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On the accurate measurement of oxygen self-diffusivities and surface exchange coefficients in oxides via SIMS depth profiling
Solid State Ionics 144 Ž2001. 71–80 www.elsevier.comrlocaterssi
On the accurate measurement of oxygen self-diffusivities and surface exchange coefficients in oxides via SIMS depth profiling P. Fielitz ) , G. Borchardt FB Physik, Metallurgie und Werkstoffwissenschaften, AG Thermochemie and Mikrokinetik, Technische UniÕersitat ¨ Clausthal, Robert-Koch-Str. 42, D-38678 Clausthal-Zellerfeld, Germany Received 26 January 2001; received in revised form 30 May 2001; accepted 13 June 2001
Abstract Oxygen diffusivity in oxides is usually determined via gasrsolid exchange experiments in an atmosphere enriched as to the concentration of the rare stable isotope 18 O. The kinetics of the 16 O and 18 O isotope exchange at the surface is usually described by the surface exchange coefficient, K. The analytical solution of the diffusion problem suggests the definition of a characteristic time constant, t ' DrK 2. This time constant determines the duration that is necessary to reach equilibrium between the 18 O gas concentration and the 18 O concentration at the surface of the solid. The simultaneous determination of D and K by secondary ion mass spectrometry ŽSIMS. depth profiling works well in a large parameter range of D and K. However, at low temperatures, the simultaneous determination of D and K may be subject to severe errors. To overcome this problem, the possibility to determine D and K separately will be discussed. q 2001 Elsevier Science B.V. All rights reserved. Keywords: Self-diffusion coefficient; Surface exchange; Isotope exchange; SIMS depth profiling
1. Introduction The most direct method of measuring the oxygen self-diffusivity, D, and the surface exchange coefficient, K, of oxide materials is isotope exchange followed by secondary ion mass spectrometry ŽSIMS. depth profiling w1–3x. The inherent accuracy of this method is based mainly on the fortuitous circumstance that the ratio 18 Or16 O of the natural isotope abundancies is about 1r500. This means the dynamic range of the measured 18 O depth profile can reach almost three orders of magnitude. Such a
dynamic range is more than necessary to determine accurate diffusion coefficients and surface exchange coefficients. However, if one looks at measured data w4,5x, one finds a large scatter in the D and K values. A first discussion about limitations to get accurate K and D data by SIMS depth profiling was published in 1992 by Chater et al. w2x. The aim of this paper is to present possibilities to overcome these limitations and to get more accurate values of D and K.
2. Solution of the diffusion problem ) Corresponding author. Tel.: q49-5323-72-2634; fax: q495323-72-3184. E-mail address: [email protected] ŽP. Fielitz..
The exchange kinetics of oxygen isotopes at the solid surface is an essential boundary condition of
0167-2738r01r$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 3 8 Ž 0 1 . 0 0 8 9 3 - 1
P. Fielitz, G. Borchardtr Solid State Ionics 144 (2001) 71–80
72
Fig. 1. Dependence of the 18 O concentration at the surface csŽ t . on the annealing time, t ŽEq. Ž7. for cg s 1 and c` s 0.002.. Solid circles show the working range for calculating trt ratios from measured surface concentrations.
the diffusion process. In thermodynamical equilibrium, the 16 O isotope flux out of the sample equals the 18 O isotope flux into the sample Žneglecting the very low concentration of 17O.. This means the total oxygen concentration is constant:
For the isotopes of a given element, differences in sputter yield and ionisation yield can be neglected. Thus, it is justified because of Eq. Ž1. to convert the measured intensity signals into 18 O atomic fractions:
cO16 Ž x ,t . q cO18 Ž x ,t . s cOtot s constant.
cs
Ž 1.
Under most experimental conditions, the oxygen incorporation can be phenomenologically described as a first-order reaction: in j18 s Ž cg y cs . K ,
Ž 2.
where K is the surface exchange coefficient. This flux must be equal to the diffusional flux of 18 O into the bulk of the sample, i.e. at x s 0: Ec D
ž / Ex
s Ž cg y cs . K ,
Ž 3.
I Ž 18 O . I Ž 18 O . q I Ž 16 O .
c Ž x ,t . y c` s Ž c g y c` . erfc
Et
sD
E c Ž x ,t . Ex
2
Ž 4.
with respect to this boundary condition ŽEq. Ž3...
x
x
žs/ t
t
x
t
ž ( / ž (/
where D is the self-diffusivity of oxygen. To calculate the expected 18 O depth profile, one has to solve the diffusion equation: Ec Ž x ,t .
Ž 5.
For the sake of simplicity, the term AconcentrationB is used instead of Aatomic fractionB throughout the text following below. The analytical solution of the diffusion problem is given by Crank w6x:
xs 0
2
.
yexp 2
q
s
t
t
erfc
q
s
t
,
Ž 6.1 . with:
s ' 2'Dt and t '
D K2
,
Ž 6.2 .
P. Fielitz, G. Borchardtr Solid State Ionics 144 (2001) 71–80
where cg is the constant 18 O gas concentration, c` the natural abundance of 18 O in the sample Žfor x s `., s the diffusion length, and t a characteristic time constant. The physical meaning of t is obvious if one considers the time dependence of the 18 O concentration at the surface of the solid Ž csŽ t . s cŽ0, t ..: cs Ž t . y c` s Ž cg y c` . 1 y exp
t
t
ž / ž( / t
erfc
t
.
Ž 7. For large or small trt ratios, the following approximate equations are useful Žwith errors less than 10%.:
ž
cs Ž t . y c` ( Ž cg y c` . 1 y
1
'p
t
(/ t
for t G 3t ,
Ž 8. cs Ž t . y c` ( Ž cg y c` .
2
'p
(
t
t
for t F 0.01t .
