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High resolution X-ray diffraction of one- and two-dimensional periodic surface gratings
P. van der Sluis
To cite this version:
P. van der Sluis. High resolution X-ray diffraction of one- and two-dimensional periodic surface gratings. Journal de Physique III, EDP Sciences, 1994, 4 (9), pp.1639-1647. �10.1051/jp3:1994230�.
�jpa-00249213�
Classification Physic-s Abstracts
61.10D 68.20 68.65
High resolution X-ray diffraction of one- and two-dimensional
periodic surface gratings
P, van der Sluis
Philips Research Laboratories, Prof. Holstlaan 4, 5656 AA Eindhoven, The Netherlands
(Received J9 Noi'ember 1993, revised 6 April 1994, accepted 30 May 1994)
Abstract. From one- and two-dimensional periodic surface gratings in Si and InP very clear satellite peaks are obtained in a rocking curve provided the appropriate diffraction geometry is
chosen. For (001) oriented Si, InP or GaAs substrates with corrugations in the [110] direction, measured with CuKa
i,
the optimal geometry is best approximated by the 113 reflection using the high angle of incidence. Due to beam compression, caused by the asymmetry of the reflection, this reflection is also suited to obtain two-dimensional reciprocal space maps from one- and two dimensional surface gratings. A narrow slit can be used to obtain adequate detector resolution. The maps reveal in detail the shape of the grating. They are compared with simple model calculations based on Fourier transformation of the shape of the grating-crystal assembly.
1. Introduction.
Recently, there has been a growing interest in low-dimensional semiconductor structures like quantum wires and quantum dots. Periodic one-dimensional arrays of wires and two-
dimensional arrays of dots can be studied with diffraction. Since the dimensions of the
structures lie in the nanometre range and the periodicities are usually below I ~Lm, X-rays are a suitable probe. The extra periodicity parallel to the surface yields satellite peaks close to Bragg peaks originating from the three-dimensional periodicity of the crystal lattice. These satellites
have been reported previously in rocking curves from gratings in Si ii], InP [2] and GaAs [3], using different diffraction set-ups. Recently, also two-dimensional reciprocal space maps have been reported showing how gratings modify the scattered intensity in the neighbourhood of a
Bragg peak [4]. In addition, modification of the X-ray reflectivity at grazing angles has been reported [5].
In our experiments, we used a four-crystal monochromator which yields a beam which is nearly parallel and monochromatic (CuKa,) [6]. We will show that by choosing the
appropriate diffraction geometry on the sample and by employing appropriate slits in front of
the detector, rocking curves and two-dimensional reciprocal space maps can be recorded with very high signal-to-noise ratios of one- and two-dimensional surface gratings. We also propose
a simple model for the interpretation of the scattering near the Bragg peak of a crystal with a surface grating of arbitrary shape.
1640 JOURNAL DE PHYSIQUE III N° 9
2. Theory.
The 113 reflection is an asymmetric reflection on a (00 Ii oriented crystal. Such a reflection can be measured with either a high angle of incidence or with a low angle of incidence. Figure I
shows the Ewald sphere construction [7] for and the geometry in real space of the l13
reflection of an (001) oriented InP crystal with a grating perpendicular to the ii10] direction.
The asymmetric I13 reflection is shown here with high angle of incidence. For crystals like Si and GaAs a similar geometry will be obtained. The extra periodicity of the grating gives rise to satellite peaks in the ii10] direction, parallel to the surface. The satellite peaks and the Bragg peaks are extended in the [001] direction due to the abruptness of the surface. For flat substrates this is known as Crystal Truncation Rod (CTR) scattering [8]. In figure I the CTR
scattering of the grating is shown as vertical lines. The low angular resolution of the detector is
depicted by a section of the Ewald sphere la small arc perpendicular to the diffracted beam).
