~, and the product k~N,kvf~i. The results of this fitting procedure are shown in Fig. l; the two components (TvN, and av-~, of the electron-hole conductivity which results from the formation of V~i and V~i, respectively, at 1000°C are also shown in the figure. The two fitting parameters kv-N~and the product kv'N~kv~ were subsequently leastsquares fit to an Arrhenius-type equation, yielding
45
N, "x.",,. \ " x
IO-I
10-2
.
i
I 0.6
~
i
I 0.7
i
I\ 0.8
(x")
Fig. 2. Electrical conductivity (r in NiO as a function of temperature. The heavy curves show the fit of the experimental data to our defect model; the light curves show the contributions to the electrical conductivity for .Oo2 = 1.89 × 10-4 atm by holes introduced through the presence of singly and doubly charged cation vacancies. The data are from the work of Farhi and Petot-Ervas [26].
I (kv-n~)0 = (1.07 ± 0.43) × 105,
10°
kcal Hkv;, = (43.0 ± 1.2) mole ' (kvN,k~)o = (4.44 ± 0.27) × 105,
"7E .g b
kcai HkvN,k~, = (42.15 ± 0 . 1 7 ) - mole "
10-1
O'V~I i(100C
10-2
o'v~ ~ (ioo(
-6
I
-4
-2
log Pozl*tm)
0
Fig. 1. Electrical conductivity tr in NiO as a function of oxygen partial pressure; the heavy curves indicate the fits of the data to eqn (15) of our defect model. The light curves show the contributions to the electrical conductivity at 1000°C by holes introduced through the presence of singly and doubly charged cation vacancies. The data are from the work of Farhi and Petot-Ervas [26].
The fit of the model to the temperature dependence of a is shown in Fig. 2; the two components arN, and av-N, of the electron-hole conductivity which results from the formation of VNi and V~, respectively, for Po2 = 1.89 × 10 -4 atm are also shown in the figure. Although the ratio [V'~d/[V'~i] may be determined from eqns (7), (9), (11), (13) and (14), the determination of the absolute values of these defect concentrations requires a knowledge of ~h. In Section 4, absolute values of ~he and of the individual defect concentrations are deduced from measurements of the deviation from stoichiometry. 4. DEFECT CONCENTRATIONS AND THE DEVIATION FROM STOICHIOMETRY The deviation from stoichiometry 6, in number of atomic defects per cation-sublattice site, can be written in the form
46
N. L. PETERSON a n d C. L. WILEY
6 = [ V ~ j + [V~i] + [V~J.
T (°C)
(16) 1400
From a knowledge of 6, the quantity (/~e) can be determined from the equation
~-
I
1200
I
I
1000
[
Ip0z(01m)
i
kv~,[l
kwN,)]p~
+ - k - ~ (or +
(~he) =
'zx.~
"k-,.
w t.) b--
Osburn and Vest [7] have reported the most extensive measurements of 6 as a function of oxygen partial pressure ( 10-4-10- ~ atm) and of temperature (900-1400°C). Their results are shown in Fig. 3 in the form of an Arrhenius plot. The results of Koel [32] cover an extensive range of Po2 (10-7-1 atm) over a limited temperature range (1200-1350°C) and are in excellent agreement with the results of Osburn and Vest [7]. The Arrhenius plots for a are rather straight and have nearly the same effective activation enthalpy at all Po~ (see Fig. 2); the slight curvatures in the isobars arise from the coexistence of V~ and V~. However, the Arrhenius plots for /~ show considerably more curvature, with smaller effective activation enthalpies at lower Po~ (see Fig. 3). Although V~ will contribute to 6 but not to ~r, contributions from V~ would be more important at higher Po~. Unfortunately, this behavior would not explain the differences in the temperature dependencies of a and (5; nor would it explain the differences in the Po~ dependencies. However, the discrepancies in the temperature and Po~ dependencies of a and ~ could be explained if the reference point for the absolute value of /~ were a little high, or if a constant vacancy concentration, independent of Po, and T, were introduced by impurities. If a defect concentration of 12 parts per million (ppm) is subtracted from each value of shown in Fig. 3, one obtains the points shown in Fig. 4. The uniform subtraction of 14 ppm defects from the data of Fig. 3 tends to cause the curvature in the Arrhenius plots to be concave downward (particularly T (°C} iooo
1200
I
I--03
I
I
I
I
I
Po2(Otm)
I-
cx 1.0' I0 -I o 1.01.10 "3 1.1.10 -4
A ,5 0
10-4
0
0 []
0 O
0
0
0 0
0
z~
v
[3
0 [3
W
Io
I 0.6
,
I 0.7
0
1.1-10 -4
(17)
6
1400
I
~ 10 • I0-I o 1.01.10-3
,
I 0.8
,
Fig. 3. Reported (uncorrected) values o f the deviation from s t o i c h i o m e t r y t5 in N i O as a function of t e m p e r a t u r e . The d a t a are from the work of O s b u r n a n d Vest [7].
