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X-Ray Topography
342 X-Ray Topography
X-Ray Topography J Baruchel and J Ha¨rtwig, ESRF, Grenoble, France & 2005, Elsevier Ltd. All Rights Reserved.
V
Introduction A major use of X-rays, ever since they were discovered over a century ago, has been and still is the visualization of the inside of systems, opaque for other probes. Whatever the imaging process used, it implies an inhomogeneous response of the sample to the beam. Recently, X-ray imaging techniques have dramatically progressed, in connection with the improvements in sources (synchrotron radiation), detectors, and computer image processing. Among these new features, three-dimensional imaging with spatial resolution in the micrometer range (microtomography), the sensitivity enhancement associated with phase contrast imaging, and new X-ray microbeam-based imaging approaches such as fluorescence or diffraction microtomography are discussed below. This section is devoted to another way of obtaining X-ray images, based on Bragg diffraction: the ‘‘X-ray topographic’’ techniques (the ‘‘historical’’ name is retained: it is widely used, but misleading because it suggests that these techniques are surface techniques, whereas very often they are used to probe the bulk of single-crystal samples). They were mostly developed in the late 1950s and 1960s to visualize defects and distortions in crystals. Work performed on silicon made the routine production of large perfect crystals, as used by the microelectronics industry, possible. These techniques are nondestructive and sensitive to a large range of distortions (typically in the 10 3–10 8 range). Understanding the main contrast mechanisms implies knowing how crystal imperfections modify the diffraction behavior of the underlying perfect crystal. The dynamical theory of diffraction describes this behavior. The specific contributions of synchrotron radiation (and, briefly, neutrons) are outlined.
Principle of the Topographic Techniques X-ray topography is an imaging technique for single crystals based on Bragg diffraction. The local variations of the amplitude or direction of the diffracted beam provide information on crystal inhomogeneities such as defects, domains, or phases. This information is often averaged out, and lost, when using usual diffraction methods.
Detector C S
Figure 1 Principle of the topographic methods.
Figure 1 schematically represents the main idea of all the diffraction imaging methods. The single-crystal sample C is set to diffract a fraction of the incoming beam, according to Bragg’s law, 2d sin yB ¼ l. The diffracted beam is recorded on a position-sensitive detector (usually X-ray films, nuclear plates or, increasingly, CCD cameras equipped with a highresolution scintillator). If a small volume V within C behaves differently from the matrix, intensity variations (‘‘contrast’’) can be produced on an area S of the two-dimensional image produced by the diffracted beam. This contrast only occurs if the platelet-shaped crystal displays inhomogeneities. The inhomogeneities that can be visualized using X-ray diffraction topography include defects such as dislocations (Figure 2), twins, domain walls, inclusions, impurity distribution, and macroscopic deformations, such as bending or acoustic waves, present within the single-crystal sample. Figure 3 is an example of such a topographic image. It shows the image recorded using the 444 reflection from a flux-grown platelet-shaped galliumsubstituted yttrium iron garnet crystal (Y3Gax Fe5 xO12, with xB1, Ga–YIG) using the MoKa1 radiation from a standard X-ray generator. In addition to a distorted region on the upper corner, associated with the crystal fixation, various features are observable on this topograph: dislocations, growth striations and growth sectors, magnetic domains, dissolution bands, and the initial stage of a crack. These features are observed because they entail a distortion of the crystalline lattice. It is worth giving the typical value of these distortions, which include either a lattice rotation dy, or a lattice parameter variation Dd/d, or a combination of both. For the growth striations, which are the main observable features, this distortion is in the 10 4 range, whereas the variation of magnetostriction associated with the
X-Ray Topography 343
(Figure 1), two ‘‘wavefields’’ propagate within the crystal if the direction of the incident beam propagation vector lies within an angle oh around the exact Bragg position (oh the intrinsic width of the diffraction curve, often called the Darwin width, being of the order of 10 5–10 6 rad):
b
oh ¼ ð2l2 Cp jFh jr0 Þ=ðpVc sin 2yB Þ
Figure 2 Schematic drawing of a dislocation, characterized by its line direction (the border of the incomplete atomic plane in this case) and the Burgers vector b (in this case – ‘‘edge’’ dislocation – horizontal and perpendicular to the line direction).
