ROUGHNESS-CAUSED DISPERSION OF HIGH FREQUENCY
SURFACE ACOUSTIC WAVES ON CRYSTAL MATERIALS
Colm M. Flannery ' ' and H. von Kiedrowski
^aul-Drude-Institut fur Festkorperelektronik, D-10117 Berlin, Germany
Engineering, Colorado School of Mines, Golden CO 80401, USA
ABSTRACT The effect of surface roughness on adhesion and tribological properties of films and
interfaces is of key importance. We investigate the dispersive effect of roughness on surface acoustic
wavepackets (30-200 MHz frequency range) for different nanometer roughnesses on silicon (001)
and (111) surfaces. The dispersion effect is significant and available theory does not adequately
predict the real SAW dispersion. Previously unknown dispersive effects on anisotropic crystal
surfaces are demonstrated and an empirical modelling approach is discussed.
INTRODUCTION
The surface state determines the useful properties of many materials, in particular the
tribological, elastic and adhesion properties. Every surface possesses a roughness and this
can have a critical effect on material performance and any devices in which it is employed.
Propagating Surface Acoustic Waves (SAW), which have most of their energy confined to
a one wavelength depth below the surface, are very sensitive to the surface state and are
affected by any change, such as presence of a liquid or solid film, residual stresses,
subsurface damage, and by the level of roughness [1]. Many techniques rely on surface
acoustic waves but little attention is paid to the possible perturbing effects that roughness
may have: the important fields of SAW devices requires ever more accurate knowledge of
acoustic parameters to design high quality devices operating at GHz frequencies[2];
Brillouin Spectroscopy regularly relies on measurement of SAW velocity dispersion at high
frequency to obtain elastic constants[3]; Non-Destructive testing (NDT) techniques
frequently inspect structures of very large roughness. But in none of these areas is attention
given to the roughness perturbation which at high frequencies or large roughnesses may
deleteriously affect results. In this work we show experimentally that the effect is
significant and should not be ignored.
SAW Theory
SAWs propagating on a rough surface will become dispersed: with increasing
frequency the velocity will slow and scattering will increase, attenuating the wave. Here we
concentrate on the more important velocity dispersion effect. For long wavelengths,
as CD -> 0, the roughness will not affect the wave and there should be no effect on the wave;
the velocity should equal the Rayleigh wave velocity. As co increases, the wavelength
CP657, Review of Quantitative Nondestructive Evaluation Vol. 22, ed. by D. O. Thompson and D. E. Chimenti
© 2003 American Institute of Physics 0-7354-0117-9/03/$20.00
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shortens and the wave will 'see' more of the roughness. For CD -» ooone can think of the
wave as having to traverse the surface profile and will be slowed, travelling the same
horizontal distance in a longer time. Thus SAW velocity slows with roughness.
Despite extensive theory [4-6], there exist very few experimental studies of SAW
interaction with roughness, the few published works [7,8] were done at low frequencies (a
few MHz) on isotropic materials and results were not convincing. The best experimental
work was done by Huang and Achenbach [9] on Aluminium at frequencies 2-7 MHz. They
demonstrated linear dispersion and increasing attenuation for increasing frequency and
roughness. Results were not quantitatively compared to theory, the most useful papers not
yet being published. Here we measure dispersion at high frequencies (100s of MHz) and on
relevant crystal materials with submicron roughness levels. The directional dependence of
dispersion is also investigated, something not yet experimentally studied.
The theory of SAW interaction with roughness is extremely complex and rather
difficult to understand. There are contradictions between a number of papers. However the
result of several more recent papers is, in the experimentally-realistic long wavelength limit
(and ignoring offset) the SAW phase velocity dependence on frequency is
v(a>)
S2
-^-oc ——2-o£l
v0
a
Where v(co) is SAW velocity at frequency co, VQ is velocity at zero frequency (velocity on a
smooth surface), 5 is rms roughness and a is transverse correlation length of the roughness.
Q is a constant which is a function of the elastic properties of the material. All authors
assumed a random Gaussian roughness distribution described by 8 and a. The important
feature to note is that the velocity dispersion is linear with frequency and depends on the
square of the rms roughness. On anisotropic materials, where elastic parameters vary
depending on direction, v,v0 and Q will also be functions of SAW propagation direction \\j.
Here we concentrate on the work of Kosachev and Shchegrov who have made the most
comprehensive study to date [5,6]. For their work the phase velocity dependence takes the
explicit form
CD
it is useful to express as the fractional change in velocity
(2)
one notes the square dependence on rms roughness but the inverse linear one on transverse
correlation length.