Ž 9. In Fig. 1, the time dependence of the 18 O concentration at the surface is plotted for a gas concentration, cg s 1, and a natural abundance, c` s 0.002. The defined characteristic time constant, t , in Eq. Ž6.2. determines the duration that is necessary to reach equilibrium between the 18 O gas concentration and the 18 O concentration at the surface of the solid. The annealing time, t, is given for each isotope exchange experiment at annealing temperature, T, and pressure, p. This means the characteristic time constant t s t ŽT, p . can be calculated by measuring the 18 O surface concentration only. From Fig. 1, one can estimate that for trt ) 10 or trt - 10y6 , the error is no longer acceptable.
3. Determination of D and K 3.1. Simultaneous determination of D and K Measured SIMS depth profiles can be fitted by Eq. Ž6.1. to get both unknown parameters D and K simultaneously. However, this works well for both parameters only in a limited range. In order to be in this working parameter range, it is very crucial to
73
choose an adequate annealing time. Fig. 1 shows that for t ) 10t , the determination of the characteristic time constant, t Žand, thus, K ., may be subject to a significant error. The value 10t is chosen here for discussion and depends on the accuracy of the measured values of cs and cg . A quantitative estimation of the resulting error can be easily done via Eq. Ž8.. To determine an accurate value of the diffusion coefficient, the dynamic range of the 18 O concentration should comprise at least one order of magnitude w2x. This estimation depends also on the data quality. From Fig. 1 Ždotted lines., it follows that the trt ratio should be greater than 3 = 10y4 . Both limitations lead to the following range of adequate annealing times: 3 = 10y4t F t F 10t .
Ž 10 .
If one assumes a range of practical values of t from 100 s to 1 month Žf 3 = 10 6 s., one gets for the interval defined by Eq. Ž10. the measurable range of t from 10 to 10 10 s Žf 317 years.. An additional boundary is set by the limitations in measuring the diffusion length, s , of the isotopes. Kilner and De Souza w3x have estimated that 30 nm is an acceptable minimum for the diffusion length that can be measured using a 10 keV Xe primary beam for SIMS analysis. This minimum is caused by ion beam mixing only. Further possible reasons for an inaccurate measurement of the diffusion length are surface roughness or the necessity to deposit a conductive layer on the surface to prevent charge effects, respectively. As a working limit for depth profiling, Kilner and De Souza w3x have taken 10 mm as the maximum for a high-quality Žfully dense. sample. This maximum is caused by crater base roughening. A usual way to overcome this limitation of depth profiling is the line scan method. The working maximum for the line scan method is about 1 mm w3x. Table 1 Working parameters for the simultaneous determination of D and K
Minimum Maximum
Annealing time, t
Ratio j ' trt
Characteristic time, t
Diffusion length, s
100 s 1 month
3=10y4 10
10 s 317 years
30 nm 5 mm, 0.5 mm
P. Fielitz, G. Borchardtr Solid State Ionics 144 (2001) 71–80
74
The working parameters of Table 1 lead to the working range of simultaneous determination of K and D Žusing Eq. Ž6.2.. that is plotted in Fig. 2 according to the following relations:
atmosphere. This can lead to large errors for the determination of t . Therefore, annealing times t F t should be realised in the parameter range near this line. Line IC–IIC: On this line, one has t s 317 years s constant and, thus, very large characteristic time constants. The problem here is to realise long annealing times to achieve a sufficient dynamic range of the 18 O concentration. In the parameter range near this line, annealing times t G 3 = 10y4t should be realised. Line IA–IB–IC: These two lines show the path of the minimal diffusion length smin s 30 nm. Therefore, in the range near this line, the depth profiles are very shallow so that ion mixing and surface roughness can lead to large errors of the determined diffusion coefficients. In this area of the parameter range, very high-quality samples and optimised SIMS techniques that allow ultra shallow depth profiling are required. Line IIA–IIB–IIC: Here, the diffusion length Ž smax s 0.5 mm. is very large so that it is necessary to use the line scan method. A qualitative estimation of the error resulting from a simultaneous determination of the D and K v
K2
1 F
tmax 2 smin
4 t max
1 F
D FDF
tmin 2 smax
4 t min
;
.
(
2 j min
smax
K F
F D
(
2 j max
smin
;
Ž 11 .
In Fig. 3, the different depth profiles at the intersection points of the different lines shown in Fig. 2 are plotted. Both figures illustrate qualitatively the latent problems for given Žor expected. values of the parameters D and K. Line IA–IIA: This line shows the line of constant characteristic time Žt s 10 s.. This means in the parameter range of D and K near this line, one has short characteristic time constants. Thus, it is necessary to realise short annealing times of the samples. This may require an optimised furnace with sufficiently high heating rates. At annealing times t s 10t , the 18 O concentration at the surface reaches about 80% of the 18 O concentration of the gas v
v
v
Fig. 2. Working range of simultaneous determination of D and K by SIMS measurements. The depth profiles at the six intersection points ŽIA, IB, IC, IIA, IIB, IIC. are shown in Fig. 3.
P. Fielitz, G. Borchardtr Solid State Ionics 144 (2001) 71–80
75
Fig. 3. Depth profiles at the six intersection points in Fig. 2. For the calculation Eq. Ž6.1. was used with the shown parameters. The x-axis for the shallow depth profiles IA, IB and IC is located at the upper edge of the plot.