10-4
-
.,43.
z
o
10-5 ¢,o t:3
=~
~¢m(I.1.10 -4 otto)
,-,
\
Io-~
( 0.6
,
i 0.7 103
\
< 0.8
J
7- (K-ql Fig. 4. Temperature dependence of the corrected deviation from stoichiometry (5 in NiO; the measured values have been corrected by the uniform subtraction of 12 ppm from each datum point reported by Osburn and Vest [7]. The dashed lines show the Arrhenius behavior of the data. The heavy solid curves show the values of ~5 predicted by our defect model under the assumption that the only atomic defects present are singly and doubly charged cation vacancies. The light curves show the contributions to ~ by singly and doubly charged cation vacancies for po~ = 1.1 × 10-4 atm. at the lower Po2), while the uniform subtraction of 10 ppm defects does not entirely remove the curvature that is apparent in Fig. 3. The correction of 12 ppm defects to the observed values of ~ yields nearly linear isobars for the Arrhenius plots for /~, with nearly the same effective activation enthalpies at all Po2 (see Fig. 4)--as was expected, based on the electrical-conductivity data. The dashed lines in Fig. 4 represent the temperature dependencies of the corrected/~ data; the heavy solid lines show the values of t5 calculated from the "reduced" equilibrium constants determined from the experimental values of a. These calculations of 6 are based on the assumptions that the only atomic defects present are V~i and V~i and that ~h is temperature independent. If neutral cation vacancies V~ are present, they will contribute to ~ but not to a, and their contribution to t3 will be larger at higher values of Po2. Hence, if V~ are present, the vertical separation between the dashed line for high Po~ and the dashed line for low Po2 must be greater than the vertical separation between the corresponding heavy solid lines. This is contrary to the observations in Fig. 4; therefore we must conclude that V ~ q i is at best a minority defect in NiO and need not be considered further in the present analysis. The near parallelism of the dashed and heavy solid lines for a given Po2 in Fig. 4 requires ~h to be
Point defects and diffusion in NiO independent of temperature. Recent analyses of electrical-conductivity data, performed for constant thermoelectric coefficient in CoO by Chen et al. [33], indicate that /~h is independent of temperature in CoO over the range 1000-1200°C; preliminary measurements on NiO lead to the same conclusion [34]. Using the values of kw~ and the product kv~k~.N~, deduced from the electrical-conductivity data, and the values of 5 shown in Fig. 4, we can use eqn (17) to calculate values of/~he. The mean of these values is (hne) = 2115 __+ 53 (~2 cm)-L If ~ e is allowed to be temperature dependent with an Arrhenius-type behavior, the fitting of the values of ~ne calculated from eqn (17) gives ~ne an effective activation enthalpy of 0.62 ___ 0.81 kcal/mole. This result is statistically consistent with the assumption that ~ is temperature independent. Since ~ is found to be temperature independent, the activation enthalpies which describe the temperature dependencies of Kv-~, and the product K~'N,Kv~, are given by the previously determined quantities Hkv',.,, and H~;,~,~,. Once V~i is excluded from consideration, eqns (1) and (2) can be combined to give 1
= O2(g) ~ ~ o + V~i + h'. z
(18)
Application of the law of mass action to eqn (18) yields gv,,
[ VNi][h'] p~
(19)
/~ ~1/2 V'niPO2 - - ,
(20)
and [ Vtqi]
[h']
47 11500
I
1400
I
I
_
I
r {*C) 1200
I
I000
I
I
I
[v.J
~__ 10-3
i
io-4
_
\
,.'5, Q.
