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The influence of surface diffusion on the results of surface analysis — a model calculation
Surface Science 78 (1978) 93-103 0 North-Holland Publishing Company
THE INFLUENCE OF SURFACE DIFFUSION ON THE RESULTS OF SURFACE ANALYSIS - A MODEL CALCULATION Ingomar JAGER Erich-Schmid-Institutfir Festkiirperphysik der &terreichischen Akademie der Wissenschaften, cfo MontanuniversitCtLeoben, A-8700 Leoben, Austria Received 22 February 1978; manuscript received in final form 5 May 1978
The infhtence of surface diffusion on some results of surface-analytical methods using sputtering is investigated by computer simulation of the combined effect of sputtering a.nd surface diffusion on a model “crystal” consisting of one monolayer of some species on top of a semiinfinite inert matrix. The computations show that surface diffusion is able to cause erroneous results under “typical” experimental conditions. Criteria are given to avoid such errors. A comparison with the results of an experiment on surface diffusion shows that heavy sputtering increases the surface diffusion coefficient by 2 to 3 orders of magnitude.
1. Introduction
The investigation of the constitution of the surface of a material and atomic layers adjacent to it by simultaneous or successive application of sputtering and analytical techniques is of steadily increasing importance for many branches of physics, metallurgy and technology (e.g. contact resistance, wear, corrosion and many others). There are many well known difficulties that prevent a straightforward interpretation of the results of such investigations, among them are preferred sputtering and knocking-in as well as the mixing of atoms from layers of different depths because of the evolution of a crater during sputtering. Other ones are not so well known. As an example: Ho [l] and Arita and Someno [2] recently investigated the influence of sputterenhanced diffusion within the “altered layer” on the results of sputtering binary alloys and found a strong effect. But one potential pitfall is in the author’s opinion usually underestimated: the influence of sputterenhanced surface diffusion. While it is true that in the early stages of surface sensitive techniques (see e.g. ref. [14]) investigations were done at ambient temperature, using ion beam diameters in the order of centimeters, thereby ruling out diffusional effects with more or less certainty, the trend nowadays is towards narrow beams to examine small areas and investigations at elevated temperatures to avoid difficulties connected with quenching [3]. Moreover, evidence for the diffusion of surface adatoms over macroscopic distances even at or 93
94
I. Jciger /Surface
diffusion and results of surface analysis
near room temperature is growing (e.g., refs. [3-6]), so nowadays surface diffusion as a possible source of error can no longer be ruled out by sheer argument. The aim of this paper is therefore to establish a simple model with the aid of which the influence of parameters such as the sputter rate or the coefficient of surface diffusion on certain results of surface sensitive techniques can be investigated by computer simulation without too many mathematical complications. This should give a clear insight into the problems and a guide to at least the orders of magnitude the parameters must lie within to avoid erroneous results.
2. Description of the model The model “crystal” consists of a homogeneous isotropic substrate which acts merely as a carrier for a monolayer of atoms different from those in the substrate. Only these atoms are considered to be of interest. Until otherwise stated the surface monolayer is thought of as being an infinite plane with uniform thickness and atomic concentration at the beginning. The atoms of this layer, furthermore represented by a concentration depending on space and time, are capable of surface diffusion governed by a certain diffusion coefficient D and can be removed by sputtering. The “sputtering equipment” is thought of being a source of an ion beam with circular cross-section of diameter d, uniform beam density over the whole cross-section and normal incidence on the surface, Admittedly the assumption of uniform beam density is an oversimplification since Hoffman [7] has shown that a Gaussian distribution is a very good approximation to the real beam density but it is felt that this should not alter the results of the computations very much. On the other hand, this simplification together with the assumption of an isotropic substrate (which is an oversimplification, too, cf. Bonzel and Latta [S]) greatly simplifies the calculation and provides an insight into the problems without additional geometrical complications. Additional ~sumptions are that the inevitable formation of a sputter crater or hole in the substrate can be neglected, whereas the enhancement of effusion through roughening of the sputtered surface is explicitly taken into account, and that the driving force for surface diffusion arises only from the concentration gradient via Fick’s second law. This last assumption, however, needs some explanations. It is well known that a difference in surface energy between covering monolayer and substrate provides an additional force on the migrating atoms that speeds up diffusion or slows it down depending on the relation between surface energy, chemical interactions between surface and bulk atoms and overlayer concentration. This can eventually lead to a “reverse” diffusion in the sense that concentration gradients are no longer “smeared out” but built up and this leads to the formation of adatom islands with a thickness of many monolayers and uncovered spots between them, as reported, e.g., for MO adatoms on Cu, Au or Al substrates by Tarng and Wehner [9J, Clearly for such cases the results of this paper are invalid,
I. Jliger/Surface diffusion and results of surface analysis
95
but for the case of weak driving forces (surface energy difference t chemical interactions G3RT) it turned out during the course of computations that the effect of the surface energy difference on the’kinetics of diffusion can be neglected. With all these assumptions the equations governing the model calculations can be written down. Outside of the sputtered spot Fick’s second law (in cylindrical coordinates) is valid: ct=D1(c,tc,/r),
r>df2,
(1)
with D1 being the coefficient of surface diffusion on the unperturbed surface. Suffices t and r denote differentation with respect to time and radius respectively. In the interior of the sputtered spot an additional term arises that describes the sputtering off of atoms: ct =D&+c,/r)-
c/T,
r
(2)
In this last term T denotes the time necessary ,to sputter off one equivalent of a monolayer, in this term all the parameters of sputtering such as sputter yield, ion beam density, primary energy etc. are incorporated. At the boundary of the sputtered spot the concentration c and its derivate c, are steady functions of r, i.e. c(d/2 - e) 3
c@/2 + e) ,
c,(d/2 - E) 3
c,(d/2 + Ej .