values at the intersections of the six lines in Fig. 2 is compiled in Table 2. 3.2. Correlation between D and K A usual way to detect a correlation between D and K data is to plot both parameters in a K–D diagram w4,5x. In Fig. 4, data of different perovskite oxides are shown in such a diagram. Despite the more or less large scatter of the data, one would
assume a strong correlation between the values of D and K in this diagram. The solid line has the slope 1r2. From this, one would expect that the characteristic time constant, t s DrK 2 , is virtually constant for all isotope exchange experiments at all different perovskites. To check this expectation, the same data are shown in a t –D diagram in Fig. 5. However, in this diagram, one can see a huge scatter of the data. Thus, it is impossible to see the same correlation for all data. In this plot, it is much easier to see differ-
Table 2 Qualitative estimation of the error which results if t , D and K values are determined at the intersections of the lines in Fig. 2 Žsee also Fig. 3. Intersection point
IA IB IC IIA IIB IIC
Experimental parameters
Qualitative estimation of the error
Annealing time, t
Characteristic time, t
Diffusion length, s
Characteristic time, t
Diffusion coefficient, D
Surface exchange coefficient, K s Ž Drt .1r 2
100 s 1 month 1 month 100 s 100 s 1 month
10 s 3.5 days 317 years 10 s 3.5 days 317 years
30 nm 30 nm 30 nm 0.5 mm 0.5 mm 0.5 mm
large large normal large normal normal
large large large normal normal normal
very large very large large large normal normal
76
P. Fielitz, G. Borchardtr Solid State Ionics 144 (2001) 71–80
Fig. 4. Plot of measured D and K values in a D–K diagram for a number of perovskite oxides w4x. The solid line is drawn to show the slope 1r2.
Fig. 5. Same data as in Fig. 4 plotted in a t –D diagram. The characteristic time constant t was calculated from DrK 2 .
P. Fielitz, G. Borchardtr Solid State Ionics 144 (2001) 71–80
ences between the data. This is the reason why we suggest to use a t –D diagram to check for correlations between D and K data. A temperature-dependent correlation between K and D can be detected by measuring the temperature dependence of the characteristic time constant t ŽT . s DŽT .rK 2 ŽT .. Both diffusion and isotope exchange are thermally activated processes with activation enthalpies, D H D and D H K , respectively. Consequently, the activation enthalpy, D Ht , for the characteristic time constant, t , is given by: D Ht s D H D y 2D H K .
Ž 12 .
To measure the temperature dependence, t s t ŽT ., of the characteristic time constant, it is necessary to measure the 18 O concentration at the surface only Žand the 18 O concentration in the gas atmosphere.. Thus, it is possible to measure the characteristic time constant with good accuracy. There is only a small instrumental mass fractionation that occurs during the measurement of oxygen isotope ratios by SIMS. Eiler et al. w7x have reported an instrumental mass fractionation between 3% and 7.5%. They have made 373 measurements of the 18 Oyr16 Oyratios in 40 silicate and phosphate minerals and glasses for the purpose of characterising this matrix effect. Further methodical advantages are as follows. It is possible to use static SIMS techniques for the determination of t , e.g. time-of-flight secondary ion mass spectrometers ŽTOF-SIMS.. The range of adequate annealing times becomes significantly greater Žsee Fig. 1.: v
77
moval of the first surface layers. Eq. Ž14., which one gets by inserting Eq. Ž7. into Eq. Ž6.1., can then be used to estimate the errors arising from the surface layer removal:
Y'
c Ž x ,t . y c` cs y c` erfc
s
x
x
t
t
x
t
ž s / y exp ž 2 s ( t q t / erfc ž s q ( t / 1 y exp
t
t
ž t / erfc ž ( t /
.
Ž 14 . This equation is plotted in Fig. 6 for different trt ratios. It is obvious that this function is almost independent of the trt ratios so that the following practical equation permits to estimate the surface concentration error: D cs 1.4 " 0.25 DY s (y D x for D x F 0.2 s , cs y c` s Ž 15 . where D x is the sputter depth. For practical cases with s 4 30 nm, the error in the surface concentration, produced by surface layer removal, is small. However, for s f 30 nm, the surface concentration is decreased by about 14% for a sputter depth of 3 nm, which means that, in this case, static SIMS would be absolutely necessary.
v
y6
10 t F t F 10t
Ž 13 .
because it is not necessary to have a sufficient dynamic range of the 18 O concentration to determine the diffusion coefficient. The time to analyse the surface concentration is much shorter than the time to measure the whole 18 O depth profile to determine simultaneously D and K Žespecially at high temperatures and, thus, large diffusion lengths.. Therefore, it is much less time-consuming to measure the temperature dependence of t or to study the dependence of the 18 O surface concentration on the annealing time Žwhich is expected to agree with Eq. Ž7., what has to be proved.. Measuring the 18 O concentration at the surface by a dynamical SIMS techniques will result in the rev
3.3. Separate determination of D and K In Section 3.2, we discussed why it is useful to measure the characteristic time constant, t , separately. From this measured t data, one could calculate the surface exchange coefficients if D was measured independently of K. In this section, we will discuss this case. From Eq. Ž6.1., one gets the following approximate equation for long annealing times: x c Ž x ,t . y c` s Ž cs y c` . erfc for t 4 t . 2'Dt Ž 16 .
ž
/
This means one can, in principle, determine the diffusion coefficient independently of the surface exchange coefficient by choosing a sufficiently long
P. Fielitz, G. Borchardtr Solid State Ionics 144 (2001) 71–80
78
Fig. 6. Decrease of the 18 O concentration near the surface calculated for different trt ratios Žplot of Eq. Ž14...
annealing time. One would also get Eq. Ž16. by solving the diffusion equation with respect to the boundary condition cŽ0, t . s cs s constant w6x. A further simplification of Eq. Ž16. can be achieved because of the asymptotic formula w6x erfcŽ z . f Žp .y0 .5 zy1 expŽyz 2 . for large z s xrw2Ž Dt .1r2 x. This means a semilogarithmic plot of x Ž c y c` . vs. x 2 should yield curves approximated by the following equation:
ln Ž x Ž c y c` . . ( y
1 4 Dt
ž
x 2 q ln Ž cs y c` .
2'Dt
'p
/
.
