Backscattering analysis of thin films on non-flat surfaces

Nuclear Instruments and Methods in Physics Research B73 (1993) 178-190 North-Holland

Backscattering

NOMB

Beam Interactions with Materials 8 Atoms

analysis of thin films on non-flat surfaces

Paul R. Berning and Andrus Niiler US Army Ballistic Research Laboratory, Aberdeen Proving Ground, MD, 21005, USA

Received 19 June 1992 and in revised form 10 August 1992

In order to allow for the analysis of contaminant layers on powder surfaces, methods of adapting ion beam analysis techniques for use on non-flat surfaces have been developed. In this work, particular attention is given to situations where the dimensions of the surface structures are much larger than the thickness of the film, but much smaller than the ion beam spot size. It is assumed that the surface material is conformal and evenly distributed across the surface. Two methods are discussed; both are designed to analyze backscattered ion energy spectra through the use of available simulation programs. These methods rely on knowledge of the distribution of surface-normal to beam-direction angles present on the surface shape to be studied. Examples of this and other relevant distribution functions corresponding to several types of surface shape will be shown. The methods described here are used to study oxide layers grown thermally on small ( = 130 km) spherical titanium powder particles. We also show examples of how to empirically determine the distribution of surface tilt angles present on a surface of unknown shape when the nature of the surface film is known.

1. Introduction This report is a continuation of our previous work (ref. [I]) in which two methods for analyzing the surfaces of spherical and cylindrical particles, without the use of a microbeam, were outlined. Here we will discuss ways of extending those two methods to other types of surface shape. The impetus behind this work is the desire to analyze surface contaminants and coatings on powder particles. While powder particles can have structure on all scales, the methods to be described here are largely meant to deal with cases where the surface structures are large in comparison to the thickness of the surface films to be studied. As seen in ref. [l], this regime is much easier to deal with than regimes where the surface structures and films are of comparable size. As a result, the formulations shown here are much simpler than those found in ref. [I], and can easily be applied to arbitrary surface shapes. It is also assumed that the surface films to be studied are conformal and that the surface material is evenly distributed, as might be expected of a contaminant. Ways of extending these methods to alternate regimes, i.e. cases involving structured films or cases where the films and surface structures are of comparable size, will also be discussed.

Correspondence

to: Dr. P.R. Berning, Director, U.S. Army Ballistic Research Laboratory, Attn.: SLCBR-TB-EP/P. Berning, Aberdeen Proving Ground, MD 21005-5066, USA.

0168-583X/93/$06.00

Traditionally, backscattering techniques have not been widely used to analyze non-flat or rough surfaces, mainly because of the complexity of simulating spectra arising from such surfaces. We have found that the level of complexity can be enormously reduced if one is willing to use a scattering angle very close to 180”. All of our methods assume that this is the case. In the first method, referred to as the “polyhedron approximation”, an energy spectrum is approximated by an appropriately weighted sum of flat surface spectrum simulations, where each flat surface simulation assumes a paiticular tilt angle present on the non-flat surface. The weight given to each simulation is determined by a function describing the distribution of surface normal tilt angles encountered by the ion beam on the surface in question. Examples of these characteristic distribution functions are derived for several different surface shapes. The second method, referred to as the “flat model approximation”, bypasses the need for large numbers of flat surface simulations. Instead, a hypothetical flat surface elemental depth profile is constructed which yields an almost identical spectrum to the one obtained for a particular surface profile and surface structure combination. In the large structure regime a formulation of this method that is simpler than the one found in ref. [I] can be used. This formulation, like the polyhedron approximation, also depends on the surface tilt distribution. In ref. [l] it was determined that the assumptions made in this approximation made it unsuitable for some combinations of surface and bulk

0 1993 - Elsevier Science Publishers B.V. All rights reserved

P.R. Beming, A. Niiler / Backscatteringanalysisof thinfilms on non-flatsurfaces

179

composition. Here we will examine the reasons for this and also demonstrate a way to minimize this problem. The version of the flat model that incorporates this feature will be referred to as the “corrected flat model approximation”. The above mentioned methods were tested by comparing their results with the results of a method of proven efficacy: the use of a microbeam small enough so that the region analyzed may be considered flat. Rutherford backscattering spectroscopy (RBS) analysis was performed using both microscopic and macroscopic beams on oxide layers grown on spherical Ti powder particles, and the radial oxygen profiles derived using the various methods are found to agree well. The flat surface spectrum simulation program RUMP [2] was used in all analysis. Several of these spherical Ti particle samples were heated under an oxygen atmosphere and then studied with the macrobeam methods to determine the temperature dependence of the growth rate of the TiO, layer. Finally, we show two examples of how, if the surface elemental profiles are known, the shape of a macrobeam derived backscattering spectrum from a surface of unknown shape can be used to determine the characteristic distribution of surface tilts for that surface.

entrance paths into the sample and perform a simulation for each [5,6]. This is complicated by the need to describe the myriad of possible exit paths for each entrance path, particularly since the descriptions are no longer simply related to one another. The nature of each exit path depends on both the shape of the surface and on the particular position of the entrance path on it, Writing a computer program that can handle arbitrary surface shapes, surface com~sitions, and scattering geometries would be a Herculean task. In keeping with the desire to reduce the complexity of the problem, we have introduced a restriction that greatly simplifies the task of simulating spectra from non-flat surfaces: a 180” scattering angle, It is understood that a true 180” detector angle is not possible, however it can be arbitrarily close to this if an annular detector is used. The benefits of this are difficult to overstate. The primary benefit is that, in this special case, the descriptions of the elemental distributions along an entrance path and its associated exit paths are once again simply related; they coincide. A less obvious secondary benefit of this simplification is that it is possible to use a currently available flat surface simulation program to generate simulations for each possible entrance path. This is the basic tack of the “polyhedron approximation”, which is described in the following section.

