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A cratering analysis for quantitative depth profiling by ion beam sputtering
Surface Science 0 North-Holland
50 (1975) 29-52 Publishing Company
A CRATERING
ANALYSIS
FOR QUANTITATIVE
DEPTH PROFILING
BY ION BEAM SPUTTERING
D.W. HOFFMAN Scientijic Research Staff, Ford Motor Company, Dearborn, Michigan 48121, Received
31 October
1974; revised manuscript
received
18 February
U.S.A.
1975
The depth profiling of composition by sputtering with a stationary ion beam of nonuniform intensity is treated analytically. Attention is devoted to the situation where the experimental signal originates simultaneously from all or most of the sloping, sputtered crater, as in ion scattering spectrometry (ISS) or secondary ion mass spectrometry (SIMS). As a result the signal itself does not relate directly to the composition at any specific depth in the specimen. Data processing formulas for various degrees of correction are presented, starting with a very simple result for the Gaussian intensity distribution. Simulated ISS outputs and processed data points are then illustrated for a number of selected composition profiles. An alternative use of the analysis to determine the radial intensity distribution in an ion beam is presented with experimental data supporting the Gaussian beam approximation as modified by the so-called clipping factor. Finally a practical stratagem is proposed for overcoming such extraneous problems as beam intensity transients, while determining the depth profiles of the elements in specimens of several components.
1. Introduction
The depth profiling of composition near the surfaces of solids is facilitated by the use of accelerated ion beams for controlled removal of material from a specimen by sputtering. The sputtering away of material at the point of beam impingement produces a crater which gradually deepens as exposure to the beam continues. In some cases elemental analysis of the composition in the advancing crater is carried out with a separate experimental technique such as Auger spectroscopy, where a fine beam of electrons is directed at the bottom of the crater. In other cases, determination of the composition within the crater relies directly on the interaction of the ion beam with the solid by measurement of the energy losses of the scattered primary ions (ion scattering spectrometry - ISS) or the identification of material sputtered away (secondary ion mass spectrometry - SIMS). In these latter cases the measured signal generally comes from all or a substantial fraction of the crater surface. If the depth of the crater is nonuniform, the signal does not relate directly to the composition at any single depth in the specimen. Rather, the signal at any instant represents an average over the concentrations at the various depths exposed in the crater at that time.
Interpretation of the resulting data for quantitative depth profiling is therefore problematic. The situation described above arises when the intensity within the primary beam is nonuniform, as in, for example, a radial Gaussian distribution. Since the sputtering rate in a given material is most rapid where the flux of incoming ions is greatest, such a beam produces a crater with sloping sides. A number of experimental remedies have been employed to overcome this problem. One approach involves passing the primary beam through a small aperture to eliminate all but a region of near-uniform intensity at its center. Another approach is to sputter with a large beam and analyze with a small beam by intermittently focussing the ion beam to a reduced size. A third avenue is to sweep the beam in a raster pattern to produce a large, flat-bottomed crater, while gating the output signal to be measured only when the beam passes the center. The purpose of this paper is to present a theoretical approach to the problem and to show how data obtained with a stationary, nonuniform beam can be processed analytically. In the treatment that follows we shall refer specifically to ISS, but the general fornlulatl~n and results should be applicable to SIMS and any other depthprofiling techniques that derive their outputs from the ion-solid interaction. The development of this paper begins with a mathematical description of the depth profiling problem followed by an advantageous change of variables that enables the further manipulation and solution of the equations. Solutions are given for Gaussian and non-Gaussian distributions of beam intensity, the former yielding an operational data-processing procedure of extreme simplicity. A series of simulated ISS outputs are then displayed for various illustrative composition profiles, including approximate processing of the simulated data to reconstruct the sample. A different appli~tion of the analysis is then presented by means of which a known specimen can be used to make an experimental determination of the beam intensity distribution in an actual ion scattering spectrometer. Finally we propose a stratagem for handling both the cratering problem and uncontrolled intensity transients in the practical depth profiling of unknown, multicomponent specimens.
2. mathematical
formulation
of the eratering probiem
The sketch in fig. 1 illustrates the experimental configuration in ion scattering spectrometry on a cross section through the specimen in the plane of the beam axis [ 1,2] . A collimated beam of inert gas ions accelerated through a field of, say, 2 kV strikes the surface of a flat specimen at an angle of 45” to the normal. The ion beam is monoenergetic but its intensity is nonuniform with a liaif-maximum diameter on the order of a millimeter. ‘Upon impact with the specimen some of the ions scatter elastically with characteristic energy losses determined by the masses of the atoms they hit. Simultaneously the sputtering action of the ion beam removes atoms from the specimen at rates which vary over the cross section of the beam according to the distribution of intensity. The resulting erosion causes the formation of a shallow, sloping crater
D. W. Hoffman/Crater&
ion
specimen
analysis for depth profiling
31
electrostatic
Fig. 1. Cross section of a neon-sputtered crater penetrating the specimen during ion scattering spectrometry. Since the depth-to-diameter ratio (exaggerated in sketch) never exceeds 10m4, the actual scattering geometry remains essentially flat.
which grows ever deeper as irradiation proceeds. It is important to notice that the walls of the crater do not ascend abruptly to the original surface, but flatten out gradually in accord with the tails of the ion beam intensity distribution. Inert gas ions scattered at right angles to the incident beam enter a detector (electrostatic analyzer) which separates them according to their scattered velocities and records the flux of ions as a function of kinetic energy. The observation of peaks in the scattered ion flux at characteristic fractions of the primary beam energy serves to identify the various atomic species exposed to the beam at the surface of the crater. The height of a given peak is proportional to the surface coverage by the corresponding element, but the absolute proportionality factor varies from element to element. Although the depth profiling capability of 1% depends on the formation and advancement of the crater into the specimen, the depth of erosion is always sufficiently small (G 10e4 mm) relative to the diameter that the scattering geometry remains essentially flat. Since the primary beam is approximately circular in cross section, the 45” angle of impingement causes the contours of the crater to be elliptical. The electrostatic analyzer accepts ions scattered from a strip of area across the crater, as shown in fig. 2 adapted from ref. [2]. A fine slit at the entrance to the analyzer limits the observed scattering angles to a very small range near 90”. The number of ions scattered per unit time by an atom at the surface of the specimen is determined by the primary beam intensity at that point and appropriate physical constants. The electrostatic analyzer adds together a selected flux of ions coming from points lying within its field of view A’ across the crater. The flux h scattered from atoms of a given species can therefore be expressed as an integral of the form h =K
j-ma, A’
(1)
where I and Care respectively the primary beam intensity and the volume fraction of the indicated species on the element of area da, and the proportionality factor K
32
D. W. Hoffmanfcratering
analysis for depth profiling
~tglLLJ Depth-
intensity
- composltlon
1-
0.24mm
-L
Differential of area
element
Area seen by electrostatic analyzer
L’ig. 2. Aerial view of primary ion spot on 1% specimen, showing contours of constant intensity--composition within the sputtered crater and the view-band across the crater scattered ions may enter the electrostatic analyzer.
depthfrom which
includes such constants as the differential scattering cross section and the probability of neutralization *. For the purpose of discussion we shall associate h with the maximum signal or peak height produced by the electrostatic analyzer in scanning past the characteristic energy for the element in question. With the help of eq. (1) the cratering problem can now be stated more explicitly. Given measured values of h as a function of time, and some measured or assumed function for the beam intensity distribution Z, solve for the concentration C at various depths in the specimen. To proceed, it is necessary to examine the components of the integral in more detail. Consider the beam intensity. In addition to the spatial variation of intensity mentioned above, the primary ion beam also frequently exhibits some weakening with time during an experiment, owing apparently to a decrease in the pressure of the inert gas in the spectrometer. This effect introduces another uncertainty into any depth profiling observations over and above the cratering problem. For the purpose of analysis let us assume the beam to be axially symmetric with separable space and time variations of intensity in the form
were t is time, Yis a radius normal to the beam axis, lo(f) is intensity at the center of the beam, and the scaled intensity function g(r) goes from unity at the center of the beam to zero at the extremities. In other words, any temporal fading of the intensity occurs proportionately throughout the beam so that the relative distribution g(r) is preserved. *
Ignored in this formulation are minor variations in the acceptance angle of the electrostatic analyzer and the differential scattering cross-section over A’. Since the variation of these factors is essentially linear across the narrow dimension of A’, their effects average out of the result when the beam is centered on A’.
D. W. HoffmanlCratering
analysis for depth profiling
33
The specimen is taken to be a coated or otherwise treated material with lamellar variations in composition parallel to the surface. As the sputtered crater burrows down into the specimen, more and more composition layers gradually become visible to the beam. We assume that the strata exposed on the sloping walls of the crater represent the undisturbed interior of the specimen with no intermixing by the beam prior to sputtering*. We do allow however for different sputtering rates of the various atomic constituents, so that the crater may penetrate more rapidly through one layer than another. Looking down onto the cratered specimen one sees the exposed edges of the various composition strata as a set of concentric ellipses. Regardless of how fast or slow the individual layers may sputter back, the elliptical contours of composition viewed at any instant must necessarily coincide with rings of constant intensity in the beam. It is convenient to take our element of area da to be an elliptical ring between closely spaced composition contours on the specimen as indicated in fig. 2. Since the crater is always extremely shallow, it is unnecessary to account geometrically for the negligible slopes of the crater walls. We thus express da in terms of the radius of the corresponding intensity contours in the beam da = fi(2nr
dr),
(3)
where the factor of fi corrects the circular element to a 45” ellipse in which r is the semi-minor axis. Owing to the fact that da follows the composition contours, the composition thereon does not vary with angle bout the beam axis, but only with the radius and the time, therefore C = C(r, t). Another point of view is to consider the element of area as projected along the beam axis (2nr dr), whence the fi factor becomes a correction onto C(r, t) for the projected density of surface atoms. Finally, we notice that above a certain size, the element da becomes truncated in the view of the electrostatic analyzer, the part beyond the strip A’ not contributing to the observed ion flux. Sighting from the analyzer along the reflected beam axis as in fig. 3, we define a multiplicative clipping factor r < rc
‘(‘) = ( i$rf) sin-’ (rc/r),
r > rc
(4)
to correct da when r exceeds the clipping radius rc. Should the band of observation A’ not be centered on the crater, the clipping factor involves two different clipping radii, but its application and effect are similar. Inserting eqs. (2), (3) and (4) into eq. (1) yields a fully parameterized integral for the ES peak height according to the model developed here. In carrying out this step
*
The compositions at the sputtered surface of the crater may in fact be perturbed by preferential sputtering from those of the underlying layers 141, but this problem is generic to 1% and other approaches to depth profiling in a sputtered crater, e.g., by Auger spectroscopy. The cratering analysis relates only to the compositions that actually appear at the sputtered surface.