Ž 17 . Eq. Ž17. implies that for large penetration depths, the slope of the curves in a ŽlogŽ x Ž c y c` .., x 2 . plot is inversely proportional to 4 Dt. In Fig. 7, the corresponding function logwŽ xrs . Ž cŽ x, t . y c` .rŽ cg y c` .x is plotted vs. Ž xrs . 2 . The different curves are calculated by Eq. Ž6.1. for different trt ratios. One can see that the slope of the curves for x G s depends very weakly on the trt ratios. Thus, if one fits measured data by using Eq. Ž16. for x G s only, one obtains the correct value of the diffusion coefficient, D, and does not need any
information on the surface exchange coefficient, K. From Fig. 7, one can estimate that the expected error of this method is less than 10%. If one assumes an error of crater depth measurement of 10%, one can estimate an overall error of the diffusion coefficient of 30%. From a usually used ŽlogŽ c y c` ., x . plot, it is, however, not evident whether the condition x G s is fulfilled or not. Therefore, it is useful to use a ŽlogŽ x Ž c y c` .., x 2 . plot and to check for linearity in this plot. Fig. 8 shows measured 18 O SIMS depth profiles in such a ŽlogŽ x Ž c y c` .., x 2 . plot from single crystalline mullite at different temperatures and orientations w8x. Also shown are solid lines for the best fit of the measured data according to Eq. Ž16.. At all temperatures, the 18 O concentration at the surface had reached almost the 18 O concentration of the gas phase so that it was not possible to calculate the surface exchange coefficients from this data. The separate measurement of D and t Žand, thus, . K is especially useful at low temperatures. For a simultaneous determination of D and K, one cannot choose an arbitrarily long annealing time Ž t F 10t . so that one has the limitation s F 2Ž10 Dt .1r2 s 2Ž10.1r2 DrK for the diffusion length of the 18 O
P. Fielitz, G. Borchardtr Solid State Ionics 144 (2001) 71–80
79
Fig. 7. Calculated 18 O depth profiles in a ŽlogwŽ xrs . P Ž cŽ x, t . y c` .rŽ cg y c` .x, Ž xrs . 2 . plot for different trt ratios Žusing Eq. Ž6.1... The solid lines show the slopes of the curves for large xrs ratios.
Fig. 8. SIMS measurements of 18 O depth profiles from single crystalline mullite at different temperatures and orientations w8x. Also shown are solid lines for the best fit of the measured data according to Eq. Ž16.. See text for the explanation of axis scales.
80
P. Fielitz, G. Borchardtr Solid State Ionics 144 (2001) 71–80
isotope. Thus, if one decreases the temperature to determine the temperature dependence of D and K, there will be a temperature where the simultaneous determination of both parameters starts to become subject to severe errors Žnear the line IA–IB in Fig. 2.. To overcome this problem, in a first step, an annealing time should be chosen, which is sufficiently short Ž t F t . to determine the characteristic time constant with a small error by measuring the 18 O surface concentration only. Then, in a next step, the same sample should be annealed at the same temperature for a much longer time to get a sufficiently large diffusion length, s , and to determine an accurate diffusion coefficient. From both independently Žand accurately. measured parameters t and D, one can then calculate the surface exchange coefficient, K s Ž Drt .1r2 .
4. Summary Oxygen tracer diffusivities in oxides are usually determined via gasrsolid exchange experiments in an atmosphere enriched as to the concentration of the rare stable isotope 18 O. The resulting 18 O diffusion profile in the oxide may be influenced by the kinetics of the 16 O and 18 O isotope exchange at the surface. This kinetics is described by the surface exchange coefficient, K. The analytical solution of the diffusion problem suggests the definition of a characteristic time constant, t ' DrK 2 . This time constant determines the duration that is necessary to reach equilibrium between the 18 O gas concentration and the 18 O concentration at the surface of the solid. By measuring the temperature dependence of t , it is possible to measure the temperature-dependent correlation between diffusivity and surface exchange. The characteristic time constant, t , can be determined by measuring the 18 O gas concentration and the 18 O concentration at the surface of the solid only. This can be done with good accuracy because there
is only a small instrumental mass fractionation that occurs during the measurement of oxygen isotope ratios by SIMS w7x. A further advantage is the possibility to use static SIMS techniques for this task, e.g. TOF-SIMS. The simultaneous determination of D and K works well in a large parameter range. However, at low temperatures, the simultaneous determination of D and K is subject to large errors because one cannot choose an arbitrarily long annealing time to get a sufficient diffusion length of the 18 O isotope. To overcome this problem, it was discussed how to determine D and K separately. Acknowledgements Financial support from Deutsche Forschungsgemeinschaft ŽDFG. is gratefully acknowledged. We are indebted to an anonymous reviewer for his valuable comments. References w1x J.A. Kilner, B.C.H Steele, Solid State Ionics 12 Ž1984. 89. w2x R.J. Chater, S. Carter, J.A. Kilner, B.C.H. Steele, Solid State Ionics 53–56 Ž1992. 859. w3x J.A. Kilner, R.A. De Souza, in: F.W. Poulsen, N. Bonanos, S. Linderoth, M. Mogensen, B. Zachau-Christiansen ŽEds.., Proceedings of the 17th Risø International Symposium on Materials Science: High Temperature Electrochemistry: Ceramics and Metals. 1996, p. 41. w4x J.A. Kilner, R.A. De Souza, I.C. Fullarton, Solid State Ionics 86–88 Ž1996. 703. w5x R.H.E. van Doorn, I.C. Fullarton, R.A. de Souza, J.A. Kilner, H.J.M. Bouwmeester, A.J. Burggraaf, Solid State Ionics 96 Ž1997. 1. w6x J. Crank, The Mathematics of Diffusion. 2nd edn., Oxford Univ. Press, Oxford, 1975. w7x J.M. Eiler, C.G. Graham, J.W. Valley, Chemical Geology 138 Ž1997. 221. w8x P. Fielitz, G. Borchardt, M. Schmucker, H. Schneider, M. ¨ Wiedenbeck, D. Rhede, S. Weber, S. Scherrer, Journal of the American Ceramic Society Ž2001. in press.
