J.F. Kelly
2.1 Introduction
2.2 The
Early History of Topography
Table 2.1 A chronological summary of
laboratory topography
2.3 Development
of Instrumental Techniques
Figure 2.1 Berg-Barrett geometry
Figure 2.2 Lang camera
2.4 Topographic
Contrast
______________________________________________________________________________
List of symbols used
re – Classical electron radius
(e2/mc2)
l -
Wavelength
F – Structure factor (Fh)
D – lattice plane spacing
q - Bragg
angle (qb)
V – Volume of unit cell
VD – Volume of defect
P(l) – Power in
synchrotron beam
m - Linear
absorption coefficient
I – Integrated Intensity
I o – Incident beam intensity
t - Thickness of sample
C – Polarization factor
Cd – Contrast factor
x -
Extinction distance
h,k,l – Miller indices
R – Resolution (R s)
d – Specimen-plate distance
D – Source-specimen distance
S – Source size
2.1 Introduction
The ability “to see” into the interior of a single crystal or look at its surface by producing images formed by x-rays Bragg reflected from the lattice planes, yield knowledge about the defects and lattice misorientation of the bulk material that is difficult to obtain in any other way. The early pioneers of this topographic technique Berg, Barrett, Guinier, Ramachandran & Wooster fully appreciated the utility of the methods they were using to open up the frontiers of materials science. That these methods were to have such wide-ranging applicability particularly to the semiconductor industry late in the 20th century was perhaps less obvious in those far off days, but nonetheless a tribute to their foresight.
2.2
The Early History of Topography
Several workers
have reported that individual Laue spots obtained by x-ray diffraction from
crystals do not appear uniform in contrast, but contain a structure that maps
variations in the lattice planes causing the spots. As early as 1931 Berg
(1931) had performed experiments on rock salt with monochromatic x-rays from a
tube some distance away, in which the point-to-point variation of the
reflecting power over the specimen was recorded on a photographic plate placed
close to the crystal. The striated images that were produced were attributed to
plastic deformation that the crystals had undergone. Barrett (1931) at about
the same time concluded that the elastic strains produced by piezoelectric
oscillations could alter the reflecting power of quartz crystals, this he
demonstrated by photographing the Laue spots in a fashion similar to Berg.
So although it
was realized at this time that x-rays could be used as a tool for exploring the
inhomogeneous strain produced in materials as well as natural imperfections,
there followed a period of about ten years during which there were few advances
in the technique. In particular little attempt was made to extend these
original investigations to metallurgical specimens, until Barrett (1945)
reported results from single crystals of silicon ferrite. With fine grain
emulsion plates, the enlarged x-ray micrographs he obtained showed black lines
corresponding to regions of high reflecting power on the specimen and white
areas that represented non reflecting portions of the crystal (regions that did
not satisfy the Bragg relation n l = 2d sinq ).
It was
Ramachandran (1944) who first used the term “topograph” to describe the
appearance of Laue spots that he obtained from polished cleavage planes of
diamond using a white X-ray beam from a tungsten target. The Laue spots were
identified as a topographic map of the crystal plate exhibiting variations in
structure. With a source-specimen distance of 30 cm and the photographic plate
kept 2.5 cm from the crystal, using a pinhole source 0.3 mm in diameter
Ramachandran achieved a resolution of 25 mm (calculated from the
values given in the original paper). He realized the practical interest that
such a technique might provide in detecting lattice perfection, as
complementary to luminescence studies in general. The term became established
in the literature when Wooster & Wooster (1945) obtained “topographs” from
diamond surfaces using characteristic copper radiation rather than the bulk
stone results of Ramachandran. The specimen and photographic plate were rocked
through an angular range sufficient for all parts of the crystal to reflect the
wide beam of filtered x-rays diverging from a pinhole.
Using polychromatic x-rays in transmission Guinier and Tennevin (1949) studied both orientation and extinction contrast in aluminium. These different contrast mechanisms are discussed more fully in section 2.4. Tuomi et al. (1974) used a similar geometry when performing the first synchrotron radiation topography experiments on silicon samples.
Improvement in resolution to that obtained by Ramachandran and Wooster was achieved by Bond and Andrus (1952) who examined structural imperfections in natural quartz surfaces using a double crystal technique. Consisting of a parallel double crystal spectrometer arrangement, this particular characteristic radiation (Cu) method provides very good sensitivity to lattice misorientation.
