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Thin layer diffraction
Paul Fewster
To cite this version:
Paul Fewster. Thin layer diffraction. Journal de Physique III, EDP Sciences, 1994, 4 (9), pp.1533- 1542. �10.1051/jp3:1994220�. �jpa-00249203�
J. Phv.<. III Fianc.e 4 (1994) 1533-1542 SEPTEMBER 1994, PAGE 1533
Classification Physic.< Ah.<tract.<
61. IO
Thin layer diffraction
Paul F. Fewster
Philips Research Laboratories, Cross Oak Lane, Redhill. U.K.
(Rec.eii,ed19 November 1993, accepted II Fehittaiy 1994)
Abstract. Many of the challenges in X-ray diffraction arise from the requirement to produce detailed information on very thin layers. This paper illustrates the present limits in the analysis of X-ray diffraction profiles with examples of epitaxial and polycrystalline layers. One of the primary
uses of X-ray diffraction of epitaxial layers i~ in the determination of composition and thickness, but this can be fraught with problems for the unwary yet a very powerful technique when the
correct procedures are used. Often the assumptions concerning epitaxial quality are rather
ambitious and this paper will consider the influence of imperfect epitaxy on the subsequent interpretation. The latter part of this paper will discuss the limits in polycrystalline diffraction
analysis for the study of nano-structures and the initial stages of the recrystallisation of
amorphous Si.
The measurement of composition in thin layers.
The determination of the composition of epitaxial layers is often carried out by comparing the
lattice parameters of the thin layer and its underlying substrate. Ideally we would like to
directly extract this information from a simple diffraction profile, but unfortunately this is not
possible except in some very special cases, Fewster and Curling [I ]. The errors introduced by simply assuming Bragg's law holds for each component, e.g. layer and substrate, can lead to
errors of the order of 15 9b, Fewster [2]. The only reliable approach to take is to simulate the
diffraction profile with a dynamical diffraction model. Some typical errors associated with the determination of the lattice parameter of a AlAs layer compared with that of a GaAs substrate
by measuring the peak separation are shown in table I : this example is for the 004 reflection with CuKa radiation.
The agreement in the measured and calculated profiles using this simulation model can be very impressive which suggests that the composition can be determined very accurately from the derived relative lattice parameters. Unfortunately various assumptions are made in this derivation. Fewster [3] (I) that the lattice parameter follows a known relationship with the alloy composition, (it that the elastic constants for the alloy are known and (iii that the lattice
parameters of the alloy at the extremes of the phase are known. Generally Vegard's law is
1534 JOURNAL DE PHYSIQUE III N° 9
Table I. The difference in the diffraction peak separations expected fi.om the Bragg angles of the substrate and layer to those obtained by simulation when the full diffraction effects al-e
taken into account.
Thickness Bragg angle Simulated Difference
separation peak
expected separation
(~m) ("arc) ("arc) (fb)
2.00 364.8 366 + 0.33
1.00 364.8 365 + 0.05
0.50 364.8 364 0.22
0.025 364.8 362 0.77
0.10 364.8 344 5.70
assumed (I.e. the proportion of a constituent is linearly related to its atomic radius), which has
been found to be a good approximation for many alloy systems. In general a similar
relationship is assumed for the elastic constants, but it is well-known that alloys are often stiffer than in their pure form so this assumption is not always valid. Alloys like AlGaAs follow Vegard's law fairly -closely, within a few per cent [Al], although the various studies in the
literature are conflicting [3]. SiGe on the other hand has a larger measurable deviation from linearity and a careful study of this was carried out by Dismukes, Ekstrom And Paff [4]. In
deriving the composition this way we must also consider the absolute lattice parameter of the substrate material, which can vary from the intrinsic value with the addition of dopants above the 10'~cm-3 level by as much as 0.0006A (for GaAs). The influence of the non-
stoichiometry in the phase extent on the lattice parameter in semiconductor binary illoys is
generally small ( 0.00005 I for GaAs for instance). So the use of using a relative lattice parameter determination method for determining the composition must not have its errors
determined on absolute terms without these influences considered. In general the accuracy obtainable is usually superior and more direct than other methods and is therefore perfectly adequate for most purposes.