(3)
For the purpose of solving eqs. (1) and (2) numerically a discrete set of the values for r has to be chosen in order to approximate the differentials by differences and then, after choosing a constant value for c (t = 0), c(t = 0) = 1
for all values of r ,
(4)
a reasonable value ford - this value was taken to be d= 2000a
(9
(this value is only of importance for the numerical procedure, the results are equally valid for other diameters, if d2/4DT is unaltered), and a proper At to maintain convergence and stability, eqs. (1) and (2), can be solved numerically by a multidimensional Runge-Kutta scheme. These computations were done on the computer of the Rechenzentrum Graz and yielded the following results.
3. Results From eqs. (1) and (2), it turns out that for fixed d the results depend only on the two parameters A =d*/4DlT,
(6)
B = DzIDI ,
(7)
I. Jdger /Surface diffusion and results of surface analysis
96
where d is the actual diameter of the sputtering ion beam. The presence of the parameter B (eq. (7)) comes from the fact that everyone believes in sputter enhanced diffusion but to the author’s present knowledge there are so far no investigations dealing with this phenomenon exactly, so it is taken into account as a free parameter. Fig. 1 shows two sets of surface layer concentration versus radius curves for the choices of A and B: A = 1, B = 100 (upper set) and A = 10, B = 100 (lower set). The choice of these values is explained in section 4. The figures written near the curves denote the sputtering time elapsed in units of T. The concentration without any diffusion is shown in the left part of fig. 1 for the sake of comparison. It shows the exponential decay of the surface concentration with time ,
c = exp(-t/T)
r < d/2
(8)
(this time dependence is valid only under the assumption of one monolayer on top of an inert matrix, the case of multilayers is much more complicated), which is the solution of eq. (2) when D1 = Dz = 0. The computed curves on the right hand side of the diagram show the continuous re-supply of sputtered-off atoms by diffusion. Especially the lower curves show this phenomenon very markedly: After 1 T the concentration in the center of the sputtered spot is almost the same with and without diffusion. But after 10 T this concentration without diffusion has fallen to
c Yl
t
I
yxrttered
I
.not sputtered
A= I 8=X30
A= 10 B=lO Parameter:+
Fig. 1. Concentration of surface layer versus radius for various t/T and two different values of A. Left hand side: without diffusion for the sake of comparison.
I. Jtiger /Surface diffirsion and results of surface analysis
97
practically zero whereas approximately 0.12 of the value at the beginning remain in the presence of surface diffusion. The main part of the results (figs. 2 to 5) can be divided into two subgroups: (a) the mean value of the concentration over the whole sputtered spot is displayed,
c(r) 27rr dr
this corresponds to a simulation of the results of SIMS (fig. 2) (b) the concentration near the center of the sputtered spot is displayed; this corresponds to the usual way sputtering and AES are combined, namely a wide ion beam with a narrow investigated area in its center (fig. 3). The concentrations are plotted versus time for various values of A and B. The very lowest curve in each diagram corresponds to A = 00(no diffusion at all) for the sake of comparison. Note that any concentration versus time curve shown in the diagrams starts with a sharp decrease of the concentration until apparently the concentration gradient necessary for re-supply of atoms from the surrounding region has built up, then the concentration stays nearly (but never exactly) constant. The reason for this is the growing depletion of the region around the sputtered spot with sputtering time yielding a decrease in diffusional ,flux with time. Therefore no. state of real equilibrium can exist. On the other hand the continuous resupply of atoms from the surrounding prevents the concentration in the interior of the sputtered spot (both cm and c in the center) from becoming exactly zero within any finite time. c approaches zero asymptotically with a much longer decay time constant than T, thus simulating a much greater thickness of the surface layer. It may be of interest that the value of A at which diffusion starts to-change the time dependence of c just a little (e.g., 1 in fig. 2) differs from the value of A at which sputtering is apparently not capable to alter the surface concentration appreciably (1 OS3in fig. 2) by only three orders of magnitude. Note furthermore that an increase in B = D2 /DI by an order of magnitude corresponds roughly to an increase in A by an order of magnitude (this is clear from the definitions of A and B in the early stages of the process, later on the growing depletion of the surrounding surface area causes differences). From the figures some sort of criterion can be derived whether surface diffusion is capable of an appreciable contribution to the result of a surface sensitive technique or not. For the application of SIMS (or any other technique measuring the mean concentration over the whole sputtered area) diffusion introduces only a negligible error if A/B=d2/4D2T>1,
(10)
(which is essentially Coburn’s result [13]), whereas for AES (or any other tech-
I
t ____ _-_--__ t _______I_ ------A=.?