A micro focus x-ray tube introduced by Schulz (1954) having an apparent focal spot size of 30 mm x 3 mm enabled him to adjust the source-specimen distance (D) to the specimen-photographic plate (d) ratio to unity (see section 2.11). In this configuration (d/D ~ 1) the geometry gained sensitivity to orientation contrast as well as extinction (nowadays called diffraction) contrast. The method employed continuous radiation in a reflection mode while retaining topographic resolution (spatial resolution µ source size), a simple way of mapping misorientation textures of large single crystals.
Lang (1957a)
examined crystal sections with slit collimated penetrating characteristic
radiation (Ag Ka); these
so-called section topographs are illustrated schematically in Figure 2.2.
Linear and planar defects could be mapped with this technique by translating
the specimen known distances and taking a series of section topographs. Lang
(1957 b) used similar techniques to study low-angle boundaries in metal single
crystals grown from the melt. A summary of the techniques discussed above is
tabulated in table 2.1 for ease of reference for the reader.
Table 2.1 A chronological summary of the topographic techniques employed by various workers with laboratory equipment, original publications can be found in the references at the end of this section by author (year). The chart shows whether a white radiation beam or single wavelength x-rays were used in the investigation and the type of experimental arrangement used to obtain the topographs. Abbreviations in parenthesis: SC – slit collimated, DS – diverging source, DC- double crystal technique
|
Year |
Author |
Continuous Radiation |
Characteristic Wavelength |
Transmission |
Reflection |
|
1931 |
Berg |
|
|
|
|
|
1944 |
Ramachandran |
|
|
|
|
|
1945 |
Barrett |
|
|
|
|
|
1945 |
Wooster &
Wooster |
|
|
|
|
|
1949 |
Guinier &
Tennevin |
|
|
|
|
|
1952 |
Bond &
Andrus |
|
|
|
|
|
1954 |
Schulz |
|
|
|
|
|
1957 |
Lang |
|
|
|
(SC) |
|
1958 |
Newkirk |
|
|
|
|
|
1958 |
Lang |
|
|
|
|
|
1958 |
Bonse &
Klapper |
|
|
|
(DC) |
|
1959 |
Newkirk, Lang |
|
|
|
|
Several
experimental methods not involving diffraction have been used to observe
individual dislocations in materials for example by etching of the surface with
acid solutions or decoration with foreign atoms. The possibility of detecting
individual dislocations by x-rays, however was made independently in 1958 by
three laboratories. These methods using different diffraction geometries are
described in section 2.3. Newkirk (1958) imaged individual screw dislocations emerging
from the polished surface of a silicon crystal using the Berg-Barrett technique
(illustrated in Figure 2.1). He attributed the extra intensity diffracted from
the imperfect lattice to a reduction in the x-ray extinction contrast from that
region. Bonse and Klapper (1958) used a double crystal arrangement for
detecting the strain surrounding single dislocation outcrops at the surface of
germanium crystals. Lang (1958, 1959) studied individual dislocations in
silicon developing the dynamic principle of the projection topograph technique
as an extension to the static section topograph. Slit collimated characteristic
radiation from a distant source produces a projection of a crystal slab and its
imperfection content by linear traversing of the specimen and film together
producing a superposition of many section topographs. Very thin specimens were
not required as penetrating Mo Ka, Ag Ka and W Ka radiation was used and the
chief experimental factor that governed the topographic resolution was the
angular size of the x-ray source in the vertical plane. In the horizontal plane
the limit is affected by the presence of both the Ka and Kb images.
These early days
of high-resolution x-ray topography (1957-1962) are recalled in a personal and
idiosyncratic manner in the review by Lang (1993).
2.3 Development of Instrumental Techniques
There have been numerous reviews and books on the subject of x-ray topography and its practical realization in the laboratory, the interested reader is directed to the following comprehensive reference sources; Tanner (1976), Lang (1978,1992), Bowen and Tanner (1998), albeit that this is not an exhaustive list. It is the aim here to familiarize the reader with the main experimental techniques that have been developed as a precursor to describing the use of synchrotron topography, which as a matter of course forms the main content of the experimental work carried out in this thesis.