EXPERIMENTAL
We polished/roughened a range of initially- flat silicon (001) wafers to various levels of
roughness varying from 8 = 10-250 nm, a was in the 15-30 um range for all samples. We
chose silicon because of its technological relevance, our experience with polishing it, and
because the dispersive effect with propagation angle had already been calculated in ref. [5]
for silicon, facilitating benchmarking of results. Roughnesses were measured by a surface
profilometer, being the average of several measured profiles. There was no clear
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60
50
(a) 8:130nm
40
30
20
10
0
+0mm
-10
-20
50
1.4
1.5
100
Frequency (MHz)
1.6
Time ((is)
1.4
1.2g
1.0
N
| 0.8 4
O
o
0.6-
Jo.4
^
0.20.050
100
150
200
250
5 roughness(nm)
FIGURE l:(a) Dispersive SAW wavepacket at 5 mm propagation difference on 5 = 130 nm Si (b) Measured
SAW dispersions on (001) Si [100] dim, with linear fits (c) Fractional dispersions (normalised slope of (b))
versus roughness.
dependence of a on 6. The transverse correlation length a, is a badly-defined parameter;
there is a certain amount of arbitrariness in choosing the appropriate spatial frequency
range in which to measure. Neither is it at all clear whether it is correct to assume this
parameter as Gaussian, nor if the long wavelength assumption is correct for a in this tens of
microns range where the SAW wavelengths is 250-25 /mi (20-200 MHz).
SAWs were generated thermoelastically by light from a pulsed nitrogen laser (0.5 ns
duration, 337 nm wavelength) and focused into a line shape on the samples. The absorbed
heat energy causes a rapid expansion of the absorbing source area, generating stresses
which give rise to wideband SAWS propagating across the substrate. These were detected
at different propagation distances by a piezoelectric foil with knife-edge detector [10].
Frequency-dependent phase velocity dispersion curves were obtained via Fourier
transforms of SAWS of 10 mm path length difference.
RESULTS
SAW Dispersion
Fig. l(a) shows Rayleigh SAW wavepackets obtained at different relative propagation
distances on a 6 = 130 nm (001) sample in the [100] direction. The wavepacket spreads out
in time, indicating dispersion. Higher frequencies (shorter wavelengths), being slowed
more by the roughness, arrive later than lower frequencies (longer wavelengths). Fig. l(b)
shows obtained frequency-dependent phase velocity dispersion curves along [100] on the
samples of different roughnesses, together with linear fits to each curve. It is clear that the
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magnitude of the slope (rate of dispersion) increases with increasing roughness and that
dispersion is linearly decreasing with frequency — the first verification of the predictions
of the theory and the first measurements on anisotropic materials. In the region of [100]
direction a pure Rayleigh SAW mode propagates unattenuated (apart from the attenuation
effects of roughness); in the region of [110] the wave is a slightly different Pseudo-SAW
(PSAW) mode, with a small amount of energy leakage into the substrate, but has
essentially all the characteristics of a Rayleigh wave. Fig. l(c) plots the dispersions
(fractional absolute velocity change at 100 MHz, obtained from slopes of the linear fits) in
both [100] and [110] directions, as a function of roughness. The result is somewhat
perplexing, the dispersion increases with roughness but the behaviour is clearly not
dependent on the square of the roughness — contradicting theory. Rather, the increase is
slower; possibly linear for 8 = 40 nm and above. Also plotted are expected dispersion
values calculated from Equation (2). These theory values are about 100 times smaller than
the measurements and are in no way compatible; possible reasons for this are discussed
below.
The dispersions of the PSAW mode on the [110] direction were larger by about 10%
than those for the Rayleigh mode in the [100] direction. This is also surprising because the
penetration depth of the PSAW is longer and one expects it to be less affected by
roughness; no theory for dispersion of Pseudo SAW modes exists so this is a new result.
This dispersion increase is most probably due to the difference in elastic constants in the
[110] direction from [100].
Angular Dependence
The Rayleigh SAW dispersion is almost constant over the detectable range of
propagation angles on (001) Si; near to 22° from [100] the dispersion falls rapidly to zero,
coinciding with the degeneracy of the wave where it becomes a shear wave with plane of
vibration primarily in-plane and very long penetration depth: no longer interacting with the
surface roughness. Examining dependence of penetration depth on propagation angle for
different cuts of materials allows one to get a good feel for the relative dispersion
behaviour caused by roughness.
However it is predicted [5] that the dispersion can vary by 40 % between extremes on
the Si (111) surface, on which the Rayleigh SAW is detectable at all propagation angles. To
this end we measured SAW dispersion on a 5= 310 nm Si (111) wafer between the
velocity extremes of [1 10] and [211] angles. Fig. 2 shows the obtained dispersions as a
function of angle. The large scatter in the data can be attributed to the statistical nature of
roughness, but the pattern is clear. We found that the dispersion indeed does vary on this
1.3-
Si (111) SAW dispersion
|
1.2-
> 1-11
1.0
0
[1 -1 0]
10
20
30
angle
FIGURE 2. Variation of fractional dispersion of SAW with angle for 6 =310 nm Si (111) surface.