----
A = *ix31
.s 4
------uI____
.? 6 .5 ~---“__,,__A=l~
.4
-----
.3 .2 .I j
2
3
2
j
.k
-7
ii?-@E a 9
I. Jtiger/Surface~~~s~on and resultsof surfaceadysis
123L.567 Fii, 4. Mean concentration
c, (filled circles) and center concentration tered spot with constant con~ntration at the boundary.
99
t/J
(open circles) for a sput-
C 1
.9 .8 .7 .6 .5 .L .3 .2 .I
Fig. 5 Refiiing of a circuIar sputtered spot (sputtering on): cm @led circles) and center centration (open circles)
100
I. Jdger /Surface
nique measuring the concentration
diffusion and results of surface analysis
at the center of the sputtered
area),
A,B=-&>O.l
(11)
2
is the limit. The sign “greater than” arises from the fact that faster diffusion corresponds to a lower value of A. Due to the simplifying assumptions these criteria are only a rough guide, but due to the uncertainty of B,the sputter induced enhancement of surface diffusion, more sophisticated computations are useless. In order to see whether these criteria are fulfilled easily in usual experiments let us consider some cases where surface diffusion data are accessible. In table 1 (De and Q taken from ref. [ 1 l]), the surface diffusion coefficients for some couples of matrix and monolayer are listed for some representative temperature (within the range in which the D measurements were done) and for room temperature together with the minimum diameter dmia necessary to fulfill criterion (10) assuming static SIMS (i.e. a time for sputtering off one monolayer of about 1000 set). The values of dmin show that at room temperature the condition (10) is almost automatically met whereas it is most easily violated at elevated temperatures. Note that stronger sputtering by a factor of 100 (reducing the time constant to a typical value for depth profiling, 10 set/monolayer) causes a decrease in dmin by only a factor of 10, this factor is also achieved by using AES (criterion (11)). The simulation of another case yields the results shown in fig. 4. Consider a finegrained two phase alloy (e.g. Cu-50 at% Ag) consisting of nearly pure crystallites of X and Y with a mean grain diameter of d. Commonly in such a case one alloy component, say Y (in the example above the silver) tends to build up a monolayer on top of the crystallites on the other component (X), most probably because of the difference in surface energy mentioned earlier [4-61. Since these X crystallites are at least partially surrounded by Y crystallites providing a nearly infinite supply of Y atoms, a crude model for this type of surface diffusion is a circular spot with diameter d at the boundary of which the concentration of the element under investigation (Y) is held at the constant value of unity. Sputtering this model yields the results shown in fig. 4 for the mean concentration of Y on X (filled circles) and the
Table 1 Matrix
CU CU cu cu cvPe cu-Fe
Monolayer
Ag Ag Au Au cu cu
DQ
Q
D
(cm’lsec)
(kcal/mole)
(cm*/sec)
0.46 0.46 1.0 x 10-3 1.0 x 10-3 1.8 1.8
17.5 17.5 15.0 15.0 23.8 23.8
773 298 773 298 1093 298
5.12 X 10” 3.94 x 10-14 5.68 X 10d 6.30 X lo-t5 3.1 x 10-S 6.05 X lo--l8
dmin (mm) 1.43 2 x 104 0.15 1x10* 3.52 7.8 X lo-’
I. Jdger /Surface diffision and results of surface analysis
101
concentration in the center of a crystallite (open circles). Remember, however, that fig. 4 shows only the contribution from the Y monolayer on an X crystallite. If the area under investigation is greater than just one crystallite, the contribution from the massive Y crystallites has to be added (each contribution weighted by its relative area and the appropriate factor to take into account the penetration depth this evaluation scheme is shown extensively by Betz et al. [S]). Note furthermore, that because of the assumption of a constant concentration along the grain boundary only Dz (the diffusion coefficient in the interior of the sputtered spot) enters the calculation, therefore the parameter of the set of curves is A/B instead of A, where again A contains the unperturbed coefficient of surface diffusion and B is the sputter-induced enhancement factor.
4. Comparison with experimental results The sputtering experiments of Betz et al. [5] are selected for a comparison of the computations of this paper with experimental results mainly because they are, in the author’s opinion, the most simple and straightforward ones. They were done on a two phase Cu-Ag alloy and show the sputtering off of a monolayer Ag on Cu crystallites by an intense“ion beam and the build-up of this layer after sputtering has been stopped (fig. 6). Since Fick’s second law in cylindrical coordinates with the boundary condition c = 1 at r = d/2 has no simple analytical solution (see Crank [lo]), th e solution has to be obtained by numerical methods, this is shown in fig. 5. Again filled circles denote the mean concentration and open circles the concentration in the center of the spot. The unit of time is r = d2/4D2 .