The measurement of thickness in thin layers.
Let us now consider the determination of thicknesses in epitaxial layers and firstly the
influence that a thin layer has on the diffraction pattern. The diffraction profile of a thin parallel
sided layer will be broadened due to the limited number of diffracting planes, show
interference fringes from the interaction of the diffraction from the top and underside of the layer and of course its peak intensity will directly relate to the amount of scattering material
present. This latter aspect has been used by Chanduri and Shah [5] whi have shown that this simple method is practical (to within 0,I ~m) for layer thickness measurement down to a few microns provided that extinction effects are taken into account. The next most obvious direct method for determining this thickness is by measuring the fringe spacing and deriving the layer
thickness from a differential form of Bragg's law A y~
Aw sin 2 o ~~~
where Aw is the fringe separation, 2 o the scattering angle, A is the wavelength of the X-rays and y~ = sin (o + ~b where ~b is the angle between the diffracting planes and the surface.
N° 9 THIN LAYER- DIFFRACTION 1535
This method was proposed by Bartels and Nijman [6] but unfortunately diffraction effects will introduce an error in this measurement ([3] and tab. II) Macrander, Lau, Strege and Chu [7]
and Armin and Halliwell [8] have taken the Fourier transform of their « rocking curves » to
extract these characteristic length scales, but the underlying problem still remains. Hence the
assumption that the diffraction from the layer and the substrate can be considered independently
is flawed : therefore the diffraction process has to be considered as a whole.
Table II. The errors iesttlting fi.om determining the thickness fi.om the fi.inge spacing.
Simulated Equation (I) Error
thickness derived thickness (9b)
(~m) (~m)
2.0000 1.9597 2.02
1.0000 0.9798 2.02
0.5000 0.4899 2.02
0.2500 0.2398 4.08
The present limits in studying epitaxial layers.
The reliability in determining the composition and thicknesses by simulation has been presented by Fewster and Curling Ii- The additional advantages of this approach is that much
more information can be determined and we can push the limits of the X-ray diffraction
technique to its limits. We can now consider two examples of very thin layers that
approximately mark the bounds of sensitivity in X-ray diffraction by using very simple experimentation. The first is a single thin 25 I Ino~~Gao~~As layer on a GaAs substrate
(Fig. I). The layer does not have a well-defined peak but appears as a broadening of the low
angle diffraction tail. This profile has been simulated using this dynamical diffraction model, from which we can see that it is very sensitive to changes in the composition at the 9b level and that the thickness could be determined to within I h. If the layer is buried then the sensitivity is greatly enhanced because of the interference effect between the capping layer and
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Fig, I. The HRMCMRD diffraction profile (dot) from the 004 reflection (CuKa) of a 25 A InGaAs
layer on a GaAs substrate, with the fitted profile (line).
1536 JOURNAL DE PHYSIQUE III N° 9
the underlying substrate. An example of a 2.5 A layer on a GaAs substrate buried under a 555 I GaAs layer
gave a 004 diffraction pattern as shown in figure 2, where we can see that the fringing is very pronounced. Unfortunately though with this single scan over a limited
angular range the thickness and composition are correlated and will always lead to a small uncertainty in the evaluated composition and thickness because of the unknown spreading of the In. To evaluate this interface diffuseness we really need to collect data over a larger angular
range and this is possible with periodic structures (Fig. 3). From modelling the diffraction
profile of a Si 3-doped GaAs periodic structure it was possible to determine the average out- diffusion of Si from the &-planes and also the variations in the period. When these results were
combined with the results from a scan close to the 002 reflection, information on the site
occupancy of the Si was obtained, Hart, Fahy, Newman and Fewster [9].