D. W. Ho ffman/Cratering
34
analysis ,for depth profiling
I:&. 3. Formulation of the clipping factor for truncation electrostatic analyzer along the retlccted beam axis. it is convenient
of crater
contours
in the view of the
to substitute dp = 2r dr,
/a’?,
(5)
so that the result and others to follow can be written more compactly stituent under consideration:
for the con-
7 0
h(t) = K fi .rrlo(t) g(p) C(P,4 4~) dp. A pure specimen
[C(r, t) = l] gives the maximum
(6) peak height:
(7) The ratio of eq. (6) to eq. (7) with cancellation tional peak height
wherein the normalizing re
constant
of common factors yields the frac-
r is the integral of the beam intensity
s -
g(p)4~) dp.
*:
(9)
0
* A definition of 1 was given previously above definition is the one that applies
[ 31 that differs from eq. (9) by the factor & ~10. The to the results quoted
there as well as here.
D. W. Hoffman/Cratering
analysis for depth profiling
35
Definition of the stabilized fractional peak height by eq. (8) serves to isolate the time dependence caused by cratering itself from such extraneous matters as beam intensity transients, while also eliminating unknown physical constants. An approach to the handling of beam intensity transients in binary specimens appears elsewhere [3] and will be reconsidered at the end of this report. Meanwhile it is the solution of the cratering problem embodied in eq. (8) to which this paper is next addressed.
3. Change of integration
variable
Eq. (8) is difficult to solve because the unknown function C(p, r) appears as the kernel, i.e. a function of both time and the radial integration variable. The time t is the moment when the depth of the crater at the radius r (= 6) reaches the level where the specimen has the composition C(p, t). Our assumptions about the specimen provide, however, that the composition is, in fact, a function of only a single variable, the depth z below the original surface. Since ISS results do not reveal actual depths of penetration, but only the elapsed times of exposure to sputtering by the ion beam, it is preferable not to deal directly with z but rather with the corresponding time f(z) required for the center or bottom (p = 0) of the crater to reach the level z. We thus write the identity C(P, t) = c(o, t*)* C(t)
(10)
to quantify implicitly the desired transformation of variables. To find the explicit relationship between p, t, and f, consider time plots of the depth of penetration at the center of the beam and at the radius r where the intensity is, say, half the maximum as sketched in fig. 4. The indicated variations in sputtering rate along each curve supposedly arise from the variations in composition within the specimen. The difference in sputtering rates between the curves at a given depth arise from the difference in beam intensity. Indeed the relative rates of penetration must stand in direct proportion to the intensities at r and at the center. Consequently, where the intensity is only half the maximum it takes the crater twice as long to reach a given depth. Thus we can write the general relationship Vr = g(r)/&8
,
(11)
or t^= t g(r),
(12)
since g(0) = 1. One sees that the two plots in fig. 4 can be superimposed by shrinking the time axis of the half-maximum curve by a factor of 4. The result is a master penetration curve on the z versus t axes from which the penetration at any selected values of r and ? can be ascertained by use of eq. (12). TO accomplish the transformation of variables in eq. (8) we differentiate eq. (12) at constant t dt*= t (dg/dp) dp,
(13)
D. W, IfoffinanjCratering
36
analysis for depth profiling
beam center
half- maximum
2
Time Fig. 4. Depth of penetration versus time at points of differing intensity in the beam. Changes in rate of penetration along a given curve are caused by variations in composition with depth in the specimen.
leading to: (14) Substituting duces
eqs. (10) and (14) into eq. (8) and converting
the limits accordingly
./I)) = + i ~(f,‘t) C(t) dt^,
pro-
(16)
0
a Volterra integral equation
of the first kind with the kernel
pertaining to the clipping factor and the distribution of intensity in the beam. One sees that the composition profile C(f) to be determined now appears in the conventional format for an unknown function. Eq. (16) is entirely equivalent to eq. (8) but has a much more convenient form for manipulation and solution. Operating on eq. (9) in a simiiar manner [or setting both C and f to unity in eq. (16)] produces a companion equation for the normalizing factor:
D. W. HoffrnanlCraterimg analysis for depth profiling
dj~(;,t)dt.
31
(18)
0
4. Gaussian beam approximation
without analyzer clipping
Eq. (16) lea+ to a simple solution of the cratering problem in the Gaussian beam approximation : gG = e
-r2 = e-p
(d lndFl/g)),:
(19)
wo
= ”
Consider first a hypothetical case without any clipping (U = 1 everywhere), which is to say that the electrostatic analyzer sees the complete irradiated area (rc * m). Then eq. (17) gives rc = 1,
(21)
and eq. (16) reduces to
fit) = +j
C(l) dt”
(unclipped
Gaussian beam),
(22)
(unclipped
Gaussian beam).
(23)
0
with the immediate
solution:
This simple formula for the processing of data yields the composition at the very bottom of the crater as a function of sputtering time. As always, conversion from time to penetration distance requires some independent knowledge of sputtering rates and their variation with composition. Although eq. (23) was obtained under the unrealistic assumption of no analyzer clipping, it can in fact be usefully applied to experimental data when the effect of clipping is not too severe. The nature of the results and errors incurred by such an approximate procedure will be examined in a later section of this paper on ISS simulation. An example of data processing by means of eq. (23) is also presented in ref. [3 ] on the investigation of interfaces under ion-plated coatings. Ref. [3] gives a graphical construction for eq. (23) when the data points f(r) are evenly distributed in time. When the observation times are not evenly spaced, a more accurate processing of the discrete data points obtains from the formula *
Without
loss of generality
we now taker
in units of its value at g = l/e.
38
D. W. HoffkanfCratering
analysis for depth profiling
t I II m+l--*rn
r
m+l -tm_l
.
(234
where the interger subscript m indexes the times of observation and the corresponding peak heights in the order of observation. In other words, fl is the first peak height observation and tl is the time of that observation,,f2 is the second peak height observation and t2 is the corresponding time, etc. Eq. (23a) is merely a discrete version of eq. (23) in which the derivative of the product (tf) is first expanded term by term and the derivative off is approximated by lever-rule averaging of the differences between .f,,, and neighboring values.
5. Gaussian beam solution with clipping correction Another instance in which the Gaussian beam solution can be applied is after the data are first corrected for the effect of the clipping factor in a preliminary step. Substituting eqs. (20) and (23) into eq. (16) prodcues
f’(t) =
& /
u(t,t)(; y
dt^,
(24)
relating the clipped peak height ,f’ to the corrected peak height f. With clipping the normalizing factor becomes less than unity,
I’; =+j
(25)
u(;/t) dt*, 0
where the clipping factor has the form (2/7r) sin-l (rel&Z U(f&
),
t*< t exp (-r:),
= t^> t exp (-rz),
i 1,
(26)
in terms of the time-ratio notation. Fig. 5 shows a plot of the clipping function for an arbitrarily chosen value of the clipping radius. To solve eq. (24) for f(t) the integral is replaced by a summation over the discrete times at which the height of a particular peak was recorded on successive ISS scans. Details of this derivation are outlined in Appendix A. The resulting linear simultaneous equations take the form; m
1, - fl, =
c ~mn(f, II= 0
- fA>>
m= 1 toN.
(27)
D. W. Hoffman/Cratering
39
analysis for depth profiling
Fig. 5. Clipping function for moderate clipping [g(r,) = 0.4 ] on a Gaussian to areas to the left of the curve as illustrated. APmn correspond
beam. Coefficients
The indices, m and n, represent integer subscripts that refer to the N data points VI, f2,. . ., f,) in order of observation. The coefficients aP,,r are areas on the graph of fig. 5 as indicated by the cross-hatched region. These equations can be solved in sequence
etc., where the solution at each step feeds into the right hand side of the next. To begin, one may have to assume f. = f; . One sees from the form of this solution that the clipping correction just adds a rectifying term onto the original data for f”. Formulas for the M’s, given in Appendix A, require numerical integration of the clipping function. The actual computation is well-adapted for computer processing. The corrected values off(t) produced by eqs. (28) are then ready for processing by the Gaussian beam solution of eq. (24).
6. General solution Using methods akin to those of the previous section one can write a general solution to eq. (16) for the case of a non-Gaussian beam. If one knows g(r) by some independent measurement, values of the kernel k(?/t) are readily generated by eq. (17).
0. W. ~off~~~~rater~ng
40
an&k
for depth profilirzg
On the other hand, the cratering theory itself can be employed to determine ~(8/t) directly from ISS measurements on specialized (sharp interface) specimens fabricated for this purpose. The details of that determination will be described presently. For the moment we assume a knowledge of&/t). The general solution to eq. (16) takes the successive form
(29) etc., where the data, J; , j”,, f3, etc., are indexed in the order of increasing sputtering times, tl, t2, t3, etc., and the coefficients are defined as follows tn+l
Arrnn +-
s
K(&)
d;,
(30)
m t,,-1 with two exceptions: (31)
K(~/tm)
dt*,
Once again the indicated operations are easily computerized. If matrix methods are used, one defines also AI& = 0,
n>m,
(33)
in order to complete the square matrix of coeffients. In the application of this ap proximate solution one should bear in mind the assumption that the increment tn_l to t, be sufficiently small that corresponding changes in C, are not large. Although the general solution becomes equivalent to the Gaussian beam solution in the appropriate Iimit, experience at the author’s laboratory favors the two step procedure given previou~y in that case.