(12)
By a suitable transformation of the time scale the measured increase of c,,, (full line of fig. 6) and the computed one (crosses in fig. 6) can be brought to full coincidence using the evaluation scheme described by [S]. This is a strong evidence for the correctness of the model. From eq. (12) the diffusion coefficient can be evaluated to be Dz = 2.3 X IO-’ *
cm’/sec .
03)
Extrapolation of available literature data [ 1l] on diffusion of Ag on Cu to room temperature yields values of D2 ranging from D2 = 5.2 X lo-l5
to
3.9 X IO-l4 cm2/sec
(14)
(these data are considered to be correct within, say, one order of magnitude, since the extrapolation ranges only from 250°C to room temperature). Because of the strong pre-sputtering treatment reported by Betz et al., the difference arises most probably not from the extrapolation but from the enhancement of diffusion by sputtering. Another possible explanation for these values of D is an
102
I. Jcger /Surface diffusion and results of surface analysis
8-- Sputtering on Ag/Cu 5/95 6-L-2..
Ag 351/356 10
20
30
LO
50
60
:
Fig. 6. Auger peak-to-peak heights (APPH) (solid lines) and computed values (crosses) for sputtering a two phase alloy (from Betz et al). [ 51).
increase of temperature due to the energy of the primary electron beam of the AES equipment as reported by LeGressus et al. [ 121. This effect however can be ruled out by an approximate calculation showing that the surface temperature deviated from the temperature of the sample holder (which was kept at ambient and monitored by a thermocouple) by no more than about 1 K due to the low input power (72 W/cm’) and the high thermal conductivity of the sample. This means that a value of B = 100 is perhaps a good estimate and therefore this values was used in fig. 1. Reinserting the figures DZ = 2.3 X lo-”
cm’/sec ,
d=lPm,
T = 7.7 set/monolayer
(15)
(d and T taken from Betz et al. [5]), yields A/B = 140.
(16)
This value (cf. fig. 4) is far too high for diffusion to alter the sputter-off curve; this again is in the agreement with the experimental results showing the clean sputteroff of the Ag monolayer (fig. 6, falling part of the curve). Many other experimental results available cannot be used for the purpose of a comparison with the computed results of this paper, in part because often one essential figure (e.g. the beam diameter) is not supported and in another part because reliable data on surface diffusion are rare.
5. Conclusion By application ble of introducing
of a simple model it can be shown that surface diffusion is capanot only quantitative but even qualitative errors into the results
of common methods of surface analysis, e.g. that one monolayer of some contaminant in conjunction with surface diffusion can yield results usually observed only in the presence of surface precipates with many atomic layer thickness, especially at high temperature or in connection with narrow primary ion beams. Criteria are given in order to avoid such errors for two distinct cases labeled “SIMS” (concentration measured over the whole sputtered spot) and “AES” (concentration measured in the center of the sputtered spot). Since the enhancement of surface diffusion through sputtering enters this process quantitatively, reliable data on surface diffusion - both with and without sputtering - are desired.
References [l] P.S. Ho, in: Proc. 7th Intern. Vacuum Congr. and 3rd Intern. Conf. on Solid Surfaces, Vienna, 1977, p. 2738. [2] M. Arita and M. Someno, in: Proc. 7th Intern. Vacuum Congr. and 3rd Intern. Conf. on Solid Surfaces, Vienna, 1977, p. 2511. [3] J.J. Burton, C.R. Helms and R.S. Polizzotti, J. Vacuum Sci. Technol. 13 (1976) 204. [4] H.G. Tompkins, J. Vacuum Sci. Technol. 12 (1975) 650. [S] G. Betz, P. Braun and W. FZrber, IAP-Bericht 76/10 (1976). [6] S. Thomas, Appl. Phys. Letters 24 (1974) 1. [7f D.W. Hoffmann, Surface Sci. 50 (1975) 29. [ 81 H.P. Bonzel and E.E. Latta, in Proc. 7tlvIntem Vacuum Congr. and 3rd Intern Conf. on Solid Surfaces, Vienna, 1977, p_ 1221. [S] M.L. Tarng and G.K. Wehner, 1. AppI. Phys. 43r(I972) 2268. [ 101 J. Crank, The Mathematics of Diffusion (Clarendon, Oxford, 1975). [ 111 Diffusion Data (Trans. Tech. Publ., Aedermannsdorf, Switzerland). f 121 C. Le Gressus, D. Ma&non and R. Sopizet, Surface Sci. 68 (1977) 338. [13] J. Coburn, J. Vacuum Sci. Technol. 13 (1976) 1037. [ 141 G.S. Anderson, J. Appl. Phys. 40 (1969) 2884.