On the accurate measurement of oxygen self-diffusivities and surface exchange coefficients in oxides via SIMS depth profiling P. Fielitz ) , G. Borchardt FB Physik, Metallurgie und Werkstoffwissenschaften, AG Thermochemie and Mikrokinetik, Technische UniÕersitat ¨ Clausthal, Robert-Koch-Str. 42, D-38678 Clausthal-Zellerfeld, Germany Received 26 January 2001; received in revised form 30 May 2001; accepted 13 June 2001
Abstract Oxygen diffusivity in oxides is usually determined via gasrsolid exchange experiments in an atmosphere enriched as to the concentration of the rare stable isotope 18 O. The kinetics of the 16 O and 18 O isotope exchange at the surface is usually described by the surface exchange coefficient, K. The analytical solution of the diffusion problem suggests the definition of a characteristic time constant, t ' DrK 2. This time constant determines the duration that is necessary to reach equilibrium between the 18 O gas concentration and the 18 O concentration at the surface of the solid. The simultaneous determination of D and K by secondary ion mass spectrometry ŽSIMS. depth profiling works well in a large parameter range of D and K. However, at low temperatures, the simultaneous determination of D and K may be subject to severe errors. To overcome this problem, the possibility to determine D and K separately will be discussed. q 2001 Elsevier Science B.V. All rights reserved. Keywords: Self-diffusion coefficient; Surface exchange; Isotope exchange; SIMS depth profiling
1. Introduction The most direct method of measuring the oxygen self-diffusivity, D, and the surface exchange coefficient, K, of oxide materials is isotope exchange followed by secondary ion mass spectrometry ŽSIMS. depth profiling w1–3x. The inherent accuracy of this method is based mainly on the fortuitous circumstance that the ratio 18 Or16 O of the natural isotope abundancies is about 1r500. This means the dynamic range of the measured 18 O depth profile can reach almost three orders of magnitude. Such a
dynamic range is more than necessary to determine accurate diffusion coefficients and surface exchange coefficients. However, if one looks at measured data w4,5x, one finds a large scatter in the D and K values. A first discussion about limitations to get accurate K and D data by SIMS depth profiling was published in 1992 by Chater et al. w2x. The aim of this paper is to present possibilities to overcome these limitations and to get more accurate values of D and K.
2. Solution of the diffusion problem ) Corresponding author. Tel.: q49-5323-72-2634; fax: q495323-72-3184. E-mail address: [email protected] ŽP. Fielitz..
The exchange kinetics of oxygen isotopes at the solid surface is an essential boundary condition of
0167-2738r01r$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 3 8 Ž 0 1 . 0 0 8 9 3 - 1
P. Fielitz, G. Borchardtr Solid State Ionics 144 (2001) 71–80
72
Fig. 1. Dependence of the 18 O concentration at the surface csŽ t . on the annealing time, t ŽEq. Ž7. for cg s 1 and c` s 0.002.. Solid circles show the working range for calculating trt ratios from measured surface concentrations.
the diffusion process. In thermodynamical equilibrium, the 16 O isotope flux out of the sample equals the 18 O isotope flux into the sample Žneglecting the very low concentration of 17O.. This means the total oxygen concentration is constant:
For the isotopes of a given element, differences in sputter yield and ionisation yield can be neglected. Thus, it is justified because of Eq. Ž1. to convert the measured intensity signals into 18 O atomic fractions:
cO16 Ž x ,t . q cO18 Ž x ,t . s cOtot s constant.
cs
Ž 1.
Under most experimental conditions, the oxygen incorporation can be phenomenologically described as a first-order reaction: in j18 s Ž cg y cs . K ,
Ž 2.
where K is the surface exchange coefficient. This flux must be equal to the diffusional flux of 18 O into the bulk of the sample, i.e. at x s 0: Ec D
ž / Ex
s Ž cg y cs . K ,
Ž 3.
I Ž 18 O . I Ž 18 O . q I Ž 16 O .
c Ž x ,t . y c` s Ž c g y c` . erfc
Et
sD
E c Ž x ,t . Ex
2
Ž 4.
with respect to this boundary condition ŽEq. Ž3...
x
x
žs/ t
t
x
t
ž ( / ž (/
where D is the self-diffusivity of oxygen. To calculate the expected 18 O depth profile, one has to solve the diffusion equation: Ec Ž x ,t .
Ž 5.
For the sake of simplicity, the term AconcentrationB is used instead of Aatomic fractionB throughout the text following below. The analytical solution of the diffusion problem is given by Crank w6x:
xs 0
2
.
yexp 2
q
s
t
t
erfc
q
s
t
,
Ž 6.1 . with:
s ' 2'Dt and t '
D K2
,
Ž 6.2 .
P. Fielitz, G. Borchardtr Solid State Ionics 144 (2001) 71–80
where cg is the constant 18 O gas concentration, c` the natural abundance of 18 O in the sample Žfor x s `., s the diffusion length, and t a characteristic time constant. The physical meaning of t is obvious if one considers the time dependence of the 18 O concentration at the surface of the solid Ž csŽ t . s cŽ0, t ..: cs Ž t . y c` s Ž cg y c` . 1 y exp
t
t
ž / ž( / t
erfc
t
.
Ž 7. For large or small trt ratios, the following approximate equations are useful Žwith errors less than 10%.:
ž
cs Ž t . y c` ( Ž cg y c` . 1 y
1
'p
t
(/ t
for t G 3t ,
Ž 8. cs Ž t . y c` ( Ž cg y c` .
2
'p
(
t
t
for t F 0.01t .