2. Principies of spectra simulation

2.2. Polyhedron approximntion

2.1. General

The “polyhedron approximation”, previously described in ref. [l], is appropriate in cases where the surface structures are much larger than the surface film thickness. It relies on a further simpli~ing assumption: if an incident ion strikes a portion of the surface that has a surface-normal to beam-direction angle of 0 and a uniform layer of thickness t on it, then the ion’s path length through the layer will be t ’ = tsec8. This is only true everywhere on a surface if the surface layer is very thin compared to the overall surface structure. In other cases each entrance/exit path would have to be described explicitly. It is also assumed that the surface material to be studied is evenly distributed across the surface - that is to say that a microbeam probing at right angles anywhere on the surface would find the same surface elemental distributions everywhere. The polyhedron approximation generates spectrum simulations for hypothetical surface profiles and shapes by adding together separate flat surface simulations. However, rather than explicitly describe all the possible entrance paths, a single surface profile is assumed and simulations that assume varying surface tilts are added. Each simulation is weighted according to the prevalence of its assumed tilt angle on the surface in question. In ref. [I] these weights were obtained by an explicit approxi-

Elemental depth profiles are often obtained from a scattered particle energy spectrum by comparing it with computer simulations that assume various possible sample configurations. The simulation process generally involves describing the elemental distributions along an incoming ion’s path (so that energy loss and scattering probabilities may be assessed), followed by similar ~haracte~zation of the many possible exit paths the scattered ions may take (so that further energy loss can be determined). In the case of a flat, laterally uniform surface it is only necessary to describe one entrance path and its associated exit paths (along which the elemental distributions are simply related to those seen along the entrance path). While this is relatively straightfo~ard, it is by no means trivial to write a computer program that can handle any flat surface configuration and perform the necessary calculations quickly. Nonetheless, several such programs are available (2-4). The problems of program speed and complexity are compounded when nonflat surfaces are considered. In part this is because there is no one ion entrance path representative of the entire sample. One approach for dealing with this is to consider a large number of

180

P.R. Berning, A. Niiler / Backscattering analysis of thin films on nomj7at surfaces

mation of a sphere or cylinder by an irregular polyhedron. Now we simply determine a function P(0) describing the distribution of tilt angles on a given surface and weight each simulation according to this function. As an example, we will show how P(0) can be derived in the case of a hemisphere. We will assume that the incident ion beam enters in the --2 direction, and that the sphere can be described in cylindrical coordinates (p, 8, z) by: p2 + zz = R2, where R is the radius of the sphere. If the ion beam intensity is uniform across its width, then the “probability” that an ion will strike a particular segment of surface will be proportional to the area of its projection onto the z = 0 plane. For the section of sphere having surface normal tilt angles between /3 and 0 + de, this projection is the ring found between radial distances p and p + dp. The area of this ring is 2apdp, so that the “probability” of striking this region is: P(P)

dp =

$dp,

where the term in the denominator is included to normalize the “probability density function” P(p) to one. This function can be converted to a function in terms of 6’ by applying the relation:

p(e)

=qp)$.

In the case of a hemisphere cos e, so that:

p = R

sin 0 and dp/d8

that, if the actual profile taken normal to the surface is given by F(d), then the “apparent” profile taken in the beam direction on a segment of surface tilted by an angle 8 will be given by F(dsece). For future convenience, we will refer to the quantity (Y= sece as an “apparent thickness factor”, because a uniform flat layer of thickness t will appear to be tsece thick when tilted at an angle 0 to the beam. It is useful to calculate the “distribution of apparent thickness factors” P(a) which is related to P(e), for surface shapes to be studied. This can be done with the following relation:

(4) In the case of a hemisphere P(a)

=

this results in:

2/a3.

(5)

The shape of this function, here with its long “tail”, gives useful insight into the shapes of spectra coming from the surface it describes. Finding an (approximately) equivalent flat surface profile for a surface with actual atomic density profiles given by F,(d) (where i denotes one of the elements present and d is depth as measured at right angles to any point on the surface) and a distribution of apparent thicknesses given by P(a) is then simply a matter of performing a weighted average over all (Y:

=

R

p(e)

= 2 sin e cos 8.

(3)

Further examples of “tilt angle distributions” characteristic of other types of surface shape will be described in a later section.

As an example, we will describe the use of eq. (6) for determining the “equivalent flat profile” R(r) of a uniform film consisting of a substance “1” on a hemisphere of substance “2”. In this case:

2.3. Flat model

F,(d)

=

:

F,(d)

=

;

The flat model takes advantage of the fact that one can obtain a simple, flat surface configuration with elemental depth profiles such that a backscattered ion energy spectrum from it mimics the spectrum from a particular non-flat surface configuration. With this approach, only a single conventional flat surface simulation calculation is needed to obtain a good approximation of the desired result. As shown in ref. [l], an approximate way of constructing such an “equivalent flat surface profile” was to calculate the average concentrations as a function of depth (where “depth” is measured in the direction of the incoming ion beam). This was done via a detailed geometric analysis involving spherical and cylindrical shapes. In regimes where surface structures are much larger than the film thicknesses the process is much simplified. In these cases it can always be assumed

for d < t for d < t ford>t,

2

and P(Q) = 2/a3. results in:

RI(l) =

INi N

i

(7)

ford>t,

It can be seen that eq. (6) then for 1st

f

0

l 1

2

for I> t,

and

(0 ““‘=j”(l-(ii’)

for 15 t forI>t,

(10)

which is identical to the solution found in ref. [l] using a more complex method. As seen there, the atomic

P.R. Bern&g, A. Niiler / Backscattering analysis of thin j3n.s on non-flat surfaces

fractions f#)

181

and area1 density A(1) are then given by: Olllt

1

L

1

fdl) =

1+$

1

f2(1)

=

12t,

2

(11)

5-l

i

i

1 -f1(0

and N,l A(l)=

Oslst

N,t+N,(I-t)+(N,-N,)

(12)

(b)

1> t. 2.4. Correcting deficiencies in the flat model approximation

As pointed out in ref. [l], the flat model approximation is somewhat naive and, as a result, breaks down in certain cases. While the simple spatial averaging procedure represents the total amount of surface material well, it does not always result in a distribution of elements representative of a truly equivalent flat surface profile. This is the case when the minimum and maximum values of the local rates of energy loss (dE/dx) in the target differ by more that a factor of two. In addition, the procedure requires assumptions concerning the densities of all the various compositions present in a target. It should be stressed that reliance of this model on density information is in no way dictated by the ion-solid interaction process, rather it is introduced as part of the effort to keep the model as conceptually simple and general as possible. There is a more appropriate way of constructing an equivalent flat surface profile: find the average composition as a function of ion energy, not depth. Another way of saying this is that any sublayer in the equivalent flat surface must have the same average composition as that found between two surfaces of constant ion energy in the non-flat surface case. Here we define a “surface of constant ion energy” as all points in the solid where incident ions have the same energy. Currently the method utilizes “surfaces of constant depth”, which correspond to “surfaces of constant ion energy” only if the rates of energy loss dE/dx do not vary with composition in the sample. Fortunately, the performance of the approximation is not degraded appreciably as long as variations in the value of dE/dx due to composition variations do not exceed a factor of 2. The differences between the simple “flat model approximation” described previously and the “corrected flat model approximation” described here are illustrated in fig. 1, which contains schematic diagrams of cross-sections of a round surface with a uniform

Fig. 1. Schematic diagrams of three surfaces that yield identical backscattered ion spectra (a) a uniform layer on a round surface, (b) a flat surface where the apparent film thickness variations are replaced by actual thickness variations, and Cc) a flat surface in which the lateral nonuniformity has been “averaged” away in some manner. The solid lines represent surfaces of constant depth and the dashed lines are surfaces of constant ion energy (assuming that surface dE/dx = 2~ bulk d E/dx).

surface layer and two “equivalent” surface configurations. The first “equivalent” surface is flat and replaces the apparent thickness variations seen on the round surface with equivalent actual film thickness variations. In the second “equivalent” surface configuration all lateral nonuniformity is averaged away, leaving what is referred to as an equivalent flat surface profile here. The solid lines shown in each diagram correspond to “surfaces of constant depth”. The dashed lines correspond to “surfaces of constant ion energy”, where it is assumed that the bulk values of d E/dx are about half that for the surface material. In the simple flat model approximation the average composition of the layer depicted in the equivalent flat profile case is the same

P.R Bermng, A. Niiler / Backscattering analysis of thin films on non-flat surfaces

182

as that between the solid lines in the other cases. In the “corrected” flat model it would have the same average composition as between the dashed lines seen in the other cases. Clearly the less appropriate “depth based” flat model underestimates the number of ‘bulk’ atoms that each layer in the equivalent profile should have. If the surface material’s dE/dx values are less than that of the bulk, then the amount of bulk material assigned to each layer will be overestimated. A simple, a priori method for ‘correcting’ this error presents itself, at least in the case of a uniform layer on a non-flat surface. Since the ratio of surface material dE/dx to bulk material dE/dx is about 2 in this example, then the amount of bulk material contained between the dashed lines will be about twice that contained between the solid lines. The underestimation or overestimation of the amount of bulk material to be assigned to each layer can then be nearly eliminated if the bulk material’s atomic density N2 is (artificially) replaced by N2*: (dE\ N~IN,XAverage/&’

L-J =Ni \ dx /z

[ I, z

0.0 -OS

0

0.5 I

(MeV) 1.0 I

1.5 I

2.0

on Li Sphere:

ltJ0

2bo

460

560

Channel

Fig. 2. Simulations of a uniform 50 nm layer of OS on a Li sphere. As the values of dE/dx differ by a factor of 10 in these substances, the “uncorrected” flat model approximation fails completely.

(13)

where li denotes the stopping cross-section of material i, defined by:

(14)

= Nier.

Energy 250

In the example shown in fig. 1: N; = ZN,. It is this value that would be inserted in the equivalent flat profile equations (11) and (12). The “corrected” equivalent flat profile equations for a uniform layer on a sphere then become:

Significantly, this “corrected” solution no longer depends on atomic density information, only on the average ratio of stopping cross-sections [e/e,] and the area1 density of the surface film N,t (the quantity l/t can be considered a “dummy” variable). Fig. 2 illustrates the benefits of this approach: it compares spectra simulated using the polyhedron approximation, the “uncorrected” flat model approximation, and the “corrected” flat0 model approximation. The example shown is for 500 A of OS on a Li sphere, a nearly worst-case scenario for the “uncorrected” flat model as (dE/dx)o, = 10.3 (dE/dx), for 2 MeV 4He [7]. As can be seen, the “corrected” flat model simulation shows nearly perfect agreement with the polyhedron approximation simulation. The Li signal is too small to be seen on the same scale as the OS yield.