THE INFLUENCE OF SURFACE DIFFUSION ON THE RESULTS OF SURFACE ANALYSIS - A MODEL CALCULATION Ingomar JAGER Erich-Schmid-Institutfir Festkiirperphysik der &terreichischen Akademie der Wissenschaften, cfo MontanuniversitCtLeoben, A-8700 Leoben, Austria Received 22 February 1978; manuscript received in final form 5 May 1978
The infhtence of surface diffusion on some results of surface-analytical methods using sputtering is investigated by computer simulation of the combined effect of sputtering a.nd surface diffusion on a model “crystal” consisting of one monolayer of some species on top of a semiinfinite inert matrix. The computations show that surface diffusion is able to cause erroneous results under “typical” experimental conditions. Criteria are given to avoid such errors. A comparison with the results of an experiment on surface diffusion shows that heavy sputtering increases the surface diffusion coefficient by 2 to 3 orders of magnitude.
1. Introduction
The investigation of the constitution of the surface of a material and atomic layers adjacent to it by simultaneous or successive application of sputtering and analytical techniques is of steadily increasing importance for many branches of physics, metallurgy and technology (e.g. contact resistance, wear, corrosion and many others). There are many well known difficulties that prevent a straightforward interpretation of the results of such investigations, among them are preferred sputtering and knocking-in as well as the mixing of atoms from layers of different depths because of the evolution of a crater during sputtering. Other ones are not so well known. As an example: Ho [l] and Arita and Someno [2] recently investigated the influence of sputterenhanced diffusion within the “altered layer” on the results of sputtering binary alloys and found a strong effect. But one potential pitfall is in the author’s opinion usually underestimated: the influence of sputterenhanced surface diffusion. While it is true that in the early stages of surface sensitive techniques (see e.g. ref. [14]) investigations were done at ambient temperature, using ion beam diameters in the order of centimeters, thereby ruling out diffusional effects with more or less certainty, the trend nowadays is towards narrow beams to examine small areas and investigations at elevated temperatures to avoid difficulties connected with quenching [3]. Moreover, evidence for the diffusion of surface adatoms over macroscopic distances even at or 93
94
I. Jciger /Surface
diffusion and results of surface analysis
near room temperature is growing (e.g., refs. [3-6]), so nowadays surface diffusion as a possible source of error can no longer be ruled out by sheer argument. The aim of this paper is therefore to establish a simple model with the aid of which the influence of parameters such as the sputter rate or the coefficient of surface diffusion on certain results of surface sensitive techniques can be investigated by computer simulation without too many mathematical complications. This should give a clear insight into the problems and a guide to at least the orders of magnitude the parameters must lie within to avoid erroneous results.
2. Description of the model The model “crystal” consists of a homogeneous isotropic substrate which acts merely as a carrier for a monolayer of atoms different from those in the substrate. Only these atoms are considered to be of interest. Until otherwise stated the surface monolayer is thought of as being an infinite plane with uniform thickness and atomic concentration at the beginning. The atoms of this layer, furthermore represented by a concentration depending on space and time, are capable of surface diffusion governed by a certain diffusion coefficient D and can be removed by sputtering. The “sputtering equipment” is thought of being a source of an ion beam with circular cross-section of diameter d, uniform beam density over the whole cross-section and normal incidence on the surface, Admittedly the assumption of uniform beam density is an oversimplification since Hoffman [7] has shown that a Gaussian distribution is a very good approximation to the real beam density but it is felt that this should not alter the results of the computations very much. On the other hand, this simplification together with the assumption of an isotropic substrate (which is an oversimplification, too, cf. Bonzel and Latta [S]) greatly simplifies the calculation and provides an insight into the problems without additional geometrical complications. Additional ~sumptions are that the inevitable formation of a sputter crater or hole in the substrate can be neglected, whereas the enhancement of effusion through roughening of the sputtered surface is explicitly taken into account, and that the driving force for surface diffusion arises only from the concentration gradient via Fick’s second law. This last assumption, however, needs some explanations. It is well known that a difference in surface energy between covering monolayer and substrate provides an additional force on the migrating atoms that speeds up diffusion or slows it down depending on the relation between surface energy, chemical interactions between surface and bulk atoms and overlayer concentration. This can eventually lead to a “reverse” diffusion in the sense that concentration gradients are no longer “smeared out” but built up and this leads to the formation of adatom islands with a thickness of many monolayers and uncovered spots between them, as reported, e.g., for MO adatoms on Cu, Au or Al substrates by Tarng and Wehner [9J, Clearly for such cases the results of this paper are invalid,
I. Jliger/Surface diffusion and results of surface analysis
95
but for the case of weak driving forces (surface energy difference t chemical interactions G3RT) it turned out during the course of computations that the effect of the surface energy difference on the’kinetics of diffusion can be neglected. With all these assumptions the equations governing the model calculations can be written down. Outside of the sputtered spot Fick’s second law (in cylindrical coordinates) is valid: ct=D1(c,tc,/r),
r>df2,
(1)
with D1 being the coefficient of surface diffusion on the unperturbed surface. Suffices t and r denote differentation with respect to time and radius respectively. In the interior of the sputtered spot an additional term arises that describes the sputtering off of atoms: ct =D&+c,/r)-
c/T,
r
(2)
In this last term T denotes the time necessary ,to sputter off one equivalent of a monolayer, in this term all the parameters of sputtering such as sputter yield, ion beam density, primary energy etc. are incorporated. At the boundary of the sputtered spot the concentration c and its derivate c, are steady functions of r, i.e. c(d/2 - e) 3
c@/2 + e) ,
c,(d/2 - E) 3
c,(d/2 + Ej .