7. ISS simulation For the purpose of illustration a number of examples have been worked out from eqs. (16) and (17) using the Gaussian beam approximation of eq. (20) and the clipping
D. W. HoffmanfCratering
41
anaiysis fordepth profiling
I Simulated
f (clipped)
-
0 d(tf
Data
f
l/dt
I
‘I_ IL c
C
Composition
*0
Composition
Profile
0
time
o
I
I
t-
time
0
I-
Profile
@
D
0
Simulated
Doto
f
-
Simulated
Eato
:n u___ I
I c
C
Composition
0
0
Composition Profile
Profile
OO
time
time
I Simulated
Dota
f
I
C
111,
m--H-l
Lomposition
OO
time
Profile
*o-
Composition Profile
time
Fig. 6. Simulated ISS peak height U, changes for various depth profiles of a given atomic component in the specimen computed with Gaussian beam clipped as in fig. 5 [ exp(-rz) = 0.41. In each frame the upper plot shows the simulated output plus a set of processed data points corrected without allowance for clipping. The lower plot shows the true depth pro’file as seen at the bottom of the crater or obtained by full correction of the simulated output with proper allowance for clipping.
D. W. HoffmanlCratering
42
analysis for depth profiling
Simulated
Simulated Data f (Clipped )
I
o
1
l--l
0
c
c
0
Composition
OO
c
Profile
OO"-------time
time
Simulated
Composition
Profile
Data
Simulated
Dato
I!!!- A-
Data
C
Composition
OO
I
OO
time
Simulated
Composition
Profile
time
xmulated
Data
Profile
Dot?
‘i: I
ioktion
Profile
time
C
A---
OO
Composition
Profile
time
Fig. 7. Simulated 1% peak height (fl changes for various depth profiles of a given atomic component in the specimen computed with Gaussian beam clipped as in fig. 5 [exp c-r;) = 0.4]. In each frame the upper plot shows the simulated output plus a set of processed data points corrected without allowance for clipping. The lower plot shows the true depth profile as seen at the bottom of the crater or obtained by full correction of the simulated output with proper allowance for clipping.
D. M! Hoffman/Cratering
analysis for depth profiling
43
factor of eq. (26). The clipping radius was placed at 40 percent of the maximum intensity (r, = dw with a resulting normalizing factor of rk = 0.824. Values of the fractional peak height were computed for various assumed composition profiles and plotted as a function of time of sputtering as indicated by the full line in the upper graph of each pair (frame) in fig. 6. The lower graphs show the assumed composition profiles as would be reproduced by application of the Gaussian beam solution with correction for clipping [eqs. (28) and (23)] to give the composition at the bottom of the crater as a function of time. The results in fig. 6 are for layers of a pure component (B) at various depths in a specimen of pure A starting with the case of a coating of B at the surface. The transition from B to the adjacent layers is assumed to be sharp (sharp interfaces). One sees that the predicted peak height changes do not reproduce the composition profiles, but rather exhibit tails that die off asymptotically toward the limiting composition at each depth level. Persistent residual signals of this sort are a common feature observed in depth profiling by ISS [2]. One sees, moreover, that the maximum predicted peak height decreases as the depth of the B layer becomes larger relative to its thickness. Also plotted on the upper graphs are points generated by eq. (23) from the simulated fcurves without the clipping correction. While this is the crudest possible data processing procedure, the results are clearly a better approximation to the composition profiles than the peak height curves. Disregard of the clipping correction gives rise to errors of two kinds. The first is a scaling error evidenced by the fact that the processed points tend to overshoot the actual composition change at a sharp interface. For a B layer at or near the surface this leads to negative processed values below the film. For the deeper layers an echo effect appears, distinct from the overshoot effect. While the square profile of the B layer is fairly reproduced by this method, it is important to notice that the interfaces obtained from the processed data are not as sharp as the specimen, as evidenced by the points plotted at the location of the interfaces in each graph. In frames E and F of fig. 6 we also show the cases of alternating A and B layers and a stepped profile with a 50-50 alloy layer between a coating of B on A. Notably the simulated peak height damps very rapidly to the average composition of the layered specimen. In the stepped specimen f merely exhibits a discontinuity in slope. In both cases, however, the rough processed points reveal the true nature of the profile to a useful degree of accuracy. Full processing with the clipping correction must, of course, reproduce the specimen profile exactly. In a cautionary vein, we warn that the processed points for the deeper layers of frame E are produced by long extrapolations from the f curve, so that any scatter in real experimental data for ,fwould become increasingly magnified (see ref. [3]). Fig. 7 shows simulated ISS peak height curves for a series of specimens coated with a fixed amount of component B and having transition profiles of various steepness at the coating-substrate interface. As in fig. 6 the first frame here depicts the case of a coating with a sharp interface, but now for comparison we include results for no clipping and severe clipping (rC = dm)). That the no-clipping curve is in fact a per-
44
D. W. HoffmanlCratering
analysis for depth profiling
feet hyperbola can easily be demonstrated by integration of eq. (23). Processed data points are also shown for these three clipping variations. As expected, the unclipped points (ge = 0) reproduce the sharp profile with good accuracy, barring only two intermediate points at the interface that result from the discreteness of the data points. Moderate clipping produces as before some negative overshoot followed by an echo of the interface. Severe clipping causes a negative overshoot almost as great as the initial peak height followed immediately by the echo of the interface. Were one to obtain such results in an experiment, the present considerations would suggest the presence of a sharp imerface, but reconstruction of the interface profile with accuracy would require the correction for clipping. For the analytical approach, therefore, it is clearly desirable the minimize the effect of clipping by the use of a focussed beam. Frames B through D of fig. 7 show linearly graded interface; of increasing thickness relative to the overall thickness of the coating with peak height simulation under moderate clipping only. As in fig. 6 the peak-height curves bear only passing resemblance to the composition profiles, with the ubiquitous tailing behavior as a function of time. By comparison the processed points reproduce the linear profiles rather well, especially if one re-scales the downward variation from unity to zero, (see dashed lines) and disregards the echo. It is interesting to notice in Frame D that the peak height variation itself becomes linear (compared with Frames A-C) over the region of the graded interface, which now comprises the whole thickness of the coating, but the slope of the simulated curve is not as steep as the change in composition. Frame E carries the sequence one step further with the surface concentration at only 50 percent B. This profile or a modification thereof could conceivably serve as a rough model for detection of solute segregation to a surface by ISS. Finally we show in Frame F the case of concentration following a cosine variation from the surface through the interface. The peak height curves from Frames C and F would no-doubt be experimentally indistinguishable. The results of this section indicate that data processing by eq. (23) offers a significant improvement over the simple time-variation of the peak height for depth profiling by KS, provided that clipping is moderate (beam well focussed) and the raw peakheight data can be corrected for intensity transients of the primary beam to yield a stabilized peak height. If the transients are known to be small, one can of course use h directly with suitable scaling of the processed results. Full application of the clipping correction has not been plotted explicitly in figs. 6 and 7 as it merely reproduces the assumed depth profiles. In practical application this step may not be necessary unless clipping is severe, but the use of this full correction leaves the non-Gaussian character of the beam as the only source of error in the results according to the model.
8. Beam shape analysis Up to this point we have regarded eq. (16) as a problem to be solved for C(Q given f(t) and the beam intensity distribution in terms of ~(?/t). Conversely, however, one
D. W. Hoffman/Cratering
45
cmlysis for depth profiling
can solve for ~(f/t) s ok) given a known specimen C(o and the corresponding observed peak heightsflt). In this way the depth profiling analysis can be turned around to determine the beam intensity distribution. A specimen profile of practical utility as a “detector” in this respect is the case of a pure coating with a sharp interface at the substrate, as illustrated in Frame A of both figs 6 and 7. Inserting the composition profile
1,
t < to
c(t) =
(34) i 0,
tat,,
into eq. (16) yields
.f(t)
=$s” K(:/t) di
(35a)
0
=- 1 rO
(to/t) ‘dd
s
(=b)
&,
where to is the moment at which the crater first penetrates to the interface, and eq. (12) effects the indicated change of variable. Differentiating with respect to go E (to/t) yields
K(fo/t) df(t)
_=-?r
d(t,lt)
Af - A(r,/t)
(36)
’
by which experimental data from a sharp-interface specimen can be processed to yield (~/r) directly. For the purpose of illustration it is instructive to carry the analysis a step further by substituting eq. (17) for K and rearranging terms to give:
df
4Po) dp, = -y&J
=r
d(l-f)
g&q
.
Since the clipping factor is unity for r < rc [see eq. (4)] , integration vides a formula for p. up to p, = r:: [~-f(toko)l po=rJ
g
of eq. (37) pro-
d(l_f) -
0
(37)
(384
go@o)
r dab%mts &W-f),
0 G p. G r(f.
(38b)
Thus a representation of the radial beam intensity distribution is obtained by plotting go = (to/t) versus r. = Go. Such a plot from the data of ref. [3] for a specimen of gold on chromium analyzed with a focussed neon beam is shown in fig. 8. Values of
46
D. W. HoffrnanlCratering
analysis for depth profiling
I o -
experlmental reference
data Gaussian
(ewr2)
(+I 9,
0 r Fig. 8. Radial distribution of intensity in an actual ion beam as determined by application cratering analysis to ISS output from a coated specimen (Au on 0) with a sharp interface.
of the
the stabilized fractional peak height were obtained by special consideration of the peak height from the chromium relative to the gold. Comparison of the data points in fig. 8 with the reference Gaussian curve [exp (-Y:) versus ro] shows rather close agreement up to the location where the sharp departure from the reference curve takes place. This departure signals the onset of the clipping effect, which occurs at rc = 0.894 and gc = 0.45 and corresponds to the “moderate” clipping factor employed in the data simulation of figs. 6 and 7. Since the data of fig. 8 adhere so closely to the Gaussian curve, one may attempt a correction of the points beyond yc by means of the special Gaussian clipping function of eq. (26). Thus: A(l-f) (2/n)
Sin-l(t’c/~o)
0.4 t > to.