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cut, the minimum being along [110] and continuously increasing with angle to a maximum
about 25-30 % higher along [2 1 1]. The overall dispersion was lower on Si (111) than on
(001), also predicted by ref. [5]. This is the first time that a directional dependence of
roughness-caused SAW dispersion has been experimentally demonstrated. The magnitude
of the relative change is similar.
DISCUSSION
These results are simultaneously encouraging and perplexing. For a wide range of
roughnesses the dispersion is clearly linear with frequency and the magnitude approaches a
velocity change of 1 % at 100 MHz; quite significant. It is reasonable to extrapolate this to
GHz frequencies, implying that the change can be large, even for nm order roughnesses.
This can have very significant implications for Brillouin spectroscopy and SAW devices. It
is well known that Brillouin spectroscopy underestimates elastic constants and roughness
effects may cause much of this. For SAW devices a velocity shift of even a few ms"1 can
mean that the designed efficiency of a filter is disimproved and optimum performance will
not be gained; with consequent commercial consequences.
That the dispersion varies significantly with propagation angle, and the relative change
is comparable to theory, has been demonstrated for the first time. Thus the main qualitative
predictions of the theory have been verified.
What is perplexing is that the magnitude of the dispersion does not increase as the
square of the roughness; the increase is much slower (Fig. l(c)). Neither are the values of
the SAW dispersion anywhere near those predicted by the theory, the measured dispersions
are 2 orders of magnitude larger than expected. One would need to assume a transverse
correlation length 100 times smaller for the theoretical predictions to be 'in the same
ballpark' and there is no justification for this. In all experimental measurements of
roughness profiles the transverse correlation length is much longer than rms roughness,
typically 100-1000 times longer [7-9]. This suggests that many assumptions of the theory
may not hold; the roughness distribution may not be Gaussian and it may not be
appropriate to assume that the acoustic wavelength is much larger than the roughness
parameter lengths, in particular the transverse correlation length. These results indicate that
theory perhaps needs considerable revision to make it applicable to normal physical
situations.
It should also be mentioned that one cannot polish/roughen a surface without causing
some damage to the underlying material. We have tried to minimise such damage but some
damage cannot be avoided. It is difficult to assign the cause for the large dispersions to
subsurface stresses or dislocations, as these effects are usually quite difficult to detect.
However these samples have been produced by standard polishing techniques, and polished
samples for SAW devices or Brillouin spectroscopy must display the same effects. This
makes the results presented here possibly more-, rather than less-, relevant. We are
currently investigating the effect of annealing, and polishing of previously-roughened
samples, to try to isolate the effect on SAW dispersion of subsurface damage from
roughness.
One can also use such results to obtain an empirical roughness-dispersion curve. In
hostile environments, where access to a sample is limited but where roughness and
corrosion effects need to be quantified and inspected, one can remotely measure the SAW
dispersion and use the empirical curve to determine the roughness or corrosion level, this
information may then be used to assess the quality of the part and whether it should be
replaced or passed according to the appropriate nondestructive testing criteria. Similarly,
empirical curves can be used to adjust parameters in modelling of SAW devices.
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1.30
1.25
1.20
Q
Q
1.15
1.10
1.05
1.00
0
10
20
30
40 [110]
M°ini
[1
10]
"
5
10
15
20
25
30
an
9le
flop)
angle
FIGURE 3: Dependence of SAW dispersion on angle for layers of varying anisotropy on (a) Si (001) (b)
Si (111). Compare to ref [5].
There is room for considerable extension to the theory. To date, only Rayleigh SAWs
on thick anisotropic substrates have been treated. We have shown that Pseudo SAWs can
interact differently than Rayleigh SAWs (modelling discussed below confirms this). Such
Pseudo SAWs play an important part in SAW devices and Brillouin measurements and
therefore deserve theoretical attention. Additionally, in layered systems, the SAW energy
can be mostly confined to the layer and the interaction with roughness is likely to be much
larger. This also deserves to be treated.