(3)
For the purpose of solving eqs. (1) and (2) numerically a discrete set of the values for r has to be chosen in order to approximate the differentials by differences and then, after choosing a constant value for c (t = 0), c(t = 0) = 1
for all values of r ,
(4)
a reasonable value ford - this value was taken to be d= 2000a
(9
(this value is only of importance for the numerical procedure, the results are equally valid for other diameters, if d2/4DT is unaltered), and a proper At to maintain convergence and stability, eqs. (1) and (2), can be solved numerically by a multidimensional Runge-Kutta scheme. These computations were done on the computer of the Rechenzentrum Graz and yielded the following results.
3. Results From eqs. (1) and (2), it turns out that for fixed d the results depend only on the two parameters A =d*/4DlT,
(6)
B = DzIDI ,
(7)
I. Jdger /Surface diffusion and results of surface analysis
96
where d is the actual diameter of the sputtering ion beam. The presence of the parameter B (eq. (7)) comes from the fact that everyone believes in sputter enhanced diffusion but to the author’s present knowledge there are so far no investigations dealing with this phenomenon exactly, so it is taken into account as a free parameter. Fig. 1 shows two sets of surface layer concentration versus radius curves for the choices of A and B: A = 1, B = 100 (upper set) and A = 10, B = 100 (lower set). The choice of these values is explained in section 4. The figures written near the curves denote the sputtering time elapsed in units of T. The concentration without any diffusion is shown in the left part of fig. 1 for the sake of comparison. It shows the exponential decay of the surface concentration with time ,
c = exp(-t/T)
r < d/2
(8)
(this time dependence is valid only under the assumption of one monolayer on top of an inert matrix, the case of multilayers is much more complicated), which is the solution of eq. (2) when D1 = Dz = 0. The computed curves on the right hand side of the diagram show the continuous re-supply of sputtered-off atoms by diffusion. Especially the lower curves show this phenomenon very markedly: After 1 T the concentration in the center of the sputtered spot is almost the same with and without diffusion. But after 10 T this concentration without diffusion has fallen to
c Yl
t
I
yxrttered
I
.not sputtered
A= I 8=X30
A= 10 B=lO Parameter:+
Fig. 1. Concentration of surface layer versus radius for various t/T and two different values of A. Left hand side: without diffusion for the sake of comparison.
I. Jtiger /Surface diffirsion and results of surface analysis
97
practically zero whereas approximately 0.12 of the value at the beginning remain in the presence of surface diffusion. The main part of the results (figs. 2 to 5) can be divided into two subgroups: (a) the mean value of the concentration over the whole sputtered spot is displayed,
c(r) 27rr dr
this corresponds to a simulation of the results of SIMS (fig. 2) (b) the concentration near the center of the sputtered spot is displayed; this corresponds to the usual way sputtering and AES are combined, namely a wide ion beam with a narrow investigated area in its center (fig. 3). The concentrations are plotted versus time for various values of A and B. The very lowest curve in each diagram corresponds to A = 00(no diffusion at all) for the sake of comparison. Note that any concentration versus time curve shown in the diagrams starts with a sharp decrease of the concentration until apparently the concentration gradient necessary for re-supply of atoms from the surrounding region has built up, then the concentration stays nearly (but never exactly) constant. The reason for this is the growing depletion of the region around the sputtered spot with sputtering time yielding a decrease in diffusional ,flux with time. Therefore no. state of real equilibrium can exist. On the other hand the continuous resupply of atoms from the surrounding prevents the concentration in the interior of the sputtered spot (both cm and c in the center) from becoming exactly zero within any finite time. c approaches zero asymptotically with a much longer decay time constant than T, thus simulating a much greater thickness of the surface layer. It may be of interest that the value of A at which diffusion starts to-change the time dependence of c just a little (e.g., 1 in fig. 2) differs from the value of A at which sputtering is apparently not capable to alter the surface concentration appreciably (1 OS3in fig. 2) by only three orders of magnitude. Note furthermore that an increase in B = D2 /DI by an order of magnitude corresponds roughly to an increase in A by an order of magnitude (this is clear from the definitions of A and B in the early stages of the process, later on the growing depletion of the surrounding surface area causes differences). From the figures some sort of criterion can be derived whether surface diffusion is capable of an appreciable contribution to the result of a surface sensitive technique or not. For the application of SIMS (or any other technique measuring the mean concentration over the whole sputtered area) diffusion introduces only a negligible error if A/B=d2/4D2T>1,
(10)
(which is essentially Coburn’s result [13]), whereas for AES (or any other tech-
I
t ____ _-_--__ t _______I_ ------A=.?