The monochromaticity of the beam results in a very thin Ewald sphere and the parallelism of the incident beam results in one single direction for the incident beam. Note that for the 113
reflection measured with high angle of incidence, the 2@ direction is nearly perpendicular to the surface. Due to the geometry of the reflection, the Ewald sphere intersects only one CTR (in the diffraction plane). By rocking the crystal, different CTRS will intersect with the Ewald
sphere separately and clearly separated peaks will be obtained. With a wide slit the whole CTR
can be recorded with an optimal signal-to-noise ratio. From the peak spacing in the rocking
curves the periodicity can be calculated. The peak intensity provides information on the shape
of the grating. The latter can be used to check the width/periodicity ratio of gratings, because a
lower peak intensity will be found if the order of the satellite reflection equals the
periodicity/width ratio.
More detailed information, however, is obtained from two-dimensional reciprocal space
maps near a Bragg peak of the substrate crystal. Inspection of the geometry of the 113 reflection measured with a high angle of incidence (Fig. I) shows that the size of the incident beam is reduced due to the diffraction. In a typical set-up the incident beam is about I mm. For InP the diffracted beam width is then only o-o13 mm. For a detector placed at a typical distance of about 173 mm, this corresponds to a resolution in 2@ of about 0.004°. This is more than
adequate to study the CTR scattering in detail. For Si and GaAs a slightly lower resolution is obtained, because the Bragg angle is somewhat larger.
CTR 10011 k
~f_ k , ,~j '~
°~
l 13
k~~ ~ Woui
[ ~~~
000
~
slits
Fig. I. Geometry of the measurement of the II 3 reflection in reciprocal and real space of a surface
grating perpendicular to Ii 10] on a (00 ii sub~trate. 1,~ is the incident beam and i~~~ is the diffracted beam.
with widths u.~~ and u'~~, respectively. D is the detector.
The two-dimensional maps contain detailed information on the shape of the surface grating
and the lattice in the grating. In the kinematical approximation, the scattering amplitude is
given by
A
=
lp ii )e'~'dv (I)
v
where p (7 is the electron density and V the scattering volume. The electron density can be
written as the convolution (denoted by *) of an infinite three-dimensional array of
functions z (&(7 7, ii the real space lattice points) with the electron density of a
,
single unit cell (U(7 ii, which is zero outside the unit cell, multiplied by the shape function (S(7)), that is unity in the crystal-grating assembly and zero everywhere else
p(71= (z8(7 7,j* U(7)) x S(7). 121
The resulting scattering amplitude is then given by
A
=
lj U(7 )e'~' dV
x z ii 7,) e'~' dV
* SIT) e'~' dV (3)
u ,
The first factor is the Fourier transform of the unit cell contents and is thus equivalent to the
structure factor Fj well known from single crystal structure determination [9]. The second
factor is again a three-dimensional array of functions, but now in reciprocal space
z (&(k k, ii Because F j has to be multiplied with z (&(k I, )), it is only necessary to
, ,
evaluate Fj at the reciprocal lattice points given by (k,). Every reciprocal lattice point with
amplitude Fj has to be convoluted with the last term in (3), which is the Fourier transform of the shape of the crystal. This means that the Fourier transform of the shape function
S(7) will be non-zero near every reciprocal lattice point. Since the three-dimensional shape of
a grating can be very complex (e.g, due to under-etch, rounded corners etc.) we calculate this Fourier transform numerically. Absorption and extinction are taken into account by weighing
the contributions in (3) with a weight decreasing exponentially with depth in the crystal. The exponent of this damping function is estimated from dynamical scattering simulations on the bulk crystal.
A complication of the used asymmetric diffraction geometry might be effects caused by
refraction of the X-rays on exit from the sample. The measured exit angle ~b is related to the
~ ~exit angle in the ~ample'' ~ by cos (~b )/cos (~b
~)
= fi. where ii is the refractive index for X- rays. The refractive index is very close to I, and for grating~ even closer to becau~e the
average electron density in the grating is a lot lower than of bulk material. The most
pronounced effects in semiconductor material can be expected for the 113 reflection of InP, since its exit angle is the smallest (0.57°). We have neglected the refraction effects because
even for our InP sample (vide infra) the corrections on the positions within the measured range would be less than 2.5 §t. If required, a first order correction of these effects can be based on the shift of intensity in reciprocal space with a refractive index calculated from the average
electron density of the grating, or measured from the angle of total reflection of X-rays near grazing incidence.