(39)
’
While these corrected points do fall closer to the Gaussian curve in fig. 8, one sees that deviation from the Gaussian distribution increases significantly at the large radii. Nevertheless the fact that the clipping correction does partially smooth out the plot of experimental points supports the validity of the clipping factor as it has been formulated. Indeed the overall behavior of the data points in fig. 8 and their general adherence to Gaussian behavior serves to substantiate the cratering analysis as a whole, while also upholding the utility of the Gaussian beam approximation.
9. A practical stratagem In the solutions and examples of the preceding sections we have used the stabilized fractional peak height f to avoid complications arising from time transients in the intensity of the primary beam. In a real 1% experiment, however, one measures h not ./?
D. W. Hoffman/Cratering
analysis for depth profiling
47
The depth profiling theory can then be applied only if one knows that the primary beam intensity remained constant (so that h is simply proportional to f’j or can compensate for changes in the beam intensity, say, from simultaneous observations of the beam current [2] or by a self-consistent normalization [3]. We now propose a special strategem for overcoming this problem in practical depth-profiling experiments. The proposed strategem involves coating any specimen to be investigated with an evaporated layer of pure gold or other suitable element before depth profiling by ISS. The thickness to be applied varies with the depth of interest in the specimen and the severity of clipping in the instrument to be used. It is necessary that the applied film of gold not diffuse into the specimen and vice versa, so that a sharp interface is maintained with the following profile, tq), t>t,,
where to is the time for initial penetration for each atomic species in the specimen:
ci=
OT
i ‘j(‘>T
of the crater through the gold. Conversely
t G t(J t>
(41)
to
etc. The coating of gold accomplishes two things by (1 j providing a reference signal from which changes in the beam intensity can be determined, while (2) also keeping the clipped portion of the crater temporarily above the underlying specimen. The requisite analysis goes back to eq. (8) which we write once for the gold
h.&
=&U@j f,,@j>
and again for each other component
(42) of interest: (43)
hj(t) = Hi(tj fi(t)> etc. We then define the ratios of the component Ri(t) Ehj(t>/hA,(t)
= “jh(f)/KA,
peak heights to that of the gold
fA,(t).
(44)
etc., where the pure peak heights cancel except for the IS ‘s. One notices by eq. (7) that the beam intensity factor Z,,(t) is thereby eliminated. To proceed we now substitute eq. (16) for & and fAU, adjusting the limits of integration in accord with the regions where the respective compositions are zero and cancelling t -’ from both sides of the result:
analysis for depth profiling
D. W. HoffmanfCratering
48
R&t) K,”
g
~(i/t)
dt = Ki i
0
K($t)
Ci(?) d?.
(45)
to
By reference to eq. (18) the left hand side can be reformulated another integral having the same limits as on the right:
R;(t)KAU [t+(~,i)di]=&@)4(nd?.
in terms of r and
(46) to
When the crater has just begin to penetrate difference
below the gold into the specimen, the time
Ar = (t - to)
(47)
is still short relative to t. Only the center of the crater has yet passed through the gold. This is, of course, the unclipped part of the crater and also where, according to the previous section, the intensity of a focussed beam approximates Gaussian behavior. That is to say:
dp
n
K@/t)=
d
ln
(l,g)
=
1,
tO
p%y
Thus Ri(t)
K,,
(tr
-
t + to) = Ki j$(i)
d?,
(49)
fo Differentiating with respect to t then yields a solution for Ci(t) in terms of the measured ratio Ri(t) and the constants Ki, KAU and r. The normalizing parameter r can however be estimated from the behavior of the gold peak near to as explained in Appendix B. The final result has the form
(50)
HAllJt @HA&W-
(aA,/dt)
1’ to
(51)
where At f (t - to), and the derivatives (dHA,/dt) and (&*u/dt) refer to the time rate of change of the height of the gold peak immediately before and after to respectively. Processing by this formula can be applied in turn to each elemental component of
D. W. HoffmanlCratering
49
analysis for depth profilinx
the sample giving the depth profile Ci(t) as seen at the bottom of the crater in units of (Ki/KAU). The normalizing constant has only to be determined once. Moreover, if r c 1, one has the simplified approximation Ki
H%u
q(t)
t0 f-G
z t0 z,
(52)
predicting that R&t) will be a linear function of t whenever Ci(t) in the specimen is constant. Whenever clipping is moderate (8, - 0.4) as in the data of the preceding section, the above processing should be valid until t exceeds (tO/gc), a point that may often be discernible in the behavior of the data. Clearly then, for a given clipping factor, deeper profiling into the sample requires a thicker layer of gold in order to increase r. and therefore the limit (to/g,). The method of depth profiling proposed above is not to be confused with ref. [3] where the focus of attention was on the possible detection of diffuse interface under a layer of gold on chromium. We believe the approach given here can provide a general method for depth profiling by ion scattering spectrometry that solves the cratering problem and circumvents the problems of clipping and beam intensity transients.
Acknowledgments The author is grateful for the time and effort freely given by numerous colleagues whose advice and encouragement have contributed substantially to this work. These persons include members of the Scientific Research Staff at the Ford Motor Company: Alan Brailsford, Robert Asaro, William Johnson, Veeravanallur Sundaram, Mary Ann Wheeler, Rao Nimmagadda, Walter Winterbottom, Edward Sickafus, and members of the ISS group at 3M Company: James McKinney, David Smith, and Robert Goff.
Appendix clipping
A. Correction
of the stabilized fractional
peak height for the effect of
Cancelling df above and below while adding and subtracting u(f/t), converts eq. (24) to the form
l-;; f’(t) =
$j-d($-) 0
jexp
(- “’ (l-u)
d(g),
(A.11
0
where the limits of integration are adjusted to eliminate vanishes. Eq. (A.l) further becomes
r;;f’(t) =f(r) -
unity with respect to
j fg du> 0
the region where (1 - U)
(A.21
50
D. W. Ho.ffman/Cratering analysis for depth profiling
after suitable integration by parts and use of eq. (12). The integral of eq. (A.2) is then broken up into steps from each time of observation to the next, up to time t. To facilitate the notation we index these times and the corresponding peak heights in order of observation by the integral subscripts m and n,
nc
r;,.r;l, =.f,,, -
Um,n
cJ
n=l
m=
fg du>
1 toN,
(A.3)
Llm,n-i
where the lower limit becomes zero for M = 1. Assuming that the time intervals are sufficiently short with respect to the variation off, we can extract a locally averaged value of the peak height from each of the sub-integrals: % r&l,
=I,
-
c
‘m,n 3 c/n_,
g du.
+r,> J
1*=1
(A.4)
‘m,n-1
Using the functionality between g and u given by eq. (26) the sub-integrals can now be individually evaluated yielding numerical coefficients Pm n defined as follows: p
m,n
-_=!m’n
exP (simi,2))2
(A.51
d”.
We note for clarity that the significance of the double index notation on u derives from its dependence on both t and ?, i.e., u(f/t) = u(fJtm> = urnan and that the meaningful range of integration in eq. (AS) stops at tn, = tm exp (-rz) unity. For convenience we therefore define P
m,n “m 3n.’c
to be independent (A.4) becomes
n>n
where Urn,,, becomes
(-4.6)
c’
of n when 11exceeds ne. With these definitions
=fm- n=l 5 w, +ft*_J (Pm
GfrLl
,I
-
pm,rz_J,
m=
and conventions
1 toN,
eq.
(A.7)
a set of linear simultaneous equations equivalent to eq. (24) and to be solved for values of f’given an observed set of the f '.By judicious juggling of indices and defining of fictitious coefficients P
m,-1
P m,m+l
= pm,0
64.8)
(= (2,
(A.9)
‘rn,m,
eq. (A.7) can be compacted
to the form
D. W. Hojyman/Cratering
analysis for depth profiling
m=l
toN,
51
(A.9)
where
pm )
(A.10) m,n = 5 Pmn+l >n-1 are partial areas to the left of the curve in fig. 5 [recall eq. (AS)] . A further simplification occurs by considering momentarily the special case of a pure, single-component specimen material for which .I” =.f= 1 at all times. Eq. (A.9) thus yields AP
(A.1 1)
for the normalizing text.
Appendix specimen
constant.
B. Experimental
Substitution
evaluation
back into eq. (A.9) leads to eq. (27) of the
of the normalizing
constant
for a gold coated
For times greater than to eq. (16) gives
f,,,(t) = &
7 K(@)dt^
03.1)
0
for the fractional height of the gold peak itself. Splitting the integral of eq. (18) at t0
rt = r
K(t/t)
dtA+ j K(@) dt*
0
enables eq. (B.l) to be rewritten
f,,(t)
(B.2)
IO
= 1- & j
K(f/t)
as:
di.
(B.3)
f0
This is advantageous since the kernel K(~/z) is unity for t^close to t. The integral can then be evaluated and the result inserted into eq. (42) for the unscaled peak height immediately after to :
h*Jt> =17’*“(t) [l-G]
(B.4)
D. W. Hoffman/Craten’ng
52
Subtracting
analysis for depth profiling
the identity
from eq. (B.4) and dividing through by (t - fn) gives:
h&(t) -(&j)
= ff*Jt> -
to
t-to
t~
~*JQ _- ~.&) l-r
(B.6)
.
Solving eq. (B.6) for r and taking the limit as (t - fo) goes to zero yields eq. (5 1) of the text. To evaluate (dH/dt) experimentally one takes the time rate of change of the gold peak immediately before to.
References [l] [2] [3] [4]
R.E. Goff and D.P. Smith, J. Vacuum Sci. Technol. 7 (1970) 72. R.E. Honig and W.L. Harrington, Thin Solid Films 19 (1973) 43. D.W. Hoffman and R. Nimmagadda, J. Vacuum Sci. Technol. 11 (1974) H. Shimizu, M. Ono and K. Nakayama, Surface Sci. 36 (1973) 817.