----
A = *ix31
.s 4
------uI____
.? 6 .5 ~---“__,,__A=l~
.4
-----
.3 .2 .I j
2
3
2
j
.k
-7
ii?-@E a 9
I. Jtiger/Surface~~~s~on and resultsof surfaceadysis
123L.567 Fii, 4. Mean concentration
c, (filled circles) and center concentration tered spot with constant con~ntration at the boundary.
99
t/J
(open circles) for a sput-
C 1
.9 .8 .7 .6 .5 .L .3 .2 .I
Fig. 5 Refiiing of a circuIar sputtered spot (sputtering on): cm @led circles) and center centration (open circles)
100
I. Jdger /Surface
nique measuring the concentration
diffusion and results of surface analysis
at the center of the sputtered
area),
A,B=-&>O.l
(11)
2
is the limit. The sign “greater than” arises from the fact that faster diffusion corresponds to a lower value of A. Due to the simplifying assumptions these criteria are only a rough guide, but due to the uncertainty of B,the sputter induced enhancement of surface diffusion, more sophisticated computations are useless. In order to see whether these criteria are fulfilled easily in usual experiments let us consider some cases where surface diffusion data are accessible. In table 1 (De and Q taken from ref. [ 1 l]), the surface diffusion coefficients for some couples of matrix and monolayer are listed for some representative temperature (within the range in which the D measurements were done) and for room temperature together with the minimum diameter dmia necessary to fulfill criterion (10) assuming static SIMS (i.e. a time for sputtering off one monolayer of about 1000 set). The values of dmin show that at room temperature the condition (10) is almost automatically met whereas it is most easily violated at elevated temperatures. Note that stronger sputtering by a factor of 100 (reducing the time constant to a typical value for depth profiling, 10 set/monolayer) causes a decrease in dmin by only a factor of 10, this factor is also achieved by using AES (criterion (11)). The simulation of another case yields the results shown in fig. 4. Consider a finegrained two phase alloy (e.g. Cu-50 at% Ag) consisting of nearly pure crystallites of X and Y with a mean grain diameter of d. Commonly in such a case one alloy component, say Y (in the example above the silver) tends to build up a monolayer on top of the crystallites on the other component (X), most probably because of the difference in surface energy mentioned earlier [4-61. Since these X crystallites are at least partially surrounded by Y crystallites providing a nearly infinite supply of Y atoms, a crude model for this type of surface diffusion is a circular spot with diameter d at the boundary of which the concentration of the element under investigation (Y) is held at the constant value of unity. Sputtering this model yields the results shown in fig. 4 for the mean concentration of Y on X (filled circles) and the
Table 1 Matrix
CU CU cu cu cvPe cu-Fe
Monolayer
Ag Ag Au Au cu cu
DQ
Q
D
(cm’lsec)
(kcal/mole)
(cm*/sec)
0.46 0.46 1.0 x 10-3 1.0 x 10-3 1.8 1.8
17.5 17.5 15.0 15.0 23.8 23.8
773 298 773 298 1093 298
5.12 X 10” 3.94 x 10-14 5.68 X 10d 6.30 X lo-t5 3.1 x 10-S 6.05 X lo--l8
dmin (mm) 1.43 2 x 104 0.15 1x10* 3.52 7.8 X lo-’
I. Jdger /Surface diffision and results of surface analysis
101
concentration in the center of a crystallite (open circles). Remember, however, that fig. 4 shows only the contribution from the Y monolayer on an X crystallite. If the area under investigation is greater than just one crystallite, the contribution from the massive Y crystallites has to be added (each contribution weighted by its relative area and the appropriate factor to take into account the penetration depth this evaluation scheme is shown extensively by Betz et al. [S]). Note furthermore, that because of the assumption of a constant concentration along the grain boundary only Dz (the diffusion coefficient in the interior of the sputtered spot) enters the calculation, therefore the parameter of the set of curves is A/B instead of A, where again A contains the unperturbed coefficient of surface diffusion and B is the sputter-induced enhancement factor.
4. Comparison with experimental results The sputtering experiments of Betz et al. [5] are selected for a comparison of the computations of this paper with experimental results mainly because they are, in the author’s opinion, the most simple and straightforward ones. They were done on a two phase Cu-Ag alloy and show the sputtering off of a monolayer Ag on Cu crystallites by an intense“ion beam and the build-up of this layer after sputtering has been stopped (fig. 6). Since Fick’s second law in cylindrical coordinates with the boundary condition c = 1 at r = d/2 has no simple analytical solution (see Crank [lo]), th e solution has to be obtained by numerical methods, this is shown in fig. 5. Again filled circles denote the mean concentration and open circles the concentration in the center of the spot. The unit of time is r = d2/4D2 .