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layer buried under a 5551GaAs layer
on a GaAs substrate, with the best fit simulated profile (line).
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a periodic Si &doped GaAs layer structure showing the quality of the fit of measured data (HRMCMRD) to the simulated profile. This made it possible to determine the various thickness through the uructure, the fluctuations in the period and the out diffusion of Si from their &planes.
N° 9 THIN LAYER DIFFRACTION 1537
Problems with imperfect layers.
These procedures described are generally adequate for
« well behaved
» samples that are nominally perfect. If however the epitaxy is less perfect and the layer is tilted with respect to
that of the substrate then a full two-dimensional diffraction map is required. The profile to be modelled will then have to be obtained by projecting this map onto the direction perpendicular
to the influence of the tilting [10]. If this is not done then the alignment of the fringes with the calculated values can be in error even for small tilts. The diffractometer used to measure the
diffraction space maps described in this paper is the High Resolution Multiple-Crystal
Multiple-Reflection diffractometer (HRMCMRD) which has been extensively discussed in
previous papers [11-13] and with examples of several applications [14,15]. It basically
consists of a 2-crystal 4-reflection monochromator matched to a -crystal 3-reflection analyzer
to create a « &-function like
» diffraction space probe. It has a nearly constant resolution over most of reciprocal space with a 10? dynamic range of intensity. This permits very high
resolution diffraction patterns to be obtained.
It is not only tilts that can disrupt our interpretation but also if the atomic planes of the thin
layer are not matched to the underlying substrate. What happens in this case is that the X-ray
wave-fields that build up inside the crystal are uncoupled and consequently the normal
dynamical diffraction theory is inadequate. This is most clearly seen in figure 4, where we can
see that the normal diffraction theory gives very poor agreement with the measurements and would require a totally unrealistic value for the 1000 h Sin x4Gen16 layer thickness to match the peak heights yet give poor alignment of the fringes. However when the wave-field
disruption is taken into account the agreement in peak heights is close to the nominal value.
This example is of a structure with a layer lattice parameter partially relaxed back to its cubic form by only 0.296, as determined by topography. By considering the detailed strain distribution close to the dislocations then we can improve upon the fit to extract their
extent [101. This arises from these distortion~ contributing to the diffuse scattering which is
also modelled (Fig. 4).
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Fig. 4. The influence of wave-field disruption on the relative peak heights in a layer which is lattice relaxed (da~h-modified dynamical theory). This can lead to an incorrect determination of the layer thickness if this i~ not taken into account (dot-dynamical theory). The continuou~ line is the measured profile.
The detailed microstructure in epitaxial layers.
Suppose now the layer relaxation is increased further then the dislocation strain-fields start to overlap. Kidd, Fewster, Andrew and Dunstan [16] have studied these effects close to the
1538 JOURNAL DE PHYSIQUE -III N° 9
« critical layer thickness» in Ino_jGaoiAs on GaAs (Fig. 5). The changes in the diffuse
scattering are quite dramatic from which it has been possible to determine the in-plane
dimensions of the dislocation strain-fields. Combining these results with simulation measure- ments to extract the dislocation strain field normal to the interface it has been possible to study
the dislocation interaction [17j. As the relaxation becomes more pronounced, beyond the
catastrophic collapse of the lattice parameter, significant shifts in the lattice parameters are
clearly visible : therefore by combining the information available in the diffraction space maps of symmetric and asymmetric reflections the full unit cell dimensions can be deduced II 4]. The diffraction space maps also contain a wealth of information on microscopic and macroscopic
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N° 9 THIN LAYER DIFFRACTION 1539
tilts and this has been combined with High Resolution Multiple-Crystal Multiple-Reflection Topography (HRMCMRT) to predict the evolution of the relaxation process at these high
strain levels [18]. The information available when these techniques are combined on the
HRMCMR diffractometer is schematically represented in figure 6. This detailed study on a
series of InGaAs/GaAs structures has lead to a « mosaic grain growth » model to account for the reduction in microscopic tilting and increasing contrast in the X-ray topographs with
thickness. This example demonstrates the power of these techniques in determining the
microstructure in these materials.