3. Distribution functions for various surface shapes

and

‘4; 0 N,t

=

0 f

NIt(I+m

(;-1)+(1-m

I ; > 1.

The following is a list of surface tilt distribution functions (P(0)), and associated distributions of apparent thickness factor (P((Y)), corresponding to several simple types of shape. The shapes considered generally fall into two classes: surfaces of translation (cylinders, gratings, etc.) and surfaces of rotation (hemispheres, spheroids, etc.). In addition to thin films on non-flat surfaces, we have also considered the related topic of structured films on flat surfaces. This would include situations where an initially uniform film is etched to form microstructures [5] or situations where “islands” spontaneously form during thin film deposition 181. In these cases the actual thickness of the structured films may vary, as opposed to the apparent thickness variation

P.R. Berning, A. Niiler / Backscattering

3.3. “Gaussoids”

seen in the case of evenly distributed films on non-flat surfaces. Spectra arising from these types of surfaces can be simulated by a weighted average of separate flat surface simulated spectra, however it is the film thickness (not the tilt angle) that changes from one simulation to the next (this is the same system needed when film thicknesses and surface structures are of comparable size). As it would be of use in these cases, we have also determined the “distributions of film thickness” (P(z)) given that the films (not the original surface) have the forms described below.

This surface is generated when a Gaussian is revolved about its axis. The maximum height is zmax, and an arbitrary minimum cutoff height zmin is assumed. This surface is described by the function: z = e-p2/20z, p = 0 -+ 2a2 ln( z,&z,,) . The disZ trycution of heights in an island shaped like this is: P(z)

e=o+-.

(sin20 + C2 c0s2ej2 ’

p(r)=&,

(x, y, z), this can be dex=0-R,

y=O+q

1

z

Tr e=o-+-,

[ sin20 + C2 c0s2eI ’

(18)

2

3/z

z=O-+CR.

(d

1 - (~/ITz,,)~

{l - (r/~z,,)~

~3zIll, e = 0 + tan-‘(

[

a2+(;2-1)

I

’

tan28 tan20

’

(20)

Irz,,/7),

cr=l+m,

(25)

This is the bell-shaped surface that is generated when a section of a sinusoidal curve is rotated about a vertical (z) axis: z = +z,,[l + cos(27rp/7)], p = 0 + r/2. cos-1

4%

(19) P(z)

=

to note that P(z) does not depend on

p(e)=c2 cose

3.2 “Sinusoids”

P(a)

(23)

=O+CR.

P(a)=

87

Inax’

’

In Cartesian coordinates scribed by: x2+z2/C2=R2,

(17)

2

(Y=l+m,

(“z+(C2-1))2’

= -sec2e

z=zin+z

~~(Zm~/~min)

(24)

2C2a

i-ye)

Z

P

2C2 sin 0 cos 9

P(,)=

1

3.4. Right elliptical cylinders

A spheroid is the surface generated when an ellipse is rotated on one of its axes. In cylindrical coordinates, they are described by: p + z2/C2 = R2, p = 0 + R. The relevant distributions are given by: =

=

It is interesting “a”.

3. I. Spheroids

P(e)

183

analysis of thin j%n.s on non-flat surfaces

Z

1

= 2 RC

(26)

3.5. Sinusoidal grating

This type of surface is referred to as a “surface with sinusoidal relief’ in ref. [9], and is also considered in ref. [6]. Here we describe it with the equation: z = iz,,(l + COS[2PX/TI). P(e) =

1

set e c0s2e - sin20 ’

7T {(7r~~.Jr)~ e=O-+tan-’

c*

z=O+CR.

\/w’

~~lll,

( 19 -

7

(27)

V1- (~/~&mx)‘(a2- 1))

cos-1 X

1 - ( 7/Tz,,)2( a = 1+

P(z)

cos-i(2z/z,,

Inax

’

(21)

\i 1+ (lTz,,/r)2,

= + ?r zmax 1-(22/z,,

z=O+z

a2 - 1)

- 1) - 1)2 ’

(22)

(29)

P.R. Berning, A. Niiler / Backscattering analysis of thin films on non-flat surfaces

184 3.6. Gaussian

“cylinder”

(4 5 I

This is described by: z = z,,, with a cutoff at zmin.

e-x2/2u2,

Spheroid ‘Sinusoid’ ---_-__ . ..li!l.i~~.ica!.c~~.~~~~ ____ Sinusoidal Grating

once again 4

t

t

(30) 3.7. Spheres

in a face-centered

cubic,

close-packed

array

0 14

1.2

1

We have considered the case of a cluster of spheres arranged into an fee pattern to see how the presence of a partially obscured layer of spheres below the topmost layer of spheres alters the overall nature of the surface. This may prove useful as we eventually wish to study aggregates of powder particles. The distribution of apparent thickness factors in this case is given by:

1.8

1.6

Apparent

Thickness

Energy

1.5

1.0

2.0

I

F

/

2

s a

P(a)

B 1 * B

=

2.2

(MeV)

I

/

0.5

2

Factor a

150

f

-

;I ;:

100

-

50

-

b z

i 0

(31) For future : Ireference, we note that this function can be approximated by the distribution function for a spheroid (eq. (18)) with C = 0.8. Examples of P((Y)‘s are shown in fig. 3a, for four different shapes: spheroid, “sinusoid”, elliptical cylinder, and sinusoidal grating. The parameters describing the specific shapes of each have been arbitrarily chosen so that each has equal maximum height and full-widthat-half-maximum. For the spheroid and elliptical cylinder this means that R = l/& and C = 6. The “sinusoid” and sinusoidal grating have z,,, = 1 and r = 2. To illustrate the differences in simulations for each of these shapes, fig. 3b depicts simulations of 30 nm of Au on a Si spheroid and a Si elliptical cylinder with the above parameters, and fig. 3c depicts simulations of 100 nm of Au on a Si “sinusoid” and a Si sinusoidal grating with the above parameters.