~
10-5
8
10-6
1
0.5
i
1
0.6
i
I
0.7
i
I
0.8
Fig. 6. Temperature dependence of the defect concentrations in NiO, as predicted from our defect model for Po~ = 0.209 atm. which include all three atomic defects V~i , VNi , and V~ can be restated for the restricted case which excludes V§~ by (a) setting kv~, = 0 when it appears alone and not as a product with other k~; and (b) setting kv~ = 1 when it occurs in a product with another k~. Arrhenius-type expressions for /~V'Ni a n d Kv-~, can be determined from the values of the fitting parameters kw,~,, [~he, and the product krN,kvg,; /('v~, = (9.92 ___0.78) × 10-2
i . e . / ( ~ = KrN,Kv~,. The remaining general equations 10-3
t
I
I
I
I
I
I
I.-
kcal / -1 × exp -(42.15 _ 0.17)-~-~ole/RTJ Kv'N~= (50.0 + 20.0) X e x p [ - ( 4 3 . 0 +_ 1.2) k c a l / R T 1 mole/ d '
J
g
10-4
a.
g
8 bt~
t/t~
:,
~0-~
-6
-4
-2
0
log poz(otm) Fig. 5. Oxygen-partial-pressure dependence of the defect concentrations in NiO, as predicted from our defect model for a temperature of 1400°C.
Absolute values of [ V~i], [ V~i], 6, and [h'] can now be calculated from eqns (9), (11), (16) and (20); these results are shown in Fig. 5 as a function of Po2 for 1400°C and in Fig. 6 as a function of temperature for Po: = 0.209 atm (air). The two components ~rN, and ~v;,~ of t5 which arise from V~i and V~qi, respectively, are shown in Fig. 4 as a function of temperature for Po~ = 1.1 X 10-4 atm. The temperature dependence of the ratio [V'~d/[V'Ni] is similar to that reported by Farhi and Petot-Ervas [26]; however, our values of this ratio are consistently about 25% lower than theirs. The slight curvature in the lines representing the Po2 and temperature dependencies of [V~i] and [V~i] results from the coupling between V'Ni and V~i through the electroneutrality condition. The coupling between charged defects can significantly alter the concentration of the minority defect. As an
48
N. L. PETERSONand C. L. WILEY
example, [ V~i] becomes virtually independent of po~ at low temperatures where [V~] ,> [V~i]. If the diffusivity of V~ is much larger than the diffusivity of V~i, then D ~ can become virtually independent of Po~ at low temperatures. Koel and Gellings [9] report Arrhenius-type expressions for Kv'N, and the product /£v~, = Kv~,KvN, deduced from their measurements of and cr as functions ofpo2 and temperature. Unfortunately, neither the experimental values of 6 and used in their calculations nor a discussion of the formalism used to deduce the equilibrium constants was included in the paper by Koel and Gellings. However, the values of the concentrations of the various defects, as calculated from our eqns (7), (9) and (11) and the values of the equilibrium constants given by Koel and Gellings, are in qualitative agreement with the values obtained in the present study. The Po~ dependence at T = 1300°C (the middle of the temperature range considered in Fig. 6) of the values of [ Vhi], [ V~i], 6, and [h'], as determined from the parameters given by Koel and Gellings, are in agreement with the results of the present analysis within 10%. Values of [V~i], [V;~i], 6, and [h'], calculated from the parameters given by Koel and Gellings, are shown in Fig. 7 as functions of temperature at Po~ = 0.209 atm. The values shown in Fig. 7 are in good agreement with the present results (Fig. 6), except for the values o f rt T:~, ~NiJ near the temperature extremes of the figures; in the temperature range actually studied by Koel and Gellings, the agreement between the two studies is good. However, we shall see that the results of studies of the tracer selfdiffusion coefficient D~i (see Section 5) and the
1600
I 10.3
I
T {*C} 1200
1400
I
I
I
I000
1
I
I
i]
g~
,0-'
Iv',,
I---
~ 10_5
.