Growth striations Growth sectors
h
Initial stage of crack
1 mm
½1
where Cp is the polarization factor (Cp ¼ cos 2yB for p polarization, i.e., for the electric field vector E in the scattering plane, and Cs ¼ 1 for s polarization, i.e., for E perpendicular to this plane), r0 is the ‘‘classical electron radius’’ (2:8 1015 m), Vc is the unit cell volume, and Fh is the structure factor corresponding to the Bragg reflection used. A wavefield corresponds to the superposition of one diffracted and one forward-diffracted wave, coupled because the perfect crystal is set for Bragg diffraction. These waves decouple outside the crystal, producing two beams, parallel to the incident and diffracted directions, respectively (Figure 4). In the case of a monochromatic spherical wave, wavefields propagate and interfere within the whole ‘‘Borrmann’’ fan or Borrmann triangle ABC (Figure 4). The characteristic coupling length between wavefields is L0 (extinction length, or Pendello¨sung period), which corresponds, in the case considered, to the distance necessary for most of the energy participating in the diffraction process to move from the transmitted to the diffracted direction, or vice versa:
Magnetic domains
L0 ¼ ðpVc cos yB Þ=ðlCp jFh jr0 Þ
½2
Dissolution bands Dislocations Figure 3 Topograph of a flux-grown Ga–YIG crystal.
magnetic domains, as well as the distortion between growth sectors, are in the 10 6 range. The distortion associated with dislocations is a function of the distance to the core; at the level of the image border, it is in the 10 5 range. Large contrast on the image can, therefore, correspond to rather weak distortions, showing that these techniques are very sensitive.
Typical values of L0 are in the 1–100 mm range. If the crystal is rotated in a monochromatic beam, or if the beam is polychromatic, the diffracted power is a function of this ‘‘variable’’ (angle, or wavelength), leading to the integrated reflectivity r, defined as the area under the diffraction curve. Figure 5 shows r for a nonabsorbing perfect crystal, as a function of t/L0, t being the thickness of the plateletshaped crystal, for both the transmission (‘‘Laue’’) and reflection (‘‘Bragg’’) cases. These curves show that *
Some Results of Dynamical Theory A few results of the dynamical theory, necessary to understand the contrast mechanisms, are given below. The discussion is restricted to symmetrical transmission geometry as shown in Figure 1. In the case of a monochromatic plane wave beam, and a perfect crystal set in Bragg diffraction position
*
*
r is proportional to t for t{L0 (‘‘kinematical’’ limit of the dynamical theory); r is nearly constant for tcL0, the saturation value being proportional to jFh j; and r displays an oscillatory behavior in the Laue case (the image produced by a low-absorption wedgeshaped crystal will, therefore, exhibit ‘‘equal thickness’’ fringes associated with the maxima and minima of the curve, and to the varying
344 X-Ray Topography
The beams diffracted by two neighboring regions I and II of the sample produce contrast on the topograph if A
2B
B
C
Figure 4 Propagation of wavefields within the ‘‘Borrmann’’ triangle ABC in the case of a perfect crystal.
Kinematical
Dynamical (Bragg case)
Dynamical (Laue case)
t /Λ Figure 5 Reflecting power r of a nonabsorbing perfect crystal as a function of the reduced thickness t /L0 for the transmission (‘‘Laue’’) and reflection (‘‘Bragg’’) cases. The kinematical approximation limit is also shown.
thickness of the crystal). Its saturation value, for tcL0 , is half of the one corresponding to the Bragg case.
Effect of Imperfections: Contrast Mechanisms For a nonabsorbing plate-shaped crystal illuminated by a parallel and polychromatic beam (a good approximation for the simplest and presently the most usual topographic technique, white beam synchrotron radiation topography), within a very simplified approach, the direction and intensity of the locally diffracted intensity depends upon * *
*
the structure factor Fh ; the angle y formed by the considered lattice planes, and the incident beam; and Y, a parameter that incorporates the modifications introduced by the crystal inhomogeneities on the perfect-crystal dynamical theory results.