Clearly from the foregoing section it has been possible to use both the continuous bremsstrahlung and characteristic lines of a laboratory source in topography experiments. The white radiation methods needless to say are of interest due to the availability of synchrotron radiation. Historically however the Berg-Barrett technique using collimated soft characteristic radiation, (Co Ka: l = 1.789Å, Cr Ka: l = 2.290Å) has been widely used. The Bragg angles from high-order reflections are large (2qB ~ 90o) and the plate can be set close to the specimen and perpendicular to the reflected rays. The technique was further developed by Newkirk (1959) reducing the ratio d/D and using fine grain emulsions to observe individual dislocations with a resolution of ~ 1mm. (Typical values given by Moore (1991) for D @ 30 cm, d @ 1mm, S = 100 mm yield a value for the resolution R = 0.3 mm). The specimen is oriented so that it Bragg reflects asymmetrically as shown in Figure 2.1.
A.R. Lang (1992) International Tables for Crystallography,
Vol. C, pp 113-123
Figure 2.1 Reproduction of figure from
Lang (1978). The Berg-Barrett reflection geometry is used
with characteristic radiation from an extended source S, slit typically 1mm
apart.
Three possible film orientations are shown: for minimum plate separation over the whole crystal length CD then position F1 is appropriate. In order to minimize image distortion then F2 parallel to the specimen surface is best while for normal incidence F3 to reduce blurring of the image should be used. In any case the plate-specimen distance should be about 1-2 mm to reduce the spread from both components of the Ka doublet.
The most popular and widely used method for transmission XRDT was developed by Lang (1959) and is called the projection topograph; it is based on the earlier section topograph Lang (1957a). In Figure 2.2 section topographs correspond to the stationary situation, projection topographs are taken by scanning the crystal and film across the beam. Great care must be taken with the traverse mechanism; this is dealt with in detail by Tanner (1976) and will not be discussed here.
With a plate shaped crystal CD, the ribbon like incident beam in a section topograph is Bragg reflected by planes normal to the surface and the multiply scattered rays will fill the triangle bounded by AB and the exit surface of the crystal. This energy flow triangle is called the Borrmann Triangle after Borrmann (1941). The diffracted wave vector Kh containing information about the lattice defects is then recorded on the photographic film (usually Ilford L4 plates) at F. The screen S prevents the incident wave vector Ko from blackening the film. The aperture allows a lattice defect at I producing diffraction contrast rays parallel to Kh to fall on the film at I1. The position of I within the crystal CD can be measured on the film I1 B1/ A1 B1, a 3D construction of the dislocation can then be made from taking a series of section topographs.
The typical beam divergence in the Lang camera is 0.5 mrad making the technique sensitive to orientation as well as extinction contrast (for more detail see section 2.4). In the case of large absorption in specimens (mt ~ 10) when anomalous transmission through the crystal occurs, energy flows parallel to the Bragg planes and the Borrmann triangle forms a narrow fan. Double crystal techniques as well as anomalous transmission have been used in topography and are outside the scope of the present survey.
A.R. Lang (1978) Techniques and
interpretation in X-ray topography. In Diffraction
and Imaging Techniques in Materials Science p 635
In order to gain more insight into the production of a topographic image it is useful to consider the main contrast mechanisms and the factors that affect and contribute to the overall
appearance and
resolution of the various techniques described so far.
2.4 Topographic Contrast
A substantial literature exists on the subject of diffraction contrast in topographic images, including specifically that due to dislocation bundles in silicon carbide Fisher et al. (1993). A general treatment of dislocation contrast in x-ray synchrotron topographs is given in the article by Tanner, Midgley and Safa (1977), and the references listed at the beginning of section 2.2. There are several physical factors which affect the range of contrast in an experiment these include x-ray wavelength, specimen absorption and the scattering factor of a particular Bragg reflection. The diffracted intensity can lie somewhere between the two extremes of an “ideally perfect crystal” and that of an “ideally imperfect crystal”. The latter term is somewhat harder in practice to realize. The diffracting power of imperfect crystals is usually much higher than that from perfect crystals.