8
0,5
0.6
~ (K-h
0.7
0.8
Fig. 7. Temperature dependence of the defect concentrations in NiO for Po2 = 0.209 atm, as predicted from the defectreaction equilibrium constants reported by Koel and Gellings [9].
chemical-diffusion coefficient /)o (see Section 6) are more compatible with the defect data shown in Fig. 6 than with that shown in Fig. 7.
5.
CATION SELF-DIFFUSION IN NiO
The defect model can now be applied to the tracer self-diffusion coefficient D*i for NiO. Since V~ and V~ are the only atomic defects of importance in NiO, we may write
D~i = D~iv, + D'iv. = DvN~f~[V~,] + Dv-NJ~[V'~],
(21a) (21b)
where DN~v~ * ~ denotes the contribution = D ~ f ~ [ VNi] to the tracer self-diffusion coefficient by a cation vacancy in the effective charge state indicated by the superscript a, D ~ , denotes the uncorrelated diffusion coefficient for a cation vacancy in the effective charge state indicated by the superscript a, and f~ denotes the correlation factor for tracer diffusion by a vacancy mechanism. Note that .f~ is a function of crystallattice geometry and does not depend upon the charge state of the vacancy when the total vacancy concentration is small, as in NiO; f~ = 0.781 [35] for diffusion on the cation sublattice of the NaCl structure. The quantity D ~ i can be written in the form
D ~ = ~ r2ZWv~i,
(22)
where Z is the number of nearest neighbors, r is the j u m p distance, and W ~ , is the j u m p frequency for V~. Volpe et al. [14, 15] and Atkinson et al. [17, 18] have provided a mutually consistent set of measurements of the cation self-diffusion in NiO, both as a function of temperature for Po2 = 1.0 atm (Fig. 8) and of oxygen partial pressure for T = 1245 and 1380°C (Fig. 9). We have globally least-squares fit the tracer-diffusion data [14, 15, 17, 18] to eqn (21b) for all available T and Po2. Using the values of [V~i] and [V~i] determined from the fitting of a and 6 in Sections 3 and 4, we obtain values of the product D~,f~. If we assume equal diffusivities for V~ and V~, we obtain a satisfactory fit to the temperature dependence of D ~ , but a totally unsatisfactory fit to the Po2 dependence of D*i (dot-dash lines in Fig. 9). Hence, Wv~ and Wv-N~ cannot be equal. This is consistent with a suggestion by Ikeda and Nii [29] that V~i is much more mobile than V~. If we assume ' unequal diffusivities for VN~ and V"Ni, we obtain equally good fits to the tracer-diffusion data, but with two very different sets (e.g. A and B) of parameters for the defect diffusivities. These two sets of parameters correspond to two equivalent chi-squared minima in parameter space. For fit A,
Dvs~f~ = (1.60 + 0.87) × e x p I - ( 4 8 . 5 -- 1.9) ~ o l e / R
cm2/sec;
49
Point defects and diffusion in NiO
T (%1 i600 1400 i200 I000 IIII I I I I I
800 I
I-
10-s
lO-~C
I0-" %=
i0-12
sented by the solid lines in Fig. 9; fit B is represented by the dashed lines. Both fits A and B are rather similar in the temperature range of the experimental data [14] (i.e. for 1245 and 1380°C), but they become more dissimilar at the higher and lower temperatures illustrated by the lines for 1155 and 1545°C. Unfortunately, the scatter in the data of Atkinson et al. [18] for 1460°C (not shown in Fig. 9) prevents selection of a preference between fits A and B. However, accurate measurements of the cation selfdiffusion in NiO as a function of Po2 at greatly different temperatures could help to identify the appropriate values for DrN~ and Dv'n~. The large difference between the activation enthalpies of migration for V~ and V~ in fit A is unexpected. An indication as to which of the two fits A or B is preferred can be obtained from considerations of the chemical-diffusion coefficient discussed in the next section. 6. CHEMICAL DIFFUSION IN NiO
iO-I~,
When one of the parameters T or Po2 is changed abruptly, the point-defect concentration at the gas/ solid interface should rapidly change to correspond to the equilibrium values consistent with the newly
IE DNi
10-14
IO-mL__L
0.5
I 0.6
0.7
0.8
0.9
I
I
I
I/
I
J
1.0
T Fig. 8. Temperature dependence of the cation self-diffusion coefficient in NiO for Po2 = 1.0 atm. The solid circles represent the data of Volpe and Reddy [14]; the open circles represent the data of Volpe, Peterson, and Reddy [15]; and the open triangles represent the data of Atkinson and Taylor [17]. The heavy curve shows the fit of the experimental data to our defect model; the light curves show the contributions to D~i by singly and doubly charged vacancies, as deduced from fit A.