[001]
'[l10]
Awjl10]
A°'[ilo]
~[001
tji
~_~ zn
~~~ [
j "~~
depth [l10]
£ - composition
Fig. 6. A schematic of the various parameters that can be measured by diffraction space mapping and topography with the HRMCMRD.
The topographic techniques discussed above are performed by placing a small grain size photographic emulsion (0.25 ~m developed grain size) in the diffracted beam from the sample.
If on the other hand the beam is placed in the beam after the analyzer then the contrast arises from a specific position on the diffraction space map defined by the HRMCMRD probe, which
is close to a &-function. Therefore we are able to investigate the origins of any scattering
observed [14]. The parasitic scattering in this instrument is very low and therefore we can
image the diffuse scattering to ascertain its origin [19]. It has been possible to characterize misfit dislocation networks, surface damage and threading dislocations. Hence this whole
combination of techniques at our disposal permits us to detect the deviations from perfection
and extract real quantitative information.
Highly imperfected epitaxial structures.
The HRMCMRD is not limited to near perfect materials, in fact it has been used on
polycrystalline material [20] and highly imperfect metallic multilayer structures, (Birch and Fewster quoted in Fewster and Andrew [2 Ii ), where the importance of using the right probe for the analysis was emphasised. The ideal situation is the combination of low resolution
1540 JOURNAL DE PHYSIQUE III N° 9
diffraction space mapping [21] and high resolution diffraction space mapping with the
HRMCMRD [I I], which is available commercially as a Materials Research Diffractometer.
An example given in figure 7 of a very weakly diffracting high T~ superconductor multilayer,
illustrates the importance of the probe size. The important parameter of interest in this example
was in the determination of the interfacial quality which could be determined by the coherently diffracting depth through the structure. This is directly related to the width of the diffraction
profile perpendicular to the interfaces. Figure 7 illustrates a series of scans under different experimental conditions, using a Philips High Resolution Diffractometer (with an
HRMCMRD modification) positioned on a synchroton beam line at LURE, Orsay [22]. The
profile with a 0. mm slit in front of the detector gave a significantly broadened profile and the diffraction broadening suggested that each layer is relaxed, since the in-depth correlation length corresponded to 200 I which is
very close to the measured period. The HRMCMRD
profile on the other hand corresponded to 8701 which is
very close to the full periodic
structure depth. Clearly the former contains significant tilt component broadening which it
cannot differentiate adequately and gave this misleading result.
HighTc superconduwr mdfik~yer
5oo
~ 25
~ ~~
250 ~
~
~ #
fl m
Jfl j
c c
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0 200 400 600
Scan angle (9''arc)
Fig. 7. The diffraction profiles close to the 007 reflection to determine the coherently diffracting
depth and hence the interface quality in a nominal (Bi~ jsr. Ca )~CUO~)
~~ ~
(Bi~(Sr, Ca)~Cu,O~)~,~~] x 5 periodic structure on MgO. The HRMCMRD profile (line) gave a
coherence length of 8701 and
a 0. mm slit gave 2001 (dash) and the measured periodicity was
1851. II.51 radiation synchrotron source.)
Thin polycrystalline layers.
Clearly as the crystallinity diminishes the intensity becomes very weak and some information becomes difficult to obtain with standard sealed X-ray sources. Suppose we wish to study very weakly diffracting polycrystalline layers do we know the limits of the techniques available
to us ? These following examples indicate the limits of detection with a standard Bragg-
Brentano focusing diffractometer with careful measurements. The first example is of 501 Ag spheres buried in an amorphous SiO~ matrix. The spheres occupy about 25 9b of the volume of the 500 h layer which is deposited on a glass substrate [23]. A first glance at the diffraction