Ž 9. In Fig. 1, the time dependence of the 18 O concentration at the surface is plotted for a gas concentration, cg s 1, and a natural abundance, c` s 0.002. The defined characteristic time constant, t , in Eq. Ž6.2. determines the duration that is necessary to reach equilibrium between the 18 O gas concentration and the 18 O concentration at the surface of the solid. The annealing time, t, is given for each isotope exchange experiment at annealing temperature, T, and pressure, p. This means the characteristic time constant t s t ŽT, p . can be calculated by measuring the 18 O surface concentration only. From Fig. 1, one can estimate that for trt ) 10 or trt - 10y6 , the error is no longer acceptable.
3. Determination of D and K 3.1. Simultaneous determination of D and K Measured SIMS depth profiles can be fitted by Eq. Ž6.1. to get both unknown parameters D and K simultaneously. However, this works well for both parameters only in a limited range. In order to be in this working parameter range, it is very crucial to
73
choose an adequate annealing time. Fig. 1 shows that for t ) 10t , the determination of the characteristic time constant, t Žand, thus, K ., may be subject to a significant error. The value 10t is chosen here for discussion and depends on the accuracy of the measured values of cs and cg . A quantitative estimation of the resulting error can be easily done via Eq. Ž8.. To determine an accurate value of the diffusion coefficient, the dynamic range of the 18 O concentration should comprise at least one order of magnitude w2x. This estimation depends also on the data quality. From Fig. 1 Ždotted lines., it follows that the trt ratio should be greater than 3 = 10y4 . Both limitations lead to the following range of adequate annealing times: 3 = 10y4t F t F 10t .
Ž 10 .
If one assumes a range of practical values of t from 100 s to 1 month Žf 3 = 10 6 s., one gets for the interval defined by Eq. Ž10. the measurable range of t from 10 to 10 10 s Žf 317 years.. An additional boundary is set by the limitations in measuring the diffusion length, s , of the isotopes. Kilner and De Souza w3x have estimated that 30 nm is an acceptable minimum for the diffusion length that can be measured using a 10 keV Xe primary beam for SIMS analysis. This minimum is caused by ion beam mixing only. Further possible reasons for an inaccurate measurement of the diffusion length are surface roughness or the necessity to deposit a conductive layer on the surface to prevent charge effects, respectively. As a working limit for depth profiling, Kilner and De Souza w3x have taken 10 mm as the maximum for a high-quality Žfully dense. sample. This maximum is caused by crater base roughening. A usual way to overcome this limitation of depth profiling is the line scan method. The working maximum for the line scan method is about 1 mm w3x. Table 1 Working parameters for the simultaneous determination of D and K
Minimum Maximum
Annealing time, t
Ratio j ' trt
Characteristic time, t
Diffusion length, s
100 s 1 month
3=10y4 10
10 s 317 years
30 nm 5 mm, 0.5 mm
P. Fielitz, G. Borchardtr Solid State Ionics 144 (2001) 71–80
74
The working parameters of Table 1 lead to the working range of simultaneous determination of K and D Žusing Eq. Ž6.2.. that is plotted in Fig. 2 according to the following relations:
atmosphere. This can lead to large errors for the determination of t . Therefore, annealing times t F t should be realised in the parameter range near this line. Line IC–IIC: On this line, one has t s 317 years s constant and, thus, very large characteristic time constants. The problem here is to realise long annealing times to achieve a sufficient dynamic range of the 18 O concentration. In the parameter range near this line, annealing times t G 3 = 10y4t should be realised. Line IA–IB–IC: These two lines show the path of the minimal diffusion length smin s 30 nm. Therefore, in the range near this line, the depth profiles are very shallow so that ion mixing and surface roughness can lead to large errors of the determined diffusion coefficients. In this area of the parameter range, very high-quality samples and optimised SIMS techniques that allow ultra shallow depth profiling are required. Line IIA–IIB–IIC: Here, the diffusion length Ž smax s 0.5 mm. is very large so that it is necessary to use the line scan method. A qualitative estimation of the error resulting from a simultaneous determination of the D and K v
K2
1 F
tmax 2 smin
4 t max
1 F
D FDF
tmin 2 smax
4 t min
;
.
(
2 j min
smax
K F
F D
(
2 j max
smin
;
Ž 11 .
In Fig. 3, the different depth profiles at the intersection points of the different lines shown in Fig. 2 are plotted. Both figures illustrate qualitatively the latent problems for given Žor expected. values of the parameters D and K. Line IA–IIA: This line shows the line of constant characteristic time Žt s 10 s.. This means in the parameter range of D and K near this line, one has short characteristic time constants. Thus, it is necessary to realise short annealing times of the samples. This may require an optimised furnace with sufficiently high heating rates. At annealing times t s 10t , the 18 O concentration at the surface reaches about 80% of the 18 O concentration of the gas v
v
v
Fig. 2. Working range of simultaneous determination of D and K by SIMS measurements. The depth profiles at the six intersection points ŽIA, IB, IC, IIA, IIB, IIC. are shown in Fig. 3.
P. Fielitz, G. Borchardtr Solid State Ionics 144 (2001) 71–80
75
Fig. 3. Depth profiles at the six intersection points in Fig. 2. For the calculation Eq. Ž6.1. was used with the shown parameters. The x-axis for the shallow depth profiles IA, IB and IC is located at the upper edge of the plot.