IO-S
T,.c;..j io-IO
Dv-n,f~ = (1.18 ± 0.43) X 10 -4
,38o / . -
[ k c a l / R T ] cm2/sec. X exp - ( 1 3 . 6 + 1.0) m o l e / / For fit B,
iO-u 1245 / ~ ~ / / / /" 1155/~ /
DrN,fo = (4.0 ___ 1.6) X 10 -2 X exp [ - ( 3 6 . 9 _ 1.0) ~kcal l I R T I ] cm2/sec;
~
D v k ~ = (2.5 +_ 1.7) X 10 -2
io-~Z kcal/RT] X e x p I - ( 3 0 . 4 _+ 2.2) m~ole/ J cm2/sec" The values of x 2 and the systematics of the deviations of the individual points from the least-squares fit are virtually identical for fits A and B. All of the calculations converged to either fit A or fit B, even though a wide variety of values for the activation enthalpies and pre-exponential factors was used as starting points in the least-squares fitting procedure. Fit A is repre-
I -6
I
I I -4 Ioq PozlOtm)
[ -2
I
0
Fig. 9. Oxygen-partial-pressure dependence of the cation self-diffusion coefficient in NiO. The data are from the work of Volpe and Reddy [14]. The solid and the dashed curves show the fits A and B, respectively, at four temperatures, including the two temperatures (1245 and 1380°C) for which experimental data are available. The dot-dash curves show the fit to the experimental data under the assumption of equal diffusivities for the singly and doubly charged cation vacancies.
50
N.L. PETERSONand C. L. WILEY
imposed conditions of temperature or oxygen partial pressure. In the interior of the sample, thermodynamic equilibrium will be achieved after some time t by the diffusion of point defects to or from the surface. The rate of achieving thermodynamic equilibrium throughout the sample is determined by the chemicaldiffusion coefficient /5, provided that the oxygen transport across the gas-solid interface is not rate limiting. When the chemical-diffusion coefficient is determined through electrical-conductivity measurements, it is denoted b y / ) , and is defined by an equation of the form of Fick's first law: ]f.[ = e(NiO)/).lV[h'][,
(23)
where ./~ = j~ ~ + j~ ~, denotes the total charge flux generated by the cation-vacancy migration, The charge flux j~f for a given defect is related to the corresponding particle flux ~b~f for that defect through the relationship
jo,.," +-elzo~d4~o,,,,
(24)
=
where zoa is the effective valence of the defect; the + sign is used for defects having a positive effective charge. The particle flux for a given defect can be determined from the equation
4~a~,= ___(NiO)Dd~.f{-Vtdef] (25)
+ Izd~,rl[def](eV~b~l -
\k.Tl)
'
where Doff is the uncorrelated diffusion coefficient for the defect; [def] is the concentration of the defect, expressed in number of the defects per cation-sublattice site; ~7~bis the gradient of the electric potential set up in the oxide by the migration of charged defects; and k~ is the Boltzmann constant. The external + sign is to be used for particle defects, e.g. holes; the external - sign is to be used for nonparticle defects, e.g. vacancies. The internal + sign is to be used for defects having a negative effective charge. Application of the coupled-current condition for NiO yields
+Jw~[.