Modelling
There is considerable potential for modelling of the roughness dispersion as a
perturbation caused by an acoustically thin layer whose elastic constants are related to the
roughness parameters. Figure 3(a) shows calculations of relative dispersion (corresponding
to Q(v|/)), normalised relative to [100], obtained by representing the roughness as an
acoustically thin silica layer. Elastic constant Cu was adjusted to give different
anisotropies, Cu and C44 were not changed. An isotropic layer shows the greatest
dependence on angle, nearly 20 % greater dispersion along [110] compared to [100], while
a layer with the same anisotropy as silicon (1.56) shows very little dependence, dispersion
along [110] is a few % smaller. The shape of the curves for the Rayleigh SAW up to 30°
compares well to Fig. 2 of ref [5], which is computed using a much more complicated
method (solving for boundary conditions on a randomly rough surface). A layer with
anisotropy of 1.3, intermediate between isotropic and silicon anisotropies, gives a
dispersion of 10 %, in line with our measurements. It is reasonable that the polishing
process should randomise the lattice structure of the roughness layer, but that some of the
crystalline structure is retained, therefore there should be some anisotropy in the roughness
layer. Similar calculations for the (111) surface (Figure 3(b)) show a behaviour which
matches our measurements well. For the Rayleigh SAW on both figures there is not a large
dependence on layer anisotropy, but the PSAW on (001) (above 30°) is much more
sensitive than Rayleigh SAW. This is a very interesting, and nonintuitive, behaviour which
deserves further investigation.
This approach is able to predict the observed essential features of dispersion on
anisotropic materials, and has the major advantage that it is computationally much simpler,
as programs to model dispersion of layered systems are widely available and used, and
extension to PSAWs and multilayered systems is straightforward; additionally this may
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also be of more relevance than a theoretical approach, which requires exceedingly complex
calculations and which might never be applicable, apart from very limited cases.
A Simplified Theory
The most widely used method to calculate SAW velocities on layered systems is the
boundary condition matrix method, thoroughly presented in the chapter of Farnell and
Adler [11]. Together with the wave equations for the substrate and layer, one solves the
determinant for the matrix of boundary conditions; derived from conditions of continuity of
stresses and displacements at the interface, zero traction at the free surface. Tiersten [12]
showed a useful simplified first-order approximation valid for the long wavelength case on
isotropic materials. A.A. Maznev has suggested [13] a simplified approach for modelling
surface roughness as a series of isolated islands on the surface of a flat substrate. The
islands act as a mass loading layer but have no extensional stiffness, they do not interact
with each other. Thus, simplifying the boundary conditions to only those of inertial
displacement (mass loading) at the interface, and following Tiersten's first-order approach,
which should be valid for this case, one obtains a dependence of the form
(3)
where h is the layer thickness (average roughness height), k is the wavenumber (= co/v), A
is a constant based only on the Poisson's ratio of the material and decreases from 0.35 to
0.13 over the range of 0.5 to 0 for Poisson's ratio. This formula shows a linear dependence
of dispersion on frequency and on roughness, is independent of density, varies only slightly
with Poisson's ratio, and is only dependent on one roughness parameter h (or average
roughness height). 8 rms roughness is usually within 10-20% of average roughness.
If one inputs values for 5, Poisson's ratio of silicon, and frequency, one finds that the
dispersion values given by Equation (3) are of the same order as those measured here for
all levels or roughness. Differences range from 20-60 %, which, especially considering the
large uncertainties in all parameters, can be considered quite reasonable correlation;
certainly standing in stark contrast to estimates from Equation (2). This is certainly quite
remarkable but one should not get too excited. The formula is for isotropic material,
involves a number of approximations and simplifications and the statistical nature of
roughness means that uncertainties in measurements can be quite large. It is probably not
correct to assume that there is no extensional stiffness, thus the values yielded may be only
coincidental. Nevertheless, it is encouraging that such a crude approach can yield estimates
for dispersion of the same order as those measured. This approach appears to have some
merit and is worth further detailed investigation.
CONCLUSIONS
We have been the first to characterise SAW dispersion due to roughness at 100 MHz
frequencies, and on anisotropic materials. The predictions of the theory have been
qualitatively verified but quantitatively do not agree. We feel that some assumptions of the
theory, that roughness is Gaussian and that SAW wavelength » roughness length may not
be valid. These dispersions are significant, and, especially at GHz frequencies, should be
taken into account for SAW device design and Brillouin spectroscopy (which is known to
underestimate constants); for NDT, results may also allow remote determination of
roughness by measuring SAW dispersion. An empirical modelling approach may also be
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used to successfully model roughness-induced SAW dispersion, especially for angular
dependence. A simplified first-order theory also estimates values of dispersion which are
comparable to measurements and is being further investigated.
ACKNOWLEDGEMENTS
C.F. acknowledges funding from EU project EFRE. The experimental work was carried
out at the Paul-Drude Institute Berlin. C.F. is now at Colorado School of Mines, and
carrying out research at NIST, Boulder, Colorado. We thank A.A. Maznev of Philips
Analytical for sharing his simplified theoretical approach with us.
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