657.
50 (1975) 29-52 Publishing Company
A CRATERING
ANALYSIS
FOR QUANTITATIVE
DEPTH PROFILING
BY ION BEAM SPUTTERING
D.W. HOFFMAN Scientijic Research Staff, Ford Motor Company, Dearborn, Michigan 48121, Received
31 October
1974; revised manuscript
received
18 February
U.S.A.
1975
The depth profiling of composition by sputtering with a stationary ion beam of nonuniform intensity is treated analytically. Attention is devoted to the situation where the experimental signal originates simultaneously from all or most of the sloping, sputtered crater, as in ion scattering spectrometry (ISS) or secondary ion mass spectrometry (SIMS). As a result the signal itself does not relate directly to the composition at any specific depth in the specimen. Data processing formulas for various degrees of correction are presented, starting with a very simple result for the Gaussian intensity distribution. Simulated ISS outputs and processed data points are then illustrated for a number of selected composition profiles. An alternative use of the analysis to determine the radial intensity distribution in an ion beam is presented with experimental data supporting the Gaussian beam approximation as modified by the so-called clipping factor. Finally a practical stratagem is proposed for overcoming such extraneous problems as beam intensity transients, while determining the depth profiles of the elements in specimens of several components.
1. Introduction
The depth profiling of composition near the surfaces of solids is facilitated by the use of accelerated ion beams for controlled removal of material from a specimen by sputtering. The sputtering away of material at the point of beam impingement produces a crater which gradually deepens as exposure to the beam continues. In some cases elemental analysis of the composition in the advancing crater is carried out with a separate experimental technique such as Auger spectroscopy, where a fine beam of electrons is directed at the bottom of the crater. In other cases, determination of the composition within the crater relies directly on the interaction of the ion beam with the solid by measurement of the energy losses of the scattered primary ions (ion scattering spectrometry - ISS) or the identification of material sputtered away (secondary ion mass spectrometry - SIMS). In these latter cases the measured signal generally comes from all or a substantial fraction of the crater surface. If the depth of the crater is nonuniform, the signal does not relate directly to the composition at any single depth in the specimen. Rather, the signal at any instant represents an average over the concentrations at the various depths exposed in the crater at that time.
Interpretation of the resulting data for quantitative depth profiling is therefore problematic. The situation described above arises when the intensity within the primary beam is nonuniform, as in, for example, a radial Gaussian distribution. Since the sputtering rate in a given material is most rapid where the flux of incoming ions is greatest, such a beam produces a crater with sloping sides. A number of experimental remedies have been employed to overcome this problem. One approach involves passing the primary beam through a small aperture to eliminate all but a region of near-uniform intensity at its center. Another approach is to sputter with a large beam and analyze with a small beam by intermittently focussing the ion beam to a reduced size. A third avenue is to sweep the beam in a raster pattern to produce a large, flat-bottomed crater, while gating the output signal to be measured only when the beam passes the center. The purpose of this paper is to present a theoretical approach to the problem and to show how data obtained with a stationary, nonuniform beam can be processed analytically. In the treatment that follows we shall refer specifically to ISS, but the general fornlulatl~n and results should be applicable to SIMS and any other depthprofiling techniques that derive their outputs from the ion-solid interaction. The development of this paper begins with a mathematical description of the depth profiling problem followed by an advantageous change of variables that enables the further manipulation and solution of the equations. Solutions are given for Gaussian and non-Gaussian distributions of beam intensity, the former yielding an operational data-processing procedure of extreme simplicity. A series of simulated ISS outputs are then displayed for various illustrative composition profiles, including approximate processing of the simulated data to reconstruct the sample. A different appli~tion of the analysis is then presented by means of which a known specimen can be used to make an experimental determination of the beam intensity distribution in an actual ion scattering spectrometer. Finally we propose a stratagem for handling both the cratering problem and uncontrolled intensity transients in the practical depth profiling of unknown, multicomponent specimens.
2. mathematical
formulation
of the eratering probiem
The sketch in fig. 1 illustrates the experimental configuration in ion scattering spectrometry on a cross section through the specimen in the plane of the beam axis [ 1,2] . A collimated beam of inert gas ions accelerated through a field of, say, 2 kV strikes the surface of a flat specimen at an angle of 45” to the normal. The ion beam is monoenergetic but its intensity is nonuniform with a liaif-maximum diameter on the order of a millimeter. ‘Upon impact with the specimen some of the ions scatter elastically with characteristic energy losses determined by the masses of the atoms they hit. Simultaneously the sputtering action of the ion beam removes atoms from the specimen at rates which vary over the cross section of the beam according to the distribution of intensity. The resulting erosion causes the formation of a shallow, sloping crater
D. W. Hoffman/Crater&
ion
specimen
analysis for depth profiling
31
electrostatic
Fig. 1. Cross section of a neon-sputtered crater penetrating the specimen during ion scattering spectrometry. Since the depth-to-diameter ratio (exaggerated in sketch) never exceeds 10m4, the actual scattering geometry remains essentially flat.
which grows ever deeper as irradiation proceeds. It is important to notice that the walls of the crater do not ascend abruptly to the original surface, but flatten out gradually in accord with the tails of the ion beam intensity distribution. Inert gas ions scattered at right angles to the incident beam enter a detector (electrostatic analyzer) which separates them according to their scattered velocities and records the flux of ions as a function of kinetic energy. The observation of peaks in the scattered ion flux at characteristic fractions of the primary beam energy serves to identify the various atomic species exposed to the beam at the surface of the crater. The height of a given peak is proportional to the surface coverage by the corresponding element, but the absolute proportionality factor varies from element to element. Although the depth profiling capability of 1% depends on the formation and advancement of the crater into the specimen, the depth of erosion is always sufficiently small (G 10e4 mm) relative to the diameter that the scattering geometry remains essentially flat. Since the primary beam is approximately circular in cross section, the 45” angle of impingement causes the contours of the crater to be elliptical. The electrostatic analyzer accepts ions scattered from a strip of area across the crater, as shown in fig. 2 adapted from ref. [2]. A fine slit at the entrance to the analyzer limits the observed scattering angles to a very small range near 90”. The number of ions scattered per unit time by an atom at the surface of the specimen is determined by the primary beam intensity at that point and appropriate physical constants. The electrostatic analyzer adds together a selected flux of ions coming from points lying within its field of view A’ across the crater. The flux h scattered from atoms of a given species can therefore be expressed as an integral of the form h =K
j-ma, A’
(1)
where I and Care respectively the primary beam intensity and the volume fraction of the indicated species on the element of area da, and the proportionality factor K
32
D. W. Hoffmanfcratering
analysis for depth profiling
~tglLLJ Depth-
intensity
- composltlon
1-
0.24mm
-L
Differential of area
element
Area seen by electrostatic analyzer
L’ig. 2. Aerial view of primary ion spot on 1% specimen, showing contours of constant intensity--composition within the sputtered crater and the view-band across the crater scattered ions may enter the electrostatic analyzer.
depthfrom which
includes such constants as the differential scattering cross section and the probability of neutralization *. For the purpose of discussion we shall associate h with the maximum signal or peak height produced by the electrostatic analyzer in scanning past the characteristic energy for the element in question. With the help of eq. (1) the cratering problem can now be stated more explicitly. Given measured values of h as a function of time, and some measured or assumed function for the beam intensity distribution Z, solve for the concentration C at various depths in the specimen. To proceed, it is necessary to examine the components of the integral in more detail. Consider the beam intensity. In addition to the spatial variation of intensity mentioned above, the primary ion beam also frequently exhibits some weakening with time during an experiment, owing apparently to a decrease in the pressure of the inert gas in the spectrometer. This effect introduces another uncertainty into any depth profiling observations over and above the cratering problem. For the purpose of analysis let us assume the beam to be axially symmetric with separable space and time variations of intensity in the form
were t is time, Yis a radius normal to the beam axis, lo(f) is intensity at the center of the beam, and the scaled intensity function g(r) goes from unity at the center of the beam to zero at the extremities. In other words, any temporal fading of the intensity occurs proportionately throughout the beam so that the relative distribution g(r) is preserved. *
Ignored in this formulation are minor variations in the acceptance angle of the electrostatic analyzer and the differential scattering cross-section over A’. Since the variation of these factors is essentially linear across the narrow dimension of A’, their effects average out of the result when the beam is centered on A’.
D. W. HoffmanlCratering
analysis for depth profiling
33
The specimen is taken to be a coated or otherwise treated material with lamellar variations in composition parallel to the surface. As the sputtered crater burrows down into the specimen, more and more composition layers gradually become visible to the beam. We assume that the strata exposed on the sloping walls of the crater represent the undisturbed interior of the specimen with no intermixing by the beam prior to sputtering*. We do allow however for different sputtering rates of the various atomic constituents, so that the crater may penetrate more rapidly through one layer than another. Looking down onto the cratered specimen one sees the exposed edges of the various composition strata as a set of concentric ellipses. Regardless of how fast or slow the individual layers may sputter back, the elliptical contours of composition viewed at any instant must necessarily coincide with rings of constant intensity in the beam. It is convenient to take our element of area da to be an elliptical ring between closely spaced composition contours on the specimen as indicated in fig. 2. Since the crater is always extremely shallow, it is unnecessary to account geometrically for the negligible slopes of the crater walls. We thus express da in terms of the radius of the corresponding intensity contours in the beam da = fi(2nr
dr),
(3)
where the factor of fi corrects the circular element to a 45” ellipse in which r is the semi-minor axis. Owing to the fact that da follows the composition contours, the composition thereon does not vary with angle bout the beam axis, but only with the radius and the time, therefore C = C(r, t). Another point of view is to consider the element of area as projected along the beam axis (2nr dr), whence the fi factor becomes a correction onto C(r, t) for the projected density of surface atoms. Finally, we notice that above a certain size, the element da becomes truncated in the view of the electrostatic analyzer, the part beyond the strip A’ not contributing to the observed ion flux. Sighting from the analyzer along the reflected beam axis as in fig. 3, we define a multiplicative clipping factor r < rc
‘(‘) = ( i$rf) sin-’ (rc/r),
r > rc
(4)
to correct da when r exceeds the clipping radius rc. Should the band of observation A’ not be centered on the crater, the clipping factor involves two different clipping radii, but its application and effect are similar. Inserting eqs. (2), (3) and (4) into eq. (1) yields a fully parameterized integral for the ES peak height according to the model developed here. In carrying out this step
*
The compositions at the sputtered surface of the crater may in fact be perturbed by preferential sputtering from those of the underlying layers 141, but this problem is generic to 1% and other approaches to depth profiling in a sputtered crater, e.g., by Auger spectroscopy. The cratering analysis relates only to the compositions that actually appear at the sputtered surface.