4. Experiment 4.1. Introduction In order to test our methods, we used them (in conjunction with 2 MeV 4He RBS) to analyze a powder particle surface with both a known shape and a

I 0

I

I 300

200

100

I 400

500

Channel Energy ,,,,,,r

0;5 -100 - - 100 - -100

nm nm nm

(Mev) 1;o

1;5

2.9

Au on Si ‘Sinusoid Au on Si Sinusoidal Crating Au on Si Flat Surface

I

1I ’ I

I_ ,~, 11 0

100

200

300

400

Channel Fig. 3. (a) Distributions of apparent thicknesses for several surfaces with shapes described in the text. (b) Simulations of 30 nm of Au on a Si spheroid, elliptical cylinder, and flat surface. (c) Simulations of 100 nm of Au on a “sinusoid”, sinusoidal grating, and flat surface.

known elemental profile. We chose to study oxide layers thermally grown on spherical Ti particles. The powder, supplied by Nuclear Metals, Inc., was nomi-

185

P.R. Beming, A. Niiler / ~ackscatter~~g analyst of thinfilms on non-flatsurfaces

In the spirit of the “corrected” flat model (see eq. [13]), we also replaced actual density values with: ETi l/302/3 NT~_,o,

I

0.1

/

/

I

/

I

0.2

0.3

0.4

0.6

0.6

“‘0 Atomic

&2/J

~

E T11-,0,

(for 2 MeV 4He). (32)

m Published values Linear fit __-___Ouadratic fit ____-.-.-Cubic fit __._._. . . _. . .._.. ____...._____..... D ‘Corrected’ vsiuea ‘Corrected’ cubic fit

0.0

=INTI

I

A cubic fit to these ?orrected” values is also shown in fig. 4. In this case, where the “uncorrected” method already works nearly perfectly, this system only improved agreement with the polyhedron approximation slightly.

0.7

Fraction

Fig. 4. Linear, quadratic, and cubic polynomial fits to the published values of titanium oxide densities. This type of function is used in the “uncorrected” flat model method; ail of the above work equally well. Also included is a cubic fit to artificial values used in the “corrected” method.

nally 99.5% pure. The powder was sieved to -80, f 200 mesh, corresponding roughly to diameters in the range 130 & 50 pm, so that all clearly fall in the “large structure” regime. Oxide layers of varying thicknesses were grown on the surfaces of these particles by heating them in a tube furnace under flowing oxygen. In one test case the oxygen (radial) depth profile was unambiguously determined through the use of the microbeam facility at the State University of New York at Albany. The results of this analysis were then compared to the results based on the use of the Ballistic Research Laboratory’s conventional macroscopic beam facility and the two methods of non-flat surface spectra simulation outlined above. These “macrobeam” methods were then used to analyze other spherical particles that had been heated at various temperatures and for various amounts of time. As stated previously, the “uncorrected” flat model appro~mation requires assumptions concerning the atomic densities of all compositions present. In the case of thermally grown oxides layers the compositions can range anywhere from pure Ti to pure TiOz, and so a somewhat arbitrary “rule” governing the densities of all compositions in this range must be assumed. Fig. 4 shows linear, quadratic and cubic polynomial functions of oxygen atomic fraction that represent fits to four published density values [lo]. Flat model simulations utilizing any of these polynomial “rules” were found to be effectively identical to simulations generated using the polyhedron approximation. This method is very effective because, in this case, the maximum and minimum values of dE/dx in this range of compositions differ by less than 15%.

4.2. Sample preparation and experimental setup The oxidized spherical particles studied here were prepared by placing a shallow layer of them in an alumina boat and inserting this into a quartz tube furnace. The temperature of the samples was monitored with a thermocouple placed in contact with the boat and typically fluctuated + 3°C during a run. Samples were heated to temperatures ranging from 400°C to 6Oo”C, for periods lasting from 2 to 6 h. In most runs the powder samples were mechanically agitated periodically in order to assure uniformity across the sample. In some early cases the oxygen flow was not monitored, in later cases the 0, flow rate was maintained at about 190 cm3/min. The oxidized Ti particles studied with the microbeam were “mounted” by pressing them into the surface of a pellet consisting of a conventional, randomly shaped Ti powder. As this was later found to cause damage to the surface of heavily oxidized Ti spheres, samples later studied with the macrobeam were “mounted” by sprinkling them onto a flat surface coated with a thick layer of graphite particles suspended in isopropanol (“carbon paint”). Care was taken to apply the spheres just as the carbon. paint appeared to dry, so as to avoid embedding them in the carbon layer (more will be said on this later). The microbeam facility at the State University of New York at Albany’s Accelerator Laboratory has been described in detail elsewhere [11,12]. We used a 2 MeV 4He+ microbeam with dimensions on the order of 1-2 p,rn and beam currents on the order of 100 pA. The resolution of the detection system was 16 keV FWHM. While no evidence of beam damage was seen, the beam was intentionally scanned over a 12 pm wide area on each spherical particle in order to avoid effects due to radiation damage or beam heating. Backscattered ion energy spectra were obtained for at least three different spherical particles on each sample, in order to verify that the oxide layer was uniform from one sphere to the next. Analysis with a macroscopic beam was performed at the U.S. Army Ballistic Research J_,aboratory’s 2.5 MV