1642 JOURNAL DE PHYSIQUE III N° 9
For one-dimensional gratings the divergence in the direction perpendicular to the diffraction
plane (axial divergence) is not important because close to the CTRS there is in that direction
zero intensity for those structures. For two-dimensional gratings the second periodicity results in satellites in the direction of the axial divergence. The axial divergence incident on the
sample and the axial acceptance angle of the detector are typically 1°-2°, enough to encompass several satellites. We therefore calculate the intensity of all satellites in three dimensions and
integrate the intensities in the direction of axial divergence. In this way again a two- dimensional intensity distribution in reciprocal space coordinates is obtained.
3. Experimental.
All diffraction curves were obtained with a Philips high resolution diffractometer (HR- I) using
a four-crystal monochromator. Normally the 220 reflection of the monochromator crystals was
used with CuKa radiation. For gratings with a periodicity larger than ~m the high resolution 440 reflection was taken. The point focus side of a long-fine-focus tube with a focus size of 0.4
by 12 mm was used. The take-off angle is 4.5°, resulting in a projected size of 0.4 by 0.95 mm (0.95 mm in the diffraction plane). The slits used for rocking curves are typically 2-4 mm and
for two-dimensional reciprocal space maps are typically 0.05-0.I mm. Figure 2 shows the
shapes of the gratings that were studied.
A:iii)~~ ~j ~ ~ fl_
fi
~~
[ii
)~ 2~~ ~ ~ ~
fi j j j
j~ jj~~
ll10~~~~'_~~ ~_f~/~ ~
W P3--
loo1] T2
~~
I 1© ~~~
i~
Fig. 2. A: Cross-section of the approximately rectangular one-dimensional grating in InP,
perpendicular to the grating direction, B Top view and cross-section of the two-dimensional grating in Si, C : Cross-section of the approximately triangular one-dimensional grating in InP, perpendicular to the
grating direction.
s
i'S w
Ew
-lo lo
3
~~
lo
i
i
Omegal')
Fig. 3. The Ii 3 rocking curve of the approximately rectangular one-dimensional grating.
4. Results and discussion.
Figure 3 shows the rocking curve of an approximately rectangular one-dimensional grating in
InP. A schematical model obtained from scanning electron microscopy (SEM) of the cleaved
sample is depicted in figure 2A. The slit in front of the detector was optimized for an optimal signal-to-noise ratio in the rocking curve. It shows the substrate peak of InP plus a total of 35
satellites. The periodicity (Pj =238.9±0.snm) is obtained from the spacing. Close
inspection of the rocking curve reveals that the ± 3th and ± 6th satellite have a slightly lower
intensity. This is caused by the rectangular width/periodicity ratio of the grating which is near 1/3. At the high angle side, less satellites are found and the background is lower. This is
because the 113 reflection on (00i) oriented InP is very asymmetric. When omega becomes larger than the diffraction angle (51.6°), the signal is diffracted into the crystal (Laue
diffraction) and is lost for detection.
Figure 4 shows the 113 rocking curves of two-dimensional gratings on a Si substrate with
periodicities ranging from 0.25-5 ~m (Fig. 28). In all cases the receiving slit was optimized
for optimal signal-to-noise ratios. The periodicity ratios of the gratings (5-2.5-1-0.5-0.25 ~m)
are easily recognized from the rocking curves. Note that for the grating with the largest periodicity, the diffraction experiment means simultaneous diffraction to the 0.18 nm lattice
spacing and the 5 000 nm grating spacing, more than 4 orders of magnitude difference.