D. W. Ho ffman/Cratering
34
analysis ,for depth profiling
I:&. 3. Formulation of the clipping factor for truncation electrostatic analyzer along the retlccted beam axis. it is convenient
of crater
contours
in the view of the
to substitute dp = 2r dr,
/a’?,
(5)
so that the result and others to follow can be written more compactly stituent under consideration:
for the con-
7 0
h(t) = K fi .rrlo(t) g(p) C(P,4 4~) dp. A pure specimen
[C(r, t) = l] gives the maximum
(6) peak height:
(7) The ratio of eq. (6) to eq. (7) with cancellation tional peak height
wherein the normalizing re
constant
of common factors yields the frac-
r is the integral of the beam intensity
s -
g(p)4~) dp.
*:
(9)
0
* A definition of 1 was given previously above definition is the one that applies
[ 31 that differs from eq. (9) by the factor & ~10. The to the results quoted
there as well as here.
D. W. Hoffman/Cratering
analysis for depth profiling
35
Definition of the stabilized fractional peak height by eq. (8) serves to isolate the time dependence caused by cratering itself from such extraneous matters as beam intensity transients, while also eliminating unknown physical constants. An approach to the handling of beam intensity transients in binary specimens appears elsewhere [3] and will be reconsidered at the end of this report. Meanwhile it is the solution of the cratering problem embodied in eq. (8) to which this paper is next addressed.
3. Change of integration
variable
Eq. (8) is difficult to solve because the unknown function C(p, r) appears as the kernel, i.e. a function of both time and the radial integration variable. The time t is the moment when the depth of the crater at the radius r (= 6) reaches the level where the specimen has the composition C(p, t). Our assumptions about the specimen provide, however, that the composition is, in fact, a function of only a single variable, the depth z below the original surface. Since ISS results do not reveal actual depths of penetration, but only the elapsed times of exposure to sputtering by the ion beam, it is preferable not to deal directly with z but rather with the corresponding time f(z) required for the center or bottom (p = 0) of the crater to reach the level z. We thus write the identity C(P, t) = c(o, t*)* C(t)
(10)
to quantify implicitly the desired transformation of variables. To find the explicit relationship between p, t, and f, consider time plots of the depth of penetration at the center of the beam and at the radius r where the intensity is, say, half the maximum as sketched in fig. 4. The indicated variations in sputtering rate along each curve supposedly arise from the variations in composition within the specimen. The difference in sputtering rates between the curves at a given depth arise from the difference in beam intensity. Indeed the relative rates of penetration must stand in direct proportion to the intensities at r and at the center. Consequently, where the intensity is only half the maximum it takes the crater twice as long to reach a given depth. Thus we can write the general relationship Vr = g(r)/&8
,
(11)
or t^= t g(r),
(12)
since g(0) = 1. One sees that the two plots in fig. 4 can be superimposed by shrinking the time axis of the half-maximum curve by a factor of 4. The result is a master penetration curve on the z versus t axes from which the penetration at any selected values of r and ? can be ascertained by use of eq. (12). TO accomplish the transformation of variables in eq. (8) we differentiate eq. (12) at constant t dt*= t (dg/dp) dp,
(13)
D. W, IfoffinanjCratering
36
analysis for depth profiling
beam center
half- maximum
2
Time Fig. 4. Depth of penetration versus time at points of differing intensity in the beam. Changes in rate of penetration along a given curve are caused by variations in composition with depth in the specimen.
leading to: (14) Substituting duces
eqs. (10) and (14) into eq. (8) and converting
the limits accordingly
./I)) = + i ~(f,‘t) C(t) dt^,
pro-
(16)
0
a Volterra integral equation
of the first kind with the kernel
pertaining to the clipping factor and the distribution of intensity in the beam. One sees that the composition profile C(f) to be determined now appears in the conventional format for an unknown function. Eq. (16) is entirely equivalent to eq. (8) but has a much more convenient form for manipulation and solution. Operating on eq. (9) in a simiiar manner [or setting both C and f to unity in eq. (16)] produces a companion equation for the normalizing factor:
D. W. HoffrnanlCraterimg analysis for depth profiling
dj~(;,t)dt.
31
(18)
0
4. Gaussian beam approximation
without analyzer clipping
Eq. (16) lea+ to a simple solution of the cratering problem in the Gaussian beam approximation : gG = e
-r2 = e-p
(d lndFl/g)),:
(19)
wo
= ”
Consider first a hypothetical case without any clipping (U = 1 everywhere), which is to say that the electrostatic analyzer sees the complete irradiated area (rc * m). Then eq. (17) gives rc = 1,
(21)
and eq. (16) reduces to
fit) = +j
C(l) dt”
(unclipped
Gaussian beam),
(22)
(unclipped
Gaussian beam).
(23)
0
with the immediate
solution:
This simple formula for the processing of data yields the composition at the very bottom of the crater as a function of sputtering time. As always, conversion from time to penetration distance requires some independent knowledge of sputtering rates and their variation with composition. Although eq. (23) was obtained under the unrealistic assumption of no analyzer clipping, it can in fact be usefully applied to experimental data when the effect of clipping is not too severe. The nature of the results and errors incurred by such an approximate procedure will be examined in a later section of this paper on ISS simulation. An example of data processing by means of eq. (23) is also presented in ref. [3 ] on the investigation of interfaces under ion-plated coatings. Ref. [3] gives a graphical construction for eq. (23) when the data points f(r) are evenly distributed in time. When the observation times are not evenly spaced, a more accurate processing of the discrete data points obtains from the formula *
Without
loss of generality
we now taker
in units of its value at g = l/e.
38
D. W. HoffkanfCratering
analysis for depth profiling
t I II m+l--*rn
r
m+l -tm_l
.
(234
where the interger subscript m indexes the times of observation and the corresponding peak heights in the order of observation. In other words, fl is the first peak height observation and tl is the time of that observation,,f2 is the second peak height observation and t2 is the corresponding time, etc. Eq. (23a) is merely a discrete version of eq. (23) in which the derivative of the product (tf) is first expanded term by term and the derivative off is approximated by lever-rule averaging of the differences between .f,,, and neighboring values.
5. Gaussian beam solution with clipping correction Another instance in which the Gaussian beam solution can be applied is after the data are first corrected for the effect of the clipping factor in a preliminary step. Substituting eqs. (20) and (23) into eq. (16) prodcues
f’(t) =
& /
u(t,t)(; y
dt^,
(24)
relating the clipped peak height ,f’ to the corrected peak height f. With clipping the normalizing factor becomes less than unity,
I’; =+j
(25)
u(;/t) dt*, 0
where the clipping factor has the form (2/7r) sin-l (rel&Z U(f&
),
t*< t exp (-r:),
= t^> t exp (-rz),
i 1,
(26)
in terms of the time-ratio notation. Fig. 5 shows a plot of the clipping function for an arbitrarily chosen value of the clipping radius. To solve eq. (24) for f(t) the integral is replaced by a summation over the discrete times at which the height of a particular peak was recorded on successive ISS scans. Details of this derivation are outlined in Appendix A. The resulting linear simultaneous equations take the form; m
1, - fl, =
c ~mn(f, II= 0
- fA>>
m= 1 toN.
(27)
D. W. Hoffman/Cratering
39
analysis for depth profiling
Fig. 5. Clipping function for moderate clipping [g(r,) = 0.4 ] on a Gaussian to areas to the left of the curve as illustrated. APmn correspond
beam. Coefficients
The indices, m and n, represent integer subscripts that refer to the N data points VI, f2,. . ., f,) in order of observation. The coefficients aP,,r are areas on the graph of fig. 5 as indicated by the cross-hatched region. These equations can be solved in sequence
etc., where the solution at each step feeds into the right hand side of the next. To begin, one may have to assume f. = f; . One sees from the form of this solution that the clipping correction just adds a rectifying term onto the original data for f”. Formulas for the M’s, given in Appendix A, require numerical integration of the clipping function. The actual computation is well-adapted for computer processing. The corrected values off(t) produced by eqs. (28) are then ready for processing by the Gaussian beam solution of eq. (24).
6. General solution Using methods akin to those of the previous section one can write a general solution to eq. (16) for the case of a non-Gaussian beam. If one knows g(r) by some independent measurement, values of the kernel k(?/t) are readily generated by eq. (17).
0. W. ~off~~~~rater~ng
40
an&k
for depth profilirzg
On the other hand, the cratering theory itself can be employed to determine ~(8/t) directly from ISS measurements on specialized (sharp interface) specimens fabricated for this purpose. The details of that determination will be described presently. For the moment we assume a knowledge of&/t). The general solution to eq. (16) takes the successive form
(29) etc., where the data, J; , j”,, f3, etc., are indexed in the order of increasing sputtering times, tl, t2, t3, etc., and the coefficients are defined as follows tn+l
Arrnn +-
s
K(&)
d;,
(30)
m t,,-1 with two exceptions: (31)
K(~/tm)
dt*,
Once again the indicated operations are easily computerized. If matrix methods are used, one defines also AI& = 0,
n>m,
(33)
in order to complete the square matrix of coeffients. In the application of this ap proximate solution one should bear in mind the assumption that the increment tn_l to t, be sufficiently small that corresponding changes in C, are not large. Although the general solution becomes equivalent to the Gaussian beam solution in the appropriate Iimit, experience at the author’s laboratory favors the two step procedure given previou~y in that case.