1. ðFh ÞI aðFh ÞII , where the differences can either reside in the modulus or the phase or the structure factor. This structure factor contrast is the main contrast mechanism in neutron diffraction topography of magnetic domains, the magnetic structure factor often being different for the various domains (Figure 6); 2. yI ayII , that is, regions I and II are misoriented – subgrains, domains, etc. – one with respect to the other: this is orientation contrast; Figure 7 shows, for instance, white or black thick lines, corresponding to the separation, or superposition, of the images of subgrains; 3. YI aYII , the two regions display extinction contrast. Imperfections modify the perfect-crystal behavior described by the dynamical theory. They can hide the interference fringes, or lead to locally enhanced or reduced diffracted intensity, depending on the absorption conditions. New images can occur through the decoupling of a wavefield and the creation of new wavefields at a defect. In the low-absorption case, the predominant contrast mechanism is the so-called direct-image process. Assume again (Figure 1) that a polychromatic, extended beam impinges on a crystal such that a small crystal volume V contains a defect (inclusion, dislocation, etc.) associated with a distortion field, which decreases with growing distance from the defect core. A wavelength range Dl=l ¼ oh =ðtan yB Þ (B10 4) participates in diffraction by the ‘‘perfect’’ matrix crystal, whereas regions around the defect are at the Bragg position for components of the incoming beam which are outside this spectral range. The defect thus leads to additional diffracted intensity on the detector. The regions that produce the direct image of a given defect are some distance away from the core of the defect. This distance depends not only on the nature of the defect, but also on the diffraction process itself. The lattice distortion acts on diffraction through an angle, the effective misorientation dy, which reflects the change in the departure from the Bragg angle that is associated with the existence of the defect: dyðrÞ ¼ ðl=sin 2yB Þ@ðh uðrÞÞ=@ðsh Þ
½3
where h is the undistorted reciprocal lattice vector, u(r) is the displacement vector, and @/@(sh) is the
X-Ray Topography 345
(a)
(b)
IV
1 mm
II
II
(1 0 1) (1 0 0) III IV
III III
(1 1 0) (0 1 1) (c)
(d)
Figure 6 The observation, by neutron diffraction topography of antiferromagnetic domains in NiO is an example of ‘‘structure factor contrast.’’ These domains are characterized by their propagation vector, which lies along one of the four equivalent 1 1 1 directions; each type of domain has a nonzero magnetic structure factor for the Bragg reflection corresponding to its propagation vector, and zero for the other three magnetic reflections: (a) 1,1, 1, (b) 1,1,1, (c) 1,1, 1, and (d) drawing of the domain configuration – type I is missing in the investigated crystal. These images were recorded – exposure times B10 h – using a 157Gd screen acting as n-b converter and X-ray film; ‘‘white’’ means more illuminated.
h 1 mm
differentiation along the reflected beam direction. It was shown that the width of the direct image corresponds to the contribution of regions such that dy is of the order of oh. This can be understood when considering that the regions where the distortion is ooh are within the diffraction range of the perfect crystal, and that those where the distortion is coh have a negligible contribution for the considered Bragg diffraction spot. This leads to ‘‘intrinsic’’ image widths, for an edge dislocation such as the one schematized in Figure 2, BL0 j . b . hj=4 ranging in the 1–100 mm scale, and therefore implies a limitation on the density of defects that can be resolved on a topograph (B104–105 cm 2 for dislocations). This also implies that direct images are only seen in crystals that are thick enough. Indeed, the dynamical theory predicts a width oh for thin ðtoL0 =pÞ crystals inversely proportional to the thickness. If the crystal is thin, only very distorted regions with a strongly reduced volume – and consequently reduced diffracted intensity – can contribute to the direct image, which cannot therefore be observable. The effective misorientation dy allows, in addition, determining on which reflections a given defect can produce an image. Equation [3] shows that dy ¼ 0 if u is perpendicular to h. The defect is not visible on the corresponding topographs. A dislocation, for instance, is usually not visible when h . b ¼ 0, b being the Burgers vector. The invisibility criteria are powerful ways of characterizing the defects. The effective misorientation can also be written as a function of the local variation in the rotation dj of the concerned lattice planes and the local relative variation in the lattice parameter, Dd/d, as dy ¼ ðDd=dÞtan yB 7dj
½4
When taking absorption into account, the situation can change drastically. Direct images do not occur in the high-absorption case (mt410), and are replaced by dips in the intensity corresponding to the disruption of a dynamical-theory-related effect, the anomalous transmission (also called the Borrmann effect). Both types of images (‘‘additional’’ intensity and ‘‘dips’’) are simultaneously present on the topographs, as well as interference fringes (‘‘dynamical’’ and ‘‘intermediary’’ images) for samples with intermediate absorption (mtB225).
Diffraction Topographic Techniques Figure 7 White beam topograph of an Fe–3%Si crystal, E E25 keV, showing orientation contrast associated with subgrains. A fir-tree magnetic domain pattern is also visible.
As already discussed, Bragg diffraction imaging may be used either in transmission or in reflection. In addition, the beam can be extended or restricted, and the image can be an integrated or a nonintegrated one.
346 X-Ray Topography
Figure 1 shows an extended, divergent, and/or nonmonochromatic beam illuminating the whole sample