We found that for a grating with a spacing of lo ~m the satellites could not be resolved any
more. From the divergence and wavelength spread of the monochromator in the 440 mode
[10], it can be expected that the satellites will become separable somewhere between 5 ~m and lo ~m.
Figure 5 shows the measurement and simulation of a two-dimensional reciprocal space map
of an approximately triangular shaped one-dimensional grating in InP (Fig. 2C). The starting
model for the calculations was obtained from SEM of a cleaved sample. The grating
periodicity (P4 obtained is 240 ± 5 nm. The main intensity lies in a cross pattern, of which the directions are approximately perpendicular to the physical surfaces in the grating. The opening angle (r~ = 95 ± 5° ) is thus directly related to this cross and can be determined accurately by matching this shape. The height (H~ = loo ± lo nm ) is derived by modelling the extend of the
intensity distribution in k~.
1644 JOURNAL DE PHYSIQUE III N° 9
~"
10k A
( 8
I
3 loo
i
-200 -loo 0 100 200
Omega (")
Fig. 4. The l13 rocking curves of two-dimensional ~quare gratings in Si with periodicities of 5 ~m (Ai, 2.5 ~m (B), ~m (C), 0.5 ~m (D) and n.25 ~m (E). For clarity. each rocking curve i~
displaced one order of magnitude.
0.5125
~
~
0.5100
-O.2425 -0.2400
k, ((')
Fig. 5. Reciprocal-space map of the one-dimensional triangular grating (upper plot), with model calculation (lower plot). Both plots have identical scales. Intensity contours at 4, 16. 64, 7 000 and 14 000 cps. The substrate peak lies at ~i ~ 0.24 Ii and k~ 0.511 2.
The final intensity distribution along the CTRS is described satisfactorily by our simple
model. The centre of the measured cross pattern, however, lies slightly below the crystal substrate peak. So the Fourier transform of the shape function is located around a reciprocal
lattice point, slightly different from the 113 substrate reciprocal lattice point. This indicates a slightly different (in this case larger) lattice in the grating. The cause of this fairly large difference (0.05 §&) in lattice constant is unknown. For e.g. small silicon crystals such an increase in lattice parameter is only observed for crystals with a size below about lo nm [I ].
This effect is not caused by refraction effects since it is also observed near the 004 reciprocal
lattice point, where refraction effects are far less important.
Two-dimensional reciprocal space maps like figure 5 can be obtained with crystals in front of the detector [4]. This offers the possibility to scan in the neighbourhood of any reflection.
The slits, however, have the advantage that the intensity is much higher, give a diffraction pattern without any so-called "analyzer streaks" (because only for a slit the transmitted
intensity outside the transparent region is zero) and that the resolution can be easily adapted to
the measurement required for rocking curves a wide slit and for two-dimensional scans a very
narrow slit.
Figure 6 shows the measurement and simulation of a two-dimensional reciprocal space map of a two-dimensional grating in Si consisting of a square two-dimensional array of tapered cylinder shaped pillars (Fig. 28). The starting model was obtained from SEM of complete (top
view) and cleaved samples (side view). The periodicities (P~
=
250 ± nm ;
P
~ = 250 ± nm) are obtained from rocking curves of the I13 reflection with a wide slit. The
map shows a shallow cross-shaped intensity distribution. The cross-shape is caused by a
tapering of the pillars. By matching the angle of the cross in the diffraction pattern, the tapering (r 13 ± 2° ) of the pillars was determined. The width (W
= 140 ± 20 nm ) at the base of the
pillars (and via the tapering angle r the height Hj is determined by matching the extend of the
intensity in the k~ direction.
The final simulation again reproduces the intensity along the CTRS well. The width of the measured CTRS is larger, probably due to a slight variation in shape and periodicity the
measurement is an average over about 2 x 10~ pillars. Also for these Si pillars a slight shift of
the grating pattern with respect to the substrate peak is found, pointing to a small difference in lattice constant.