(12)
By a suitable transformation of the time scale the measured increase of c,,, (full line of fig. 6) and the computed one (crosses in fig. 6) can be brought to full coincidence using the evaluation scheme described by [S]. This is a strong evidence for the correctness of the model. From eq. (12) the diffusion coefficient can be evaluated to be Dz = 2.3 X IO-’ *
cm’/sec .
03)
Extrapolation of available literature data [ 1l] on diffusion of Ag on Cu to room temperature yields values of D2 ranging from D2 = 5.2 X lo-l5
to
3.9 X IO-l4 cm2/sec
(14)
(these data are considered to be correct within, say, one order of magnitude, since the extrapolation ranges only from 250°C to room temperature). Because of the strong pre-sputtering treatment reported by Betz et al., the difference arises most probably not from the extrapolation but from the enhancement of diffusion by sputtering. Another possible explanation for these values of D is an
102
I. Jcger /Surface diffusion and results of surface analysis
8-- Sputtering on Ag/Cu 5/95 6-L-2..
Ag 351/356 10
20
30
LO
50
60
:
Fig. 6. Auger peak-to-peak heights (APPH) (solid lines) and computed values (crosses) for sputtering a two phase alloy (from Betz et al). [ 51).
increase of temperature due to the energy of the primary electron beam of the AES equipment as reported by LeGressus et al. [ 121. This effect however can be ruled out by an approximate calculation showing that the surface temperature deviated from the temperature of the sample holder (which was kept at ambient and monitored by a thermocouple) by no more than about 1 K due to the low input power (72 W/cm’) and the high thermal conductivity of the sample. This means that a value of B = 100 is perhaps a good estimate and therefore this values was used in fig. 1. Reinserting the figures DZ = 2.3 X lo-”
cm’/sec ,
d=lPm,
T = 7.7 set/monolayer
(15)
(d and T taken from Betz et al. [5]), yields A/B = 140.
(16)
This value (cf. fig. 4) is far too high for diffusion to alter the sputter-off curve; this again is in the agreement with the experimental results showing the clean sputteroff of the Ag monolayer (fig. 6, falling part of the curve). Many other experimental results available cannot be used for the purpose of a comparison with the computed results of this paper, in part because often one essential figure (e.g. the beam diameter) is not supported and in another part because reliable data on surface diffusion are rare.
5. Conclusion By application ble of introducing
of a simple model it can be shown that surface diffusion is capanot only quantitative but even qualitative errors into the results
of common methods of surface analysis, e.g. that one monolayer of some contaminant in conjunction with surface diffusion can yield results usually observed only in the presence of surface precipates with many atomic layer thickness, especially at high temperature or in connection with narrow primary ion beams. Criteria are given in order to avoid such errors for two distinct cases labeled “SIMS” (concentration measured over the whole sputtered spot) and “AES” (concentration measured in the center of the sputtered spot). Since the enhancement of surface diffusion through sputtering enters this process quantitatively, reliable data on surface diffusion - both with and without sputtering - are desired.
References [l] P.S. Ho, in: Proc. 7th Intern. Vacuum Congr. and 3rd Intern. Conf. on Solid Surfaces, Vienna, 1977, p. 2738. [2] M. Arita and M. Someno, in: Proc. 7th Intern. Vacuum Congr. and 3rd Intern. Conf. on Solid Surfaces, Vienna, 1977, p. 2511. [3] J.J. Burton, C.R. Helms and R.S. Polizzotti, J. Vacuum Sci. Technol. 13 (1976) 204. [4] H.G. Tompkins, J. Vacuum Sci. Technol. 12 (1975) 650. [S] G. Betz, P. Braun and W. FZrber, IAP-Bericht 76/10 (1976). [6] S. Thomas, Appl. Phys. Letters 24 (1974) 1. [7f D.W. Hoffmann, Surface Sci. 50 (1975) 29. [ 81 H.P. Bonzel and E.E. Latta, in Proc. 7tlvIntem Vacuum Congr. and 3rd Intern Conf. on Solid Surfaces, Vienna, 1977, p_ 1221. [S] M.L. Tarng and G.K. Wehner, 1. AppI. Phys. 43r(I972) 2268. [ 101 J. Crank, The Mathematics of Diffusion (Clarendon, Oxford, 1975). [ 111 Diffusion Data (Trans. Tech. Publ., Aedermannsdorf, Switzerland). f 121 C. Le Gressus, D. Ma&non and R. Sopizet, Surface Sci. 68 (1977) 338. [13] J. Coburn, J. Vacuum Sci. Technol. 13 (1976) 1037. [ 141 G.S. Anderson, J. Appl. Phys. 40 (1969) 2884.