IJ~'l = IJ~,
(26)
Assuming that Dh" ,> D ~ , one can determine the (eWk~ quantity \ k ~ ! ;
ev~ _
vW] - [h']
ksT
-
-V
ln[h'l.
(27)
This result is the same as that obtained by simply setting ~¢ = 0 . The electroneutrality condition, eqn (10), allows one to calculate the quantity ~7[h']; ~Tlh']
=
V[ V~] + 27[ V~,].
(28)
Then one can write
I/ol e(NiO)lV[h']l
(29a)
= (NiO)IV[ V~] + 2V[ V~,]I
(29b)
2(D, N,[ V~,] + 2D~ ~[ V~,])([ V~,] + 3[ V~]) 2 r I/~t 1~2
(29c) In going from eqn (29b) to (29c) no assumptions are made; one simply performs the algebraic operations indicated in eqn (29b). From eqns (21a, b), one can write (D~,. + 2D*~,..) = (D, 'N+[V~,] + 2Dv'~i[V'~])f..
(30)
Then from eqns (29c) and (30), one obtains L =
(D~i,.. + 2D'i,..)
(D,N,[V'Nd + 2D, ~,[V~,])
= 2(D*w. + 2D~w.)([V~] + 3[V~]) t D,([VNi] + 2[V~i])2
(31a)
(3 l b)
Hence. the correlation factor f. can be determined from measurements of D~, and /). and a knowledge of [V~i] and [VLi]. Among the many measurements of/5. in NiO [9, 26-30], three measurements [26, 28. 29] are in reasonable agreement with one another. Using the extensive measurements o f / 5 . by Farhi and PetotErvas [26], the values of [V~] and [V~,] determined in Section 4, the values of D*.. and D * . . obtained from fits A and B in Section 5, and eqn (31b), we have calculated values off~; the results are tabulated in columns 2 and 3 of Table 1. Since ~ is expected to be independent of temperature, fit A from the present study appears to provide the more consistent set of diffusivities for the singly and doubly charged cation vacancies. We have also used eqn (21b), values of D*, [14, 15, 17, 18], and values of [V~i] and [V~i] computed from the parameters reported by Koel and Gellings [9] to obtain values of the products DrN,f~ and DvN,f~. Again, two equally good, but distinctly different, fits were obtained, similar to the fits A and B reported in Section 5. Using eqn (31b), the results of the A and B fits mentioned earlier in this paragraph, and values of/5~ obtained from the studies by Farhi and Petot-Ervas [26], we have calculated the values of f~ reported in columns 4 and 5 of Table 1. However, the apparent temperature dependence of the values offo deduced from the defect data of Koel and Gellings is inconsistent with the expected temperature independence off~. The calculated values of J~ listed in Table 1 involve the combined interpretation of ~r, D~,, 6, and /9o. The values off~ from all four fits near the center of the temperature range investigated (1300°C) are in
51
Point defects and diffusion in NiO Table 1. The correlation factor f~ for tracer diffusion by a vacancy mechanism at po~ = 0.209, as calculated from measurements of Dg~ and/),, eqn (31b), and defect concentrations obtained either in the present study or from values of the equilibrium constants reported by Koel and Gellings [9] Present study
Study of Koel and Ge11£n~s Fit k Fit B
T (°C)
Fit A
Fit B
1000
0.66
0.40
0.62
0.47
1100
0.70
0.51
0.69
0.58
1200
0.73
0.63
0.77
0.71
1300
0.76
0.76
0.85
0.85
1400
0.77
0.87
0.93
0.99
1500
0.77
0.98
1.02
1.14
1600
0.78
1.06
1.11
1.29
1700
0.78
1.13
1.20
1.44