186

P.R. Beming, A. Niiler / Backscattering analysis of thin films on non-flat surfaces

Van de Graaff facility. In order to obtain a scattering angle as close to 180” as possible, an annular detector was used. The detector was at an angle of 178.9 + O.l”, and had an acceptance of 0.8” and a solid angle of 1.4 msr. The resolution of this detection system was typically between 20 and 30 keV FWHM. The beam diameter here was approximately 2 mm, and as such over a hundred spheres could be analyzed simultaneously. In all cases, the energy resolution and calibration were carefully determined using a variety of standards. 4.3. Experimental

Energy (MeV)

(a) 0.2

0.4 /

,

80

0.8 I

0.8 I

1.0 t

1.2 f

1.6

1.4 /

0 Sum of Batch d 5 bficrobeam Data .p -Fit to Batch # Mtcrobsam Data

4

results

The one set of samples studied with both a microbeam and a macrobeam (Batch #5) consisted of spherical Ti particles heated for 5.0 hours at 503°C in 1 atm of 0,. In this case the oxygen flow rate was not monitored, and may have been insufficient to expel all of the air in the tube furnace. Fig. 5 contains the results of both the microbeam and macrobeam anaIyses. Fig. Sa contains the RBS spectrum taken using the microbeam. It is actually the sum of three spectra taken from three separate spheres, but as these spectra appeared identical it was decided to add them together in order to improve the statistics. The conventional flat surface RBS spectrum simulation program RUMP [2] was used to determine the radial oxygen/ titanium depth profile, and the resulting simulation is also shown in fig. 5a. Fig. 5b contains the RBS spectrum taken with the macroscopic beam. It also includes simulations generated using the polyhedron approximation and flat model approximation methods described above, where the radial profile assumed is the one found via microbeam analysis. As can be seen, there is excellent agreement between the two predictions and the data. Fig. 5c contains the radial oxygen profile obtained from the fit seen in fig. 5a and also the associated “equivalent” flat surface profile used in the flat model prediction seen in fig. 5b. Two other samples were analyzed with the microbeam, but in these cases excess particles were not saved for later analysis with a macrobeam. The sum of microbeam spectra from three different spheres in Batch #6 (5.0 h at 406°C) can be found in fig. 6a, along with a fitted simulation. The radial depth profile obtained from this fit was then used to predict the shape of a macrobeam spectrum using both the methods outlined above. These predictions are compared to macrobeam data taken from a similarly prepared sample (Batch #9: 5.0 h at 405°C) in fig. 6b. Once again, the agreement is excellent. Fig. 6c contains the derived radial oxygen profile and the “equivalent” flat surface profile. When the microbeam derived spectra from several different spheres in Batch #7 (5.0 h at 598°C) were examined, they were found to differ radically from one

d0

4Ao

Channel

Energy (We,

0.8

Oi6

/

(MeV) 1.0

/

1.2

1.4

/

0 0

I

150

I

I

200

250

I

300

I

350

Channel

0

5000

10000

’ 16000

Areel Density (10’6/cm2)

Fig. 5. (a) Microbeam derived 2 MeV 4He RBS spectrum for an oxide layer on spherical Ti particles in Batch #5, and a conventional simulation used to fit it. (b) Macrobeam derived spectrum for the same batch, with nearly identical spectra simulated with the two methods described in the text, each assuming the radial profile obtained with the microbeam. (c) The oxygen radial depth profile found and its “equivalent” flat surface profile.

P.R. Berning, A. Niiler / Backscattering analysis of thin films on non-flat surfaces

Energy .2

0.4

0.6

I

I

I 0

--Fit L

Sum

(I&V)

0.8 I

1.0 I

1.2 I

1.4 I

1.6

1

of Batch 6 Uicrobsam Data to Batch # I! Microbeam Data

-,

,

0,

200

100

,\I

, 300

400

0

600

500

I

150

Channel

Energy (b)

60

5.

0.6 I

c

0 Batch IE!&b Batch

0.6 I

1.2 I

9 Macrobeam Data similar to Batch #6) 6 ,fiierobeam fit + 01 h&-on Approx. 6 ,fiicrabeam fit + Fia T Yodel Appror.

350

another. Only one could be fitted with a profile having the expected thick TiO, layer and long diffusion “tail”. Later examination under a scanning electron microscope revealed the reason for this: an approximately 1 pm thick layer of oxide had cracked and delaminated from the spheres. This probably occurred when the spheres were pressed into the surface of the Ti powder pellet, and so we abandoned this method of mounting. Several more batches of oxidized Ti spheres were prepared and analyzed using a macroscopic beam. Fig. 7 contains spectra from Batches #8, #ll, and #12, which were heated in an 0, atmosphere for 5.0 h at 498°C 502”C, and 500°C respectively. Fig. 8 contains the radial profiles derived from the fitted simulations also seen in fig. 7, as well as the profile for the

0 0 250

/

300

1.4 I

3

200

I

250

Fig. 7. Macrobeam derived RBS spectra for several batches of spherical Ti particles, all heated in 0, for 5 h at about 5OO”C, and simulations fit to them using the methods described in the text.