For an ideally perfect crystal there is attenuation of the beam set exactly for the Bragg reflection, which leads to extinction of the beam. On the other hand for an ideally imperfect crystal there is negligible extinction. This type of intensity contrast in a Bragg reflected x-ray beam arising from the point-to-point variation in perfection of the crystal lattice is known as extinction contrast (the term is synonymous with diffraction contrast). The range of extinction contrast can be made to be about 100 times in the diffracted intensity by varying the particular wavelength and thickness of the crystal. The geometry of the topographic experiment plays a part in determining the magnitude of the diffracted intensity; the wavelength spread of the radiation, the amount of collimation and the ratio of the source-specimen distance to the specimen-photographic plate distance all playing their role.
Contrast may also arise from some part of a crystal that is oriented to satisfy the Bragg relation (n l = 2d sinq) thereby diffracting x-rays, whereas an adjoining region tilted with respect to the first may not satisfy the condition. This misorientation of the lattice provides a boundary between regions of different contrast and is appropriately called orientation contrast. The choice of either an extended characteristic source or localized white beam in a topography experiment, clearly would compromise the sensitivity that it is possible to achieve using this mechanism to assess the perfection of the crystal. Using a collimated single wavelength would provide a tool for probing variations in orientation contrast more easily.
At this stage it is good to set out the conditions whereby choice of various experimental parameters may affect the topographic images obtained. For example consider two regions in a crystal that are misoriented with respect to each other and using monochromatic radiation which has been collimated so that its angular divergence is less than the angle of misorientation of the two regions. If one region is set at the Bragg angle to diffract and produces a diffracted beam then the other region will be outside the angular range to diffract and provide no reflection. By rotation of the specimen the former region can be made to lie outside the Bragg condition bringing the latter region into a position to diffract. There will be little change in the intensity pattern by changing the specimen-film distance, save loss of resolution on the photographic emulsion.
If instead white radiation is used then both regions may produce diffracted beams but at slightly different angles, which can be detected on a photographic plate some distance from the specimen. This occurs because the two regions can select different wavelengths from those available in the continuous beam. In this case altering the specimen-plate distance enables the setup to become sensitive to orientation contrast.
One can calculate the effect of wavelength spread on the quality of topographs that can be achieved. If dl is the wavelength range corresponding to the full width half maximum (FWHM) intensity of an x-ray line, then the range of corresponding Bragg angles dql can be obtained by differentiating the Bragg equation:
n l = 2d sinq 2.1
n dl = 2 Dd sinq + 2d cosq dq 2.2
Dividing 2.2 by 2.1
dl /l = (Dd /d ) + cotq dq 2.3
For a given set of Bragg planes in a perfect crystal Dd = 0
dql = tan q (dl /l) 2.4
The image of a point on the crystal will be spread say horizontally into a length dxl, proportional to the specimen-plate distance d, on the emulsion given by:
dxl = (d) dql 2.5
Lang (1978) tabulates values for dxl for both Ag and Cu characteristic lines and d spacings of 1Å and 3.5Å, putting an upper limit of d ~ 1cm to reduce image spread.
A comprehensive account of the contrast observed on images in x-ray topography has been given by Authier (1978), the treatment distinguishes the intensities of x-rays diffracted by a crystal in terms of the geometric (directions of reflections) or kinematical (intensities of reflections) theory and the dynamical (multiple diffraction) theory. An exhaustive treatment of the dynamical diffraction of x-rays by perfect crystals is given in the review by Batterman and Cole (1964). In the kinematical theory it is assumed that the incident radiation is the same for the entire volume of crystal and therefore the scattering from each element is independent from the rest, generally applicable to very thin or imperfect materials. On the other hand the dynamical theory takes into account all wave interactions within the sample especially when large single perfect crystals are used. In this case the observed integrated diffracted intensity becomes smaller than the value predicted by the kinematical approach. This reduction in the observed intensity is known as extinction.
A quantity known as the extinction length x is given by:
x = p V cos qb / l re |F| C 2.6
The extinction distance gives the depth of crystal that is imaged in an x-ray reflection topograph. This quantity is used in preference to the penetration distance 1 / m, where m is the linear absorption coefficient given by the formula for photoelectric absorption:
I = I o exp (-m t) 2.7
If the strain field locally around a defect is changed over a distance comparable to or greater than the extinction distance then the defect will cause image contrast in an x-ray topograph.
References
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