acceptable agreement with the theoretical value off~ = 0.78. It is the temperature independence of the f~ values obtained from fit A of the present study which suggests that this fit gives the preferred values of defect parameters for NiO. As noted in Section 4, [V~i] becomes virtually independent of Po2 at low temperatures where [ V~i] >> [ V~i]. If the defect parameters deduced from fit A are correct, Dv'N, >> DvN~ at low temperatures such that DNi * v. dominates D * i even though VNi " is a minority defect. Hence, D~i in NiO should be virtually independent of Po, at low temperatures (e.g. 800°C) where studies of the oxidation of nickel metal have been pursued. The effective diffusivities of cation vacancies having different effective charge states are often observed to be equal. As an example, Dieckmann [36] noted that in CoO D~o oc { [ V~o] + [ V~:o] + [ V~o] }, a condition that is not in complete agreement with the Po2 dependence o f / ) ~ in CoO reported by Petot-Ervas et al. [37] who suggest that Dye,, > Dvbo. Similar observations of equal diffusivities for the various vacancies have been reported for M n O [38] and CuaO [39]; these may be rationalized as follows. If the lifetimes of the various charge states of the cation vacancies are small compared to the mean time of stay of a vacancy, a vacancy may change its charge state many tens or hundreds of times between j u m p s so that only one activation energy, e.g. that corresponding to the energetically most favorable vacancy charge state, is required to describe the cation-vacancy exchanges. The actual charge state of the vacancy during the j u m p is unknown. In the present study of NiO, the diffusivities of V~4i and V~qi a r e distinctly different. This different behavior of vacancies in NiO relative to that in, say, CoO may be rationalized as follows. The hole concentration in NiO is about 50 times smaller than that in CoO, and ~h is slightly smaller in NiO. Hence a given vacancy would change its charge state approximately 50 times less often in NiO than in CoO, and j u m p s of V~i and V~i would occur as separable events with different activation energies. A cation
vacancy in NiO might remain in the same charge state for more than one successive jump. This explanation may be only a part of the correct interpretation, depending on the localization of the electron hole during the j u m p process. 7. CONCLUSIONS (1) A defect model for N i O is developed and quantitatively fit to the data for electrical conductivity, deviation from stoichiometry, and cation self-diffusion. (2) Both singly and doubly charged cation vacancies are required to explain the experimental observations; neutral cation vacancies are not required. (3) The diffusivities of the two types of charged cation vacancies are not equal; the doubly charged cation vacancy moves with the smaller activation enthalpy. The electron-hole mobility is temperature independent. (4) The defect data are quantitatively consistent with the chemical-diffusion data and with a (temperature-independent) correlation factor f~ = 0.75. Acknowledgements--The authors thank G. Petot-Ervas and coworker for making available the graphical results of their extensive measurements of a in NiO as functions of temperature and Po2, T. O. Mason for disclosing his preliminary results regarding the temperature independence of uh in NiO, and Janet B. Anderson for many helpful discussions regarding the least-squares fitting procedures. REFERENCES
1. Kofstad P., Non-Stoichiometry, Diffusion, and Electrical Conductivity in Binary Metal Oxides. Wiley Interscience, New York (1972). 2. Gesmundo F. and Viani F., Solid St. lonics 6, 33 (1982). 3. MitoffS. P., J. Chem. Phys. 35, 882 (1961). 4. Sockel H.-G. and Schmalzried H., Ber. Bunsenges Phys. Chem. 72, 745 (1968). 5. Tretyakov Y. D. and Rapp R. A., Trans. A I M E 245, 1235 (1969). 6. Tripp W. C. and Tallan N. M., J. Am. Ceram. Soc. 53, 531 (1970). 7. Osburn C. M. and Vest R. W., J. Phys. Chem. Solids 32, 1331 (1971).