(MeV) 1.0 I

200

Channel

10 -

150

187

300

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Channel

0.7 0.6 .-z

Batch Batch

...latchXU.LsP.?... Batch#ll (5Oo’C)

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5L

0

4 /

0.0 0

1000

#6 (603’0 C6 (4SS.C)

0.2

-....__ .-..-.-......._....___._____.__ _ e

t

0.1

2000

3000

Areal Density (10’6/cm2)

Fig. 6. (a) Microbeam spectrum for oxide layer on spherical Ti particles in Batch #6 and fitted conventional simulation. (b) Macrobeam spectrum for the similarly prepared Batch #9 and simulations based on the Batch #6 radial profile found with the microbeam. (c) The oxygen radial profile used in all simulations and its “equivalent” flat surface profile.

t

0.0

’ 0

I

2000

4000

6000

I

6000

Areal Density (10’61cm2)

Fig. 8. Radial oxygen depth profiles derived from the simulated spectra seen in fig. 7, along with the profile found for the similarly prepared Batch #5 (all were heated for 5 h in 0,).

188

P.R. Berning, A. Niiler / Backscattering analysis of thin films on non-flat surfaces

similarly prepared Batch #.5. These profiles differ primarily in the thickness of the surface TiO, layer; the diffusion “tails” are all very similar. The markedly thin TiO, layer seen in the case of Batch #5 is what suggests that the oxygen flow was insufficient in this instance. The TiO, layers in the other cases were (900 + 100) X 10” at./cm’ (= 93 k 10 nm) thick. The differences in the spectra for these cases, while small, are somewhat more pronounced than they would be if the surfaces were flat. This is because, on non-flat surfaces, the composition of the surface-most layers affects the shape of the entire spectrum, unlike a flat surface spectrum where only the regions near leading edges are affected. If the rate of TiO, film growth is limited by a diffusion process, then the observed film thicknesses can be used to derive a diffusion coefficient. Here we assume that, if the TiO, film thickness t’ is seen after a certain period of time, then the diffusion coefficient D is given by:

The temperature dependence of D often takes the form of the Arrhenius relation:

where T is the absolute temperature, k, is Boltzmann’s constant, DO is the frequency factor, and Q is the activation energy of the limiting process. A plot of In D versus l/T can be found in fig. 9, along with a fit from which values of Q = 1.24( kO.10) eV and D, = 3.4( + 11, - 2.6) X lo-’ cm’/s are derived. The density of TiO, was assumed to be 4.26 g/cm3. Published

0 0 10‘”

=

16”

=

macrobeam microbeam

analysis analysis

zt “E -z 0

0 = 1.24 Do= 16”

1

I

1.101.16

e’J

3.4x10”cmz/s

I 1.20

1

1

1.25

1.30

1000/T

I 1.35

II

1 1.40

1.45

1.50

(K-l)

Fig. 9. The temperature dependence of the diffusion coefficient for oxygen diffusion through TiO,, based on TiO, layer thicknesses found on Ti spheres heated at various temperatures. Both microbeam and macrobeam derived data are included.

Energy (MeV) 0.8

l

Batch

1.2

1.0

I

I

I

1.4

I

#5 Spheres Partially Embedded in Carbon #S Radial Profile + Sphere. cutoff at 77’ #S Radial Profile + Sphere

0

“R Channel Fig. 10. An RBS spectrum from spherical Ti particles partially embedded in a layer of carbon paint. Also included are a simulation that fit a spectrum from fully exposed spheres and a simulation that assumes that the surface’s distribution of “tilts” is the same as that for a sphere except that only surface tilts up to 77” are included.

values of the diffusion coefficient of 0 in rutile TiO, vary over several orders of magnitude, the values found here are larger than most (a survey can be found in ref. [13]). The activation energy found here is markedly smaller than published values for oxygen diffusion in TiO,, which are typically about 2.4 eV. This suggests that the bulk diffusion may be short-circuited here, possibly by grain boundary diffusion. 4.4. Determining

P(8)

empirically

Backscattered ion energy spectra from non-flat surfaces are affected by both the surface elemental depth profile and the shape of the surface. In the regime where surface structures are much larger than the surface film thickness all relevant aspects of the shape are incorporated in the distribution of tilt angles P(O). It then follows that, if the surface elemental profile is known prior to analysis with a macrobeam, then P(O) can be determined empirically from the shape of the resultant spectrum. Two simple surfaces of unknown character were looked at here, both are variants involving Ti spheres studied previously. The first case consisted of particles from Batch #5 that were applied to a layer of carbon paint that was too wet and as a result the particles became partially embedded in the carbon layer. Fig. 10 shows that the spectrum from the embedded spheres differs noticeably from the spectrum from fully exposed spheres. As it is the edges of the spheres that are being obscured, we tried to fit the spectrum by

P.R. Beming, A. Niiler / Backscattering analysis of thin films on non-flat surfaces

0.4 I

Energy

(MeV)

0.6 I

0.6 I

1.0 I

IA0

2.;0

5. Discussion 1.2 I

1.4

“,

100

A0

360

360

Channel

Fig. 11. An RBS spectrum from a loosely bound cluster of spherical Ti particles. Also included is simulation that assumes that the overall surface has the same distribution of surface tilts as an oblate spheroid, as an approximation to the distribution expected for a close-packed array of spheres. The data are much better fit by a simulation assuming an individual sphere’s distribution of surface tilts.

that P(O) was that for a sphere, but with a cutoff at some angle 0,,, < 90”:

assuming

2 sin e cos 0 P(e) =

sin*&

’

e=0+

189

ecut

(35)

As seen in fig. 10, the observed spectrum is well fit by a simulation that assumes the above P(e), with ecut = 77”. The second surface of unknown character was for a cluster of spheres. It was found that, during the oxidation process, several of the topmost layers of spheres would sometimes fuse together into clumps. The bonding between spheres in these clumps was extremely weak and probably was due to interdiffusion of the surface oxide layers at points of contact between the spheres. As pointed out in section 3.7, an ordered arrangement of spheres may have a distribution of surface tilts similar to an oblate spheroid with C = 0.8. A computer program written to simulate randomly overlapping spheres indicated that P(e) would be the same as for an individual sphere. Fig. 11 reveals that this is the case for the clusters studied: the spectrum from one of these clusters is fit very well if P(e) for a fully exposed, individual sphere is assumed. This was found to be the case for several of these clusters, from different batches. Remarkably, tilting the cluster to various angles about an axis perpendicular to the beam-detector plane (in the range f 10’) had very little effect on the shape of the spectrum obtained, and thus on the apparent P(e).