values at the intersections of the six lines in Fig. 2 is compiled in Table 2. 3.2. Correlation between D and K A usual way to detect a correlation between D and K data is to plot both parameters in a K–D diagram w4,5x. In Fig. 4, data of different perovskite oxides are shown in such a diagram. Despite the more or less large scatter of the data, one would
assume a strong correlation between the values of D and K in this diagram. The solid line has the slope 1r2. From this, one would expect that the characteristic time constant, t s DrK 2 , is virtually constant for all isotope exchange experiments at all different perovskites. To check this expectation, the same data are shown in a t –D diagram in Fig. 5. However, in this diagram, one can see a huge scatter of the data. Thus, it is impossible to see the same correlation for all data. In this plot, it is much easier to see differ-
Table 2 Qualitative estimation of the error which results if t , D and K values are determined at the intersections of the lines in Fig. 2 Žsee also Fig. 3. Intersection point
IA IB IC IIA IIB IIC
Experimental parameters
Qualitative estimation of the error
Annealing time, t
Characteristic time, t
Diffusion length, s
Characteristic time, t
Diffusion coefficient, D
Surface exchange coefficient, K s Ž Drt .1r 2
100 s 1 month 1 month 100 s 100 s 1 month
10 s 3.5 days 317 years 10 s 3.5 days 317 years
30 nm 30 nm 30 nm 0.5 mm 0.5 mm 0.5 mm
large large normal large normal normal
large large large normal normal normal
very large very large large large normal normal
76
P. Fielitz, G. Borchardtr Solid State Ionics 144 (2001) 71–80
Fig. 4. Plot of measured D and K values in a D–K diagram for a number of perovskite oxides w4x. The solid line is drawn to show the slope 1r2.
Fig. 5. Same data as in Fig. 4 plotted in a t –D diagram. The characteristic time constant t was calculated from DrK 2 .
P. Fielitz, G. Borchardtr Solid State Ionics 144 (2001) 71–80
ences between the data. This is the reason why we suggest to use a t –D diagram to check for correlations between D and K data. A temperature-dependent correlation between K and D can be detected by measuring the temperature dependence of the characteristic time constant t ŽT . s DŽT .rK 2 ŽT .. Both diffusion and isotope exchange are thermally activated processes with activation enthalpies, D H D and D H K , respectively. Consequently, the activation enthalpy, D Ht , for the characteristic time constant, t , is given by: D Ht s D H D y 2D H K .
Ž 12 .
To measure the temperature dependence, t s t ŽT ., of the characteristic time constant, it is necessary to measure the 18 O concentration at the surface only Žand the 18 O concentration in the gas atmosphere.. Thus, it is possible to measure the characteristic time constant with good accuracy. There is only a small instrumental mass fractionation that occurs during the measurement of oxygen isotope ratios by SIMS. Eiler et al. w7x have reported an instrumental mass fractionation between 3% and 7.5%. They have made 373 measurements of the 18 Oyr16 Oyratios in 40 silicate and phosphate minerals and glasses for the purpose of characterising this matrix effect. Further methodical advantages are as follows. It is possible to use static SIMS techniques for the determination of t , e.g. time-of-flight secondary ion mass spectrometers ŽTOF-SIMS.. The range of adequate annealing times becomes significantly greater Žsee Fig. 1.: v
77
moval of the first surface layers. Eq. Ž14., which one gets by inserting Eq. Ž7. into Eq. Ž6.1., can then be used to estimate the errors arising from the surface layer removal:
Y'
c Ž x ,t . y c` cs y c` erfc
s
x
x
t
t
x
t
ž s / y exp ž 2 s ( t q t / erfc ž s q ( t / 1 y exp
t
t
ž t / erfc ž ( t /
.
Ž 14 . This equation is plotted in Fig. 6 for different trt ratios. It is obvious that this function is almost independent of the trt ratios so that the following practical equation permits to estimate the surface concentration error: D cs 1.4 " 0.25 DY s (y D x for D x F 0.2 s , cs y c` s Ž 15 . where D x is the sputter depth. For practical cases with s 4 30 nm, the error in the surface concentration, produced by surface layer removal, is small. However, for s f 30 nm, the surface concentration is decreased by about 14% for a sputter depth of 3 nm, which means that, in this case, static SIMS would be absolutely necessary.
v
y6
10 t F t F 10t
Ž 13 .
because it is not necessary to have a sufficient dynamic range of the 18 O concentration to determine the diffusion coefficient. The time to analyse the surface concentration is much shorter than the time to measure the whole 18 O depth profile to determine simultaneously D and K Žespecially at high temperatures and, thus, large diffusion lengths.. Therefore, it is much less time-consuming to measure the temperature dependence of t or to study the dependence of the 18 O surface concentration on the annealing time Žwhich is expected to agree with Eq. Ž7., what has to be proved.. Measuring the 18 O concentration at the surface by a dynamical SIMS techniques will result in the rev
3.3. Separate determination of D and K In Section 3.2, we discussed why it is useful to measure the characteristic time constant, t , separately. From this measured t data, one could calculate the surface exchange coefficients if D was measured independently of K. In this section, we will discuss this case. From Eq. Ž6.1., one gets the following approximate equation for long annealing times: x c Ž x ,t . y c` s Ž cs y c` . erfc for t 4 t . 2'Dt Ž 16 .
ž
/
This means one can, in principle, determine the diffusion coefficient independently of the surface exchange coefficient by choosing a sufficiently long
P. Fielitz, G. Borchardtr Solid State Ionics 144 (2001) 71–80
78
Fig. 6. Decrease of the 18 O concentration near the surface calculated for different trt ratios Žplot of Eq. Ž14...
annealing time. One would also get Eq. Ž16. by solving the diffusion equation with respect to the boundary condition cŽ0, t . s cs s constant w6x. A further simplification of Eq. Ž16. can be achieved because of the asymptotic formula w6x erfcŽ z . f Žp .y0 .5 zy1 expŽyz 2 . for large z s xrw2Ž Dt .1r2 x. This means a semilogarithmic plot of x Ž c y c` . vs. x 2 should yield curves approximated by the following equation:
ln Ž x Ž c y c` . . ( y
1 4 Dt
ž
x 2 q ln Ž cs y c` .
2'Dt
'p
/
.