52
N. L. PETERSON and C. L. WILEY
8. Sockel H.-G. and llschner B., Z. Phys. Chem. (N.F.) 74, 284 (1971). 9. Koel G. J. and Gellings P. J., Oxid. Met. 5, 185 (1972). 10. Lindner R. and Akerstr6m A., Z. Phys. Chem. (N.F.) 6, 162 (1956); Disc. Faraday Soc. 23, 133 (1957). 11. Shim M. T. and Moore W. J., J. Chem. Phys. 26, 802 (1957). 12. Choi J. S. and Moore W. J., a( Phys. Chem. 66, 1308 (1962). 13. Klotsman S. M., Timofeyev A. N. and Trakhtenberg I. S., Fiz. Metal. Metalloved 14, 428 (1962); Phys. Metals Metallogrv. 14, 91 (1962). 14. Volpe M. L. and Reddy J., J. Chem. Phys. 53, 1117 (1970). 15. Volpe M. L., Peterson N. L. and Reddy J., Phys. Rev. B3, 1417 (1971). 16. Ikeda Y., Nii K., Beranger G. and Lacombe P., Trans. Japan Inst. Met. 15, 441 (1974). 17. Atkinson A. and Taylor R. I., J. Mater. Sci. 13, 427 (1978); Phil. Mag. A39, 581 (1979). 18. Atkinson A., Hughes A. E. and Hammou A., Phil. Mag. A43, 1071 (1981). 19. Lesage B., Huntz A. M. and Lacombe P., J. Phys. Chem. Solids 42, 705 (1981). 20. Meyer M., Barbezat S., El Houch C. and Talon R., J. Phys. Colloq. (C6) 327 (1980). 21. Dubois C., Monty C. and Philibert J., Phil. Mag. A46, 419 (1982). 22. Uno R., J. Phys. Soc. Japan 22, 1502 (1967). 23. Bransky I. and Tallan N. M., J. Chem. Phys. 49, 1243 (1968). 24. Eror N. G. and Wagner, Jr. J. B., Phys. Status Solidi 35, 641 (1969). 25. Meier G. H. and Rapp R. A., Z. Phys. Chem. (N.F.) 74, 168 (1971). 26. Farhi R. and Petot-Ervas G., J. Phys. Chem. Solids 39, 1169 (1978); 39, 1175 (1978).
27. Price J. P. and Wagner, Jr. J. B., Z. Phys. Chem. (N.F.) 49, 257 (1966). 28. Deren J. and Mrowec S., J. Mater. Sei. 8, 545 (1973). 29. Ikeda Y. and Nii K., Trans. Japan Inst. Met. 17, 419 (1976). 30. Nowotny J. and Sadowski A., J. Am. Ceram Soc. 62, 24 (1979). 31. We report the data from Ref. [26] for only values of /)o2 > 10 4 atm. However, Farhi and Petot-Ervas have made an additional eleven measurements of a for Po2 < 10-4 atm. A fit of all their available data to eqn (15) is slightly worse than that using only the data for Po2 > 1 0 - 4 atm; for the fit to all of the data, only seventyfive percent of the data points are within 3% of the fits to eqn (15). However, both fits are of nearly the same quality and both fits provide consistent and satisfactory agreement with measurements of 6, D*~, and D,. In our reported analysis, we have arbitrarily chosen to consider only the ~ data for Po2 > 10-4 arm. This choice is based, in part, on our own difficulties in obtaining accurate values ofpo2 for Po2 < 10-6 atm. 32. Koel G. J., Ph.D. dissertation, Technical University of Twente (1971). 33. Chen H. C., Gartstein E. and Mason T. 0., J. Phys. Chem. Solids 43, 991 (1982). 34. Mason T. O., private communication (1983). 35. Compaan K. and Haven Y., Trans. Faraday Soc. 52, 786 (1956); 54, 1498 (1958). 36. Dieckmann R., Z. Phys. Chem. (N.F.) 107, 189 (1977). 37. Petot-Ervas G., Radji O., Sossa B. and Ochin P., Rad. Effects 75, 301 (1983). 38. Peterson N. L. and Chen W. K., J. Phys. Chem. Solids 43, 29 (1982). 39. Peterson N. L. and Wiley C. L., J. Phys. Chem. Solids" 45, 281 (1984).