We have continued to develop the two techniques for adapting backscattering analysis to non-flat surfaces which were first introduced in ref. [l]. Here we have extended them for use on surfaces of arbitrary shape, in cases where the surface structures are much larger than the film thicknesses to be studied and a function P(e) describing the distribution of surface tilts is known. Several examples of this function have been given. Both methods have been tested against a more conventional analysis method based on the use of a microbeam, in an RBS study of oxide layers on spherical Ti particles. These methods can be used in conjunction with any backscattering technique for which a flat surface spectra simulation program exists. The “polyhedron approximation” method described here is largely the same as in ref. [l], except that here it makes a more direct use of the distribution of surface tilts P(e). The “flat model approximation” differs from the formulation found in ref. [l] in that the method of generating an equivalent flat surface profile is less involved, and it also now depends on P(e). The reasons behind its limitations have now been explained, and methods for widening the range of situations for which it is useful have been demonstrated. As the results obtained through the use of the two techniques are largely identical, the choice of which to use depends on the speed and level of convenience one desires. When using a highly flexible and high speed simulation program such as RUMP [2], the time it takes to simulate a non-flat surface spectrum with each method is about the same: on the order of 10 min on an 80286 based IBM compatible PC. In the case of the polyhedron approximation this time is spent generating and adding a large number of flat surface simulations, which is quickly and easily done with RUMP. In the case of the flat model the time is spent performing the necessary numerical integration in order to find the equivalent flat surface profiles (see eq.(6)) and then the one flat surface simulation based on these. The flat model method would be a better choice if a slower, less flexible simulation program were to be used, as it is less affected by the speed and convenience of the program chosen. We have also shown that one can use these methods to determine the distribution of surface tilts P(e) empirically if the nature of the surface elemental profile is known, say through the use of a microbeam. Thus one can not only analyze surface profiles on non-flat surfaces of known shape, one can derive certain aspects of the shape of a surface if the surface profile is known. It has been pointed out previously that backscattering can be used to investigate the roughness of surfaces [14-161, however this was for cases where the surface was uncontaminated and where the surface structures

190

P.R. Berning, A. Niiler / Backscattering analysis of thin films on non-flat surfaces

were small compared to the range of the backscattering technique used. Our methods take advantage of the presence of a conformal thin surface layer and is only suitable for structures much larger than the film thickness. The two methods outlined here can be extended so as to allow study of films on structures of comparable size to the film thickness, however each combination of surface profile and structure would require special treatment. This is because the elemental profile and the surface shape can not be treated as separate issues in these cases. A simulation can be built up with many flat surface simulations as before, but here the “apparent” profiles seen by beam particles all along the surface would all have to be explicitly determined and described (similar to the system in ref. [5]). Once again, the use of a nearly 180” detector angle would allow one to use a conventional flat surface simulation program to generate the spectrum associated with each path into the surface.

Acknowledgements

We would like to thank Wayne Skala at SUNY Albany and Laszlo Kecskes at the Ballistic Research Laboratory for their technical assistance. This work was performed under a grant from the National Research Council.

References [l] P.R. Berning and A. Niiler, Nucl. Instr. and Meth. B63 (1992) 434. [2] L.R. Doolittle, Nucl. Instr. and Meth. B9 (1985) 344. [3] A. Niiler and L.J. Kecskes, Nucl. Instr. and Meth. B40/41 (1989) 838. [4] Gy. Vizkelethy, Nucl. Instr. and Meth. B45 (1990) 1. [5] X.S. Guo, W.A. Lanford, and K.P. Rodbell, Nucl. Instr. and Meth. B45 (1990) 157. [6] R.D. Edge, Nucl. Instr. and Meth. B33 (1988) 582. [7] J.F. Ziegler, J.P. Biersack, and U. Littmark, The Stopping and Range of Ions in Solids (Pergamon, New York, 1985). [B] L.C. Feldman and M. Zinke-Allmang, J. Vat. Sci. Technot. A8 (1990) 3033. [9] V.S. Shorin and A.N. Sosnin, Nucl. Instr. and Meth. B53 (1991) 199. [lo] R.C. Weast (ed.), CRC Handbook of Chemistry and Physics (CRC Press, Cleveland, 1978) pp. B171-2. [ll] W.G. Morris, H. Bakhru, and A.W. Haberl, Nucl. Instr. and Meth. BlO/ll (1985) 697. [12] R.E. Benenson, P. Berning, and H. Bakhru, Nucl. Instr. and Meth. B45 (1990) 519. [13] Z. Liu and G. Welsch, Metall. Trans. 19A (1988) 1121. [14] U. Bill and R.D. Edge, IEEE Trans. Nucl. Sci. NS26 (1979) 1812. [15] R.D. Edge and U. Bill, Nucl. Instr. and Meth. 168 (1980) 157. [16] A.R. Knudson, Nucl. Instr. and Meth. 168 (1980) 163.