Ž 17 . Eq. Ž17. implies that for large penetration depths, the slope of the curves in a ŽlogŽ x Ž c y c` .., x 2 . plot is inversely proportional to 4 Dt. In Fig. 7, the corresponding function logwŽ xrs . Ž cŽ x, t . y c` .rŽ cg y c` .x is plotted vs. Ž xrs . 2 . The different curves are calculated by Eq. Ž6.1. for different trt ratios. One can see that the slope of the curves for x G s depends very weakly on the trt ratios. Thus, if one fits measured data by using Eq. Ž16. for x G s only, one obtains the correct value of the diffusion coefficient, D, and does not need any
information on the surface exchange coefficient, K. From Fig. 7, one can estimate that the expected error of this method is less than 10%. If one assumes an error of crater depth measurement of 10%, one can estimate an overall error of the diffusion coefficient of 30%. From a usually used ŽlogŽ c y c` ., x . plot, it is, however, not evident whether the condition x G s is fulfilled or not. Therefore, it is useful to use a ŽlogŽ x Ž c y c` .., x 2 . plot and to check for linearity in this plot. Fig. 8 shows measured 18 O SIMS depth profiles in such a ŽlogŽ x Ž c y c` .., x 2 . plot from single crystalline mullite at different temperatures and orientations w8x. Also shown are solid lines for the best fit of the measured data according to Eq. Ž16.. At all temperatures, the 18 O concentration at the surface had reached almost the 18 O concentration of the gas phase so that it was not possible to calculate the surface exchange coefficients from this data. The separate measurement of D and t Žand, thus, . K is especially useful at low temperatures. For a simultaneous determination of D and K, one cannot choose an arbitrarily long annealing time Ž t F 10t . so that one has the limitation s F 2Ž10 Dt .1r2 s 2Ž10.1r2 DrK for the diffusion length of the 18 O
P. Fielitz, G. Borchardtr Solid State Ionics 144 (2001) 71–80
79
Fig. 7. Calculated 18 O depth profiles in a ŽlogwŽ xrs . P Ž cŽ x, t . y c` .rŽ cg y c` .x, Ž xrs . 2 . plot for different trt ratios Žusing Eq. Ž6.1... The solid lines show the slopes of the curves for large xrs ratios.
Fig. 8. SIMS measurements of 18 O depth profiles from single crystalline mullite at different temperatures and orientations w8x. Also shown are solid lines for the best fit of the measured data according to Eq. Ž16.. See text for the explanation of axis scales.
80
P. Fielitz, G. Borchardtr Solid State Ionics 144 (2001) 71–80
isotope. Thus, if one decreases the temperature to determine the temperature dependence of D and K, there will be a temperature where the simultaneous determination of both parameters starts to become subject to severe errors Žnear the line IA–IB in Fig. 2.. To overcome this problem, in a first step, an annealing time should be chosen, which is sufficiently short Ž t F t . to determine the characteristic time constant with a small error by measuring the 18 O surface concentration only. Then, in a next step, the same sample should be annealed at the same temperature for a much longer time to get a sufficiently large diffusion length, s , and to determine an accurate diffusion coefficient. From both independently Žand accurately. measured parameters t and D, one can then calculate the surface exchange coefficient, K s Ž Drt .1r2 .
4. Summary Oxygen tracer diffusivities in oxides are usually determined via gasrsolid exchange experiments in an atmosphere enriched as to the concentration of the rare stable isotope 18 O. The resulting 18 O diffusion profile in the oxide may be influenced by the kinetics of the 16 O and 18 O isotope exchange at the surface. This kinetics is described by the surface exchange coefficient, K. The analytical solution of the diffusion problem suggests the definition of a characteristic time constant, t ' DrK 2 . This time constant determines the duration that is necessary to reach equilibrium between the 18 O gas concentration and the 18 O concentration at the surface of the solid. By measuring the temperature dependence of t , it is possible to measure the temperature-dependent correlation between diffusivity and surface exchange. The characteristic time constant, t , can be determined by measuring the 18 O gas concentration and the 18 O concentration at the surface of the solid only. This can be done with good accuracy because there
is only a small instrumental mass fractionation that occurs during the measurement of oxygen isotope ratios by SIMS w7x. A further advantage is the possibility to use static SIMS techniques for this task, e.g. TOF-SIMS. The simultaneous determination of D and K works well in a large parameter range. However, at low temperatures, the simultaneous determination of D and K is subject to large errors because one cannot choose an arbitrarily long annealing time to get a sufficient diffusion length of the 18 O isotope. To overcome this problem, it was discussed how to determine D and K separately. Acknowledgements Financial support from Deutsche Forschungsgemeinschaft ŽDFG. is gratefully acknowledged. We are indebted to an anonymous reviewer for his valuable comments. References w1x J.A. Kilner, B.C.H Steele, Solid State Ionics 12 Ž1984. 89. w2x R.J. Chater, S. Carter, J.A. Kilner, B.C.H. Steele, Solid State Ionics 53–56 Ž1992. 859. w3x J.A. Kilner, R.A. De Souza, in: F.W. Poulsen, N. Bonanos, S. Linderoth, M. Mogensen, B. Zachau-Christiansen ŽEds.., Proceedings of the 17th Risø International Symposium on Materials Science: High Temperature Electrochemistry: Ceramics and Metals. 1996, p. 41. w4x J.A. Kilner, R.A. De Souza, I.C. Fullarton, Solid State Ionics 86–88 Ž1996. 703. w5x R.H.E. van Doorn, I.C. Fullarton, R.A. de Souza, J.A. Kilner, H.J.M. Bouwmeester, A.J. Burggraaf, Solid State Ionics 96 Ž1997. 1. w6x J. Crank, The Mathematics of Diffusion. 2nd edn., Oxford Univ. Press, Oxford, 1975. w7x J.M. Eiler, C.G. Graham, J.W. Valley, Chemical Geology 138 Ž1997. 221. w8x P. Fielitz, G. Borchardt, M. Schmucker, H. Schneider, M. ¨ Wiedenbeck, D. Rhede, S. Weber, S. Scherrer, Journal of the American Ceramic Society Ž2001. in press.