Especially for two-dimensional gratings, the extra intensity obtained with a receiving slit if compared with a crystal in front of the detector is important, because only 15 §& of the grating
volume is Si, while for one-dimensional gratings this factor is usually near 50 §&.
5. Conclusions.
With a carefully selected wide slit (2-4 mm) 113 rocking curves can be measured from periodic
surface gratings on (001) oriented InP, Si and GaAs substrates with a very high signal-to-noise
ratio. By replacing this wide slit with a very narrow slit (0.05-0. I mm) two-dimensional
reciprocal space maps of the scattered intensity near the 113 Bragg peak on these substrates
can be obtained. From these maps the detailed shape of the grating and the lattice parameter in the grating can be obtained from Fourier transformation of a shape function which describes
the outer contours of the crystal-grating assembly. The relative orientation of these outer
surfaces can be determined most accurately. The model can be applied to describe the intensity distribution of one- and two-dimensional gratings of arbitrary shape as long as dynamical
diffraction effects are negligible. The lattice parameter in the grating was always found to be
slightly larger than in the substrate.
1646 JOURNAL DE PHYSIQUE III NO 9
'&
I
~ ,
~~
o.55
Fig. 6. Reciprocal-space map of the two-dimensional square grating of tapered pillars (upper plot), with model calculation (lower plot). Both plots have identical scales. Intensity contours at 3, 8, 32, 15 000 and 30 000 cps. The substrate peak lies at kjj 0.2605 and k~ 0.5526.
Acknowledgments.
I am indebted to J. J. M. Binsma, T, van Dongen and P. Huisman for providing the InP
samples and M. J. Verheijen and J. Haisma for providing the Si samples.
References
Ill Aristov V. V., Erko A. I., Nikulin A. Yu., Snigirev A. A., Observation of X-ray Bragg Diffraction
on the Periodic Surface Relief of a Perfect Silicon Crystal, Opt. Commun. 58 (1986) 300-302.
[2] Macrander A. T., Slusky S. E. G., X-Ray Diffraction from Corrugated Crystalline Surfaces and Interfaces, Appl. Phys. Let'. 56 (1990) 443-445.
[3] Tapfer L., Grambow P., X-Ray Bragg Diffraction on Periodic Surface Gratings, Appl. Fhys. A 50 (1990) 3-6.
[4] Gailhanou M., Baumbach T., Marti U., Silva P. C., Reinhart F. K., Ilegems M., X-Ray Diffraction
Reciprocal Space Mapping of a GaAs Surface Grating, Appl. Fhys. Let'. 62 (1993) 1623-1625.
[5] Tolan M., Konig G., Briigemann L., Press W., Brinkop F., Kotthaus J. P., X-Ray Diffraction from Laterally Structured Surfaces Total External Reflection and Grating Truncation Rods, Europhys. Let'. 20 (1992) 223-228.
[6] Bartels W. J., High Resolution X-ray diffractometer, Philips Tech. Rev. 41 (1983) 183-185.
[7] Stout G. H., Jensen L. H., X-ray Structure Determination (The Macmillan Company, London, 1972) p. 31-37.
[8] Harada J., Evaluation of the Roughness of a Crystal Surface by X-ray Scattering. I. Theoretical Considerations, Acta Ciyst. A 48 (1992) 764-771.
[9] Stout G. H., Jensen L. H., X-ray Structure Determination (The Macmillan Company, London, 1972) p. 221-222.
[10] Van der Sluis P., Three-dimensional Resolution Function of the Four-reflection Monochromator : a Route to a Higher Intensity, J. Appl. Cryst. 27 (1994) 50-55.
ii Veptek S., Iqbal Z., Oswald H. R., Sarott F.-A., Wagner J. J., Webb A. P., Lattice Dilatation of Small Silicon Crystallites- Implications for Amorphous Silicon, Sol. State Commun. 39 (1981) 509-512.