7. ISS simulation For the purpose of illustration a number of examples have been worked out from eqs. (16) and (17) using the Gaussian beam approximation of eq. (20) and the clipping
D. W. HoffmanfCratering
41
anaiysis fordepth profiling
I Simulated
f (clipped)
-
0 d(tf
Data
f
l/dt
I
‘I_ IL c
C
Composition
*0
Composition
Profile
0
time
o
I
I
t-
time
0
I-
Profile
@
D
0
Simulated
Doto
f
-
Simulated
Eato
:n u___ I
I c
C
Composition
0
0
Composition Profile
Profile
OO
time
time
I Simulated
Dota
f
I
C
111,
m--H-l
Lomposition
OO
time
Profile
*o-
Composition Profile
time
Fig. 6. Simulated ISS peak height U, changes for various depth profiles of a given atomic component in the specimen computed with Gaussian beam clipped as in fig. 5 [ exp(-rz) = 0.41. In each frame the upper plot shows the simulated output plus a set of processed data points corrected without allowance for clipping. The lower plot shows the true depth pro’file as seen at the bottom of the crater or obtained by full correction of the simulated output with proper allowance for clipping.
D. W. HoffmanlCratering
42
analysis for depth profiling
Simulated
Simulated Data f (Clipped )
I
o
1
l--l
0
c
c
0
Composition
OO
c
Profile
OO"-------time
time
Simulated
Composition
Profile
Data
Simulated
Dato
I!!!- A-
Data
C
Composition
OO
I
OO
time
Simulated
Composition
Profile
time
xmulated
Data
Profile
Dot?
‘i: I
ioktion
Profile
time
C
A---
OO
Composition
Profile
time
Fig. 7. Simulated 1% peak height (fl changes for various depth profiles of a given atomic component in the specimen computed with Gaussian beam clipped as in fig. 5 [exp c-r;) = 0.4]. In each frame the upper plot shows the simulated output plus a set of processed data points corrected without allowance for clipping. The lower plot shows the true depth profile as seen at the bottom of the crater or obtained by full correction of the simulated output with proper allowance for clipping.
D. M! Hoffman/Cratering
analysis for depth profiling
43
factor of eq. (26). The clipping radius was placed at 40 percent of the maximum intensity (r, = dw with a resulting normalizing factor of rk = 0.824. Values of the fractional peak height were computed for various assumed composition profiles and plotted as a function of time of sputtering as indicated by the full line in the upper graph of each pair (frame) in fig. 6. The lower graphs show the assumed composition profiles as would be reproduced by application of the Gaussian beam solution with correction for clipping [eqs. (28) and (23)] to give the composition at the bottom of the crater as a function of time. The results in fig. 6 are for layers of a pure component (B) at various depths in a specimen of pure A starting with the case of a coating of B at the surface. The transition from B to the adjacent layers is assumed to be sharp (sharp interfaces). One sees that the predicted peak height changes do not reproduce the composition profiles, but rather exhibit tails that die off asymptotically toward the limiting composition at each depth level. Persistent residual signals of this sort are a common feature observed in depth profiling by ISS [2]. One sees, moreover, that the maximum predicted peak height decreases as the depth of the B layer becomes larger relative to its thickness. Also plotted on the upper graphs are points generated by eq. (23) from the simulated fcurves without the clipping correction. While this is the crudest possible data processing procedure, the results are clearly a better approximation to the composition profiles than the peak height curves. Disregard of the clipping correction gives rise to errors of two kinds. The first is a scaling error evidenced by the fact that the processed points tend to overshoot the actual composition change at a sharp interface. For a B layer at or near the surface this leads to negative processed values below the film. For the deeper layers an echo effect appears, distinct from the overshoot effect. While the square profile of the B layer is fairly reproduced by this method, it is important to notice that the interfaces obtained from the processed data are not as sharp as the specimen, as evidenced by the points plotted at the location of the interfaces in each graph. In frames E and F of fig. 6 we also show the cases of alternating A and B layers and a stepped profile with a 50-50 alloy layer between a coating of B on A. Notably the simulated peak height damps very rapidly to the average composition of the layered specimen. In the stepped specimen f merely exhibits a discontinuity in slope. In both cases, however, the rough processed points reveal the true nature of the profile to a useful degree of accuracy. Full processing with the clipping correction must, of course, reproduce the specimen profile exactly. In a cautionary vein, we warn that the processed points for the deeper layers of frame E are produced by long extrapolations from the f curve, so that any scatter in real experimental data for ,fwould become increasingly magnified (see ref. [3]). Fig. 7 shows simulated ISS peak height curves for a series of specimens coated with a fixed amount of component B and having transition profiles of various steepness at the coating-substrate interface. As in fig. 6 the first frame here depicts the case of a coating with a sharp interface, but now for comparison we include results for no clipping and severe clipping (rC = dm)). That the no-clipping curve is in fact a per-
44
D. W. HoffmanlCratering
analysis for depth profiling
feet hyperbola can easily be demonstrated by integration of eq. (23). Processed data points are also shown for these three clipping variations. As expected, the unclipped points (ge = 0) reproduce the sharp profile with good accuracy, barring only two intermediate points at the interface that result from the discreteness of the data points. Moderate clipping produces as before some negative overshoot followed by an echo of the interface. Severe clipping causes a negative overshoot almost as great as the initial peak height followed immediately by the echo of the interface. Were one to obtain such results in an experiment, the present considerations would suggest the presence of a sharp imerface, but reconstruction of the interface profile with accuracy would require the correction for clipping. For the analytical approach, therefore, it is clearly desirable the minimize the effect of clipping by the use of a focussed beam. Frames B through D of fig. 7 show linearly graded interface; of increasing thickness relative to the overall thickness of the coating with peak height simulation under moderate clipping only. As in fig. 6 the peak-height curves bear only passing resemblance to the composition profiles, with the ubiquitous tailing behavior as a function of time. By comparison the processed points reproduce the linear profiles rather well, especially if one re-scales the downward variation from unity to zero, (see dashed lines) and disregards the echo. It is interesting to notice in Frame D that the peak height variation itself becomes linear (compared with Frames A-C) over the region of the graded interface, which now comprises the whole thickness of the coating, but the slope of the simulated curve is not as steep as the change in composition. Frame E carries the sequence one step further with the surface concentration at only 50 percent B. This profile or a modification thereof could conceivably serve as a rough model for detection of solute segregation to a surface by ISS. Finally we show in Frame F the case of concentration following a cosine variation from the surface through the interface. The peak height curves from Frames C and F would no-doubt be experimentally indistinguishable. The results of this section indicate that data processing by eq. (23) offers a significant improvement over the simple time-variation of the peak height for depth profiling by KS, provided that clipping is moderate (beam well focussed) and the raw peakheight data can be corrected for intensity transients of the primary beam to yield a stabilized peak height. If the transients are known to be small, one can of course use h directly with suitable scaling of the processed results. Full application of the clipping correction has not been plotted explicitly in figs. 6 and 7 as it merely reproduces the assumed depth profiles. In practical application this step may not be necessary unless clipping is severe, but the use of this full correction leaves the non-Gaussian character of the beam as the only source of error in the results according to the model.
8. Beam shape analysis Up to this point we have regarded eq. (16) as a problem to be solved for C(Q given f(t) and the beam intensity distribution in terms of ~(?/t). Conversely, however, one
D. W. Hoffman/Cratering
45
cmlysis for depth profiling
can solve for ~(f/t) s ok) given a known specimen C(o and the corresponding observed peak heightsflt). In this way the depth profiling analysis can be turned around to determine the beam intensity distribution. A specimen profile of practical utility as a “detector” in this respect is the case of a pure coating with a sharp interface at the substrate, as illustrated in Frame A of both figs 6 and 7. Inserting the composition profile
1,
t < to
c(t) =
(34) i 0,
tat,,
into eq. (16) yields
.f(t)
=$s” K(:/t) di
(35a)
0
=- 1 rO
(to/t) ‘dd
s
(=b)
&,
where to is the moment at which the crater first penetrates to the interface, and eq. (12) effects the indicated change of variable. Differentiating with respect to go E (to/t) yields
K(fo/t) df(t)
_=-?r
d(t,lt)
Af - A(r,/t)
(36)
’
by which experimental data from a sharp-interface specimen can be processed to yield (~/r) directly. For the purpose of illustration it is instructive to carry the analysis a step further by substituting eq. (17) for K and rearranging terms to give:
df
4Po) dp, = -y&J
=r
d(l-f)
g&q
.
Since the clipping factor is unity for r < rc [see eq. (4)] , integration vides a formula for p. up to p, = r:: [~-f(toko)l po=rJ
g
of eq. (37) pro-
d(l_f) -
0
(37)
(384
go@o)
r dab%mts &W-f),
0 G p. G r(f.
(38b)
Thus a representation of the radial beam intensity distribution is obtained by plotting go = (to/t) versus r. = Go. Such a plot from the data of ref. [3] for a specimen of gold on chromium analyzed with a focussed neon beam is shown in fig. 8. Values of
46
D. W. HoffrnanlCratering
analysis for depth profiling
I o -
experlmental reference
data Gaussian
(ewr2)
(+I 9,
0 r Fig. 8. Radial distribution of intensity in an actual ion beam as determined by application cratering analysis to ISS output from a coated specimen (Au on 0) with a sharp interface.
of the
the stabilized fractional peak height were obtained by special consideration of the peak height from the chromium relative to the gold. Comparison of the data points in fig. 8 with the reference Gaussian curve [exp (-Y:) versus ro] shows rather close agreement up to the location where the sharp departure from the reference curve takes place. This departure signals the onset of the clipping effect, which occurs at rc = 0.894 and gc = 0.45 and corresponds to the “moderate” clipping factor employed in the data simulation of figs. 6 and 7. Since the data of fig. 8 adhere so closely to the Gaussian curve, one may attempt a correction of the points beyond yc by means of the special Gaussian clipping function of eq. (26). Thus: A(l-f) (2/n)
Sin-l(t’c/~o)
0.4 t > to.
(39)
’
While these corrected points do fall closer to the Gaussian curve in fig. 8, one sees that deviation from the Gaussian distribution increases significantly at the large radii. Nevertheless the fact that the clipping correction does partially smooth out the plot of experimental points supports the validity of the clipping factor as it has been formulated. Indeed the overall behavior of the data points in fig. 8 and their general adherence to Gaussian behavior serves to substantiate the cratering analysis as a whole, while also upholding the utility of the Gaussian beam approximation.
9. A practical stratagem In the solutions and examples of the preceding sections we have used the stabilized fractional peak height f to avoid complications arising from time transients in the intensity of the primary beam. In a real 1% experiment, however, one measures h not ./?
D. W. Hoffman/Cratering
analysis for depth profiling
47
The depth profiling theory can then be applied only if one knows that the primary beam intensity remained constant (so that h is simply proportional to f’j or can compensate for changes in the beam intensity, say, from simultaneous observations of the beam current [2] or by a self-consistent normalization [3]. We now propose a special strategem for overcoming this problem in practical depth-profiling experiments. The proposed strategem involves coating any specimen to be investigated with an evaporated layer of pure gold or other suitable element before depth profiling by ISS. The thickness to be applied varies with the depth of interest in the specimen and the severity of clipping in the instrument to be used. It is necessary that the applied film of gold not diffuse into the specimen and vice versa, so that a sharp interface is maintained with the following profile, tq), t>t,,
where to is the time for initial penetration for each atomic species in the specimen:
ci=
OT
i ‘j(‘>T
of the crater through the gold. Conversely
t G t(J t>
(41)
to
etc. The coating of gold accomplishes two things by (1 j providing a reference signal from which changes in the beam intensity can be determined, while (2) also keeping the clipped portion of the crater temporarily above the underlying specimen. The requisite analysis goes back to eq. (8) which we write once for the gold
h.&
=&U@j f,,@j>
and again for each other component
(42) of interest: (43)
hj(t) = Hi(tj fi(t)> etc. We then define the ratios of the component Ri(t) Ehj(t>/hA,(t)
= “jh(f)/KA,
peak heights to that of the gold
fA,(t).
(44)
etc., where the pure peak heights cancel except for the IS ‘s. One notices by eq. (7) that the beam intensity factor Z,,(t) is thereby eliminated. To proceed we now substitute eq. (16) for & and fAU, adjusting the limits of integration in accord with the regions where the respective compositions are zero and cancelling t -’ from both sides of the result:
analysis for depth profiling
D. W. HoffmanfCratering
48
R&t) K,”
g
~(i/t)
dt = Ki i
0
K($t)
Ci(?) d?.
(45)
to
By reference to eq. (18) the left hand side can be reformulated another integral having the same limits as on the right:
R;(t)KAU [t+(~,i)di]=&@)4(nd?.
in terms of r and
(46) to
When the crater has just begin to penetrate difference
below the gold into the specimen, the time
Ar = (t - to)
(47)
is still short relative to t. Only the center of the crater has yet passed through the gold. This is, of course, the unclipped part of the crater and also where, according to the previous section, the intensity of a focussed beam approximates Gaussian behavior. That is to say:
dp
n
K@/t)=
d
ln
(l,g)
=
1,
tO
p%y
Thus Ri(t)
K,,
(tr
-
t + to) = Ki j$(i)
d?,
(49)
fo Differentiating with respect to t then yields a solution for Ci(t) in terms of the measured ratio Ri(t) and the constants Ki, KAU and r. The normalizing parameter r can however be estimated from the behavior of the gold peak near to as explained in Appendix B. The final result has the form
(50)
HAllJt @HA&W-
(aA,/dt)
1’ to
(51)
where At f (t - to), and the derivatives (dHA,/dt) and (&*u/dt) refer to the time rate of change of the height of the gold peak immediately before and after to respectively. Processing by this formula can be applied in turn to each elemental component of
D. W. HoffmanlCratering
49
analysis for depth profilinx
the sample giving the depth profile Ci(t) as seen at the bottom of the crater in units of (Ki/KAU). The normalizing constant has only to be determined once. Moreover, if r c 1, one has the simplified approximation Ki
H%u
q(t)
t0 f-G
z t0 z,
(52)
predicting that R&t) will be a linear function of t whenever Ci(t) in the specimen is constant. Whenever clipping is moderate (8, - 0.4) as in the data of the preceding section, the above processing should be valid until t exceeds (tO/gc), a point that may often be discernible in the behavior of the data. Clearly then, for a given clipping factor, deeper profiling into the sample requires a thicker layer of gold in order to increase r. and therefore the limit (to/g,). The method of depth profiling proposed above is not to be confused with ref. [3] where the focus of attention was on the possible detection of diffuse interface under a layer of gold on chromium. We believe the approach given here can provide a general method for depth profiling by ion scattering spectrometry that solves the cratering problem and circumvents the problems of clipping and beam intensity transients.
Acknowledgments The author is grateful for the time and effort freely given by numerous colleagues whose advice and encouragement have contributed substantially to this work. These persons include members of the Scientific Research Staff at the Ford Motor Company: Alan Brailsford, Robert Asaro, William Johnson, Veeravanallur Sundaram, Mary Ann Wheeler, Rao Nimmagadda, Walter Winterbottom, Edward Sickafus, and members of the ISS group at 3M Company: James McKinney, David Smith, and Robert Goff.
Appendix clipping
A. Correction
of the stabilized fractional
peak height for the effect of
Cancelling df above and below while adding and subtracting u(f/t), converts eq. (24) to the form
l-;; f’(t) =
$j-d($-) 0
jexp
(- “’ (l-u)
d(g),
(A.11
0
where the limits of integration are adjusted to eliminate vanishes. Eq. (A.l) further becomes
r;;f’(t) =f(r) -
unity with respect to
j fg du> 0
the region where (1 - U)
(A.21
50
D. W. Ho.ffman/Cratering analysis for depth profiling
after suitable integration by parts and use of eq. (12). The integral of eq. (A.2) is then broken up into steps from each time of observation to the next, up to time t. To facilitate the notation we index these times and the corresponding peak heights in order of observation by the integral subscripts m and n,
nc
r;,.r;l, =.f,,, -
Um,n
cJ
n=l
m=
fg du>
1 toN,
(A.3)
Llm,n-i
where the lower limit becomes zero for M = 1. Assuming that the time intervals are sufficiently short with respect to the variation off, we can extract a locally averaged value of the peak height from each of the sub-integrals: % r&l,
=I,
-
c
‘m,n 3 c/n_,
g du.
+r,> J
1*=1
(A.4)
‘m,n-1
Using the functionality between g and u given by eq. (26) the sub-integrals can now be individually evaluated yielding numerical coefficients Pm n defined as follows: p
m,n
-_=!m’n
exP (simi,2))2
(A.51
d”.
We note for clarity that the significance of the double index notation on u derives from its dependence on both t and ?, i.e., u(f/t) = u(fJtm> = urnan and that the meaningful range of integration in eq. (AS) stops at tn, = tm exp (-rz) unity. For convenience we therefore define P
m,n “m 3n.’c
to be independent (A.4) becomes
n>n
where Urn,,, becomes
(-4.6)
c’
of n when 11exceeds ne. With these definitions
=fm- n=l 5 w, +ft*_J (Pm
GfrLl
,I
-
pm,rz_J,
m=
and conventions
1 toN,
eq.
(A.7)
a set of linear simultaneous equations equivalent to eq. (24) and to be solved for values of f’given an observed set of the f '.By judicious juggling of indices and defining of fictitious coefficients P
m,-1
P m,m+l
= pm,0
64.8)
(= (2,
(A.9)
‘rn,m,
eq. (A.7) can be compacted
to the form
D. W. Hojyman/Cratering
analysis for depth profiling
m=l
toN,
51
(A.9)
where
pm )
(A.10) m,n = 5 Pmn+l >n-1 are partial areas to the left of the curve in fig. 5 [recall eq. (AS)] . A further simplification occurs by considering momentarily the special case of a pure, single-component specimen material for which .I” =.f= 1 at all times. Eq. (A.9) thus yields AP
(A.1 1)
for the normalizing text.
Appendix specimen
constant.
B. Experimental
Substitution
evaluation
back into eq. (A.9) leads to eq. (27) of the
of the normalizing
constant
for a gold coated
For times greater than to eq. (16) gives
f,,,(t) = &
7 K(@)dt^
03.1)
0
for the fractional height of the gold peak itself. Splitting the integral of eq. (18) at t0
rt = r
K(t/t)
dtA+ j K(@) dt*
0
enables eq. (B.l) to be rewritten
f,,(t)
(B.2)
IO
= 1- & j
K(f/t)
as:
di.
(B.3)
f0
This is advantageous since the kernel K(~/z) is unity for t^close to t. The integral can then be evaluated and the result inserted into eq. (42) for the unscaled peak height immediately after to :
h*Jt> =17’*“(t) [l-G]
(B.4)
D. W. Hoffman/Craten’ng
52
Subtracting
analysis for depth profiling
the identity
from eq. (B.4) and dividing through by (t - fn) gives:
h&(t) -(&j)
= ff*Jt> -
to
t-to
t~
~*JQ _- ~.&) l-r
(B.6)
.
Solving eq. (B.6) for r and taking the limit as (t - fo) goes to zero yields eq. (5 1) of the text. To evaluate (dH/dt) experimentally one takes the time rate of change of the gold peak immediately before to.
References [l] [2] [3] [4]
R.E. Goff and D.P. Smith, J. Vacuum Sci. Technol. 7 (1970) 72. R.E. Honig and W.L. Harrington, Thin Solid Films 19 (1973) 43. D.W. Hoffman and R. Nimmagadda, J. Vacuum Sci. Technol. 11 (1974) H. Shimizu, M. Ono and K. Nakayama, Surface Sci. 36 (1973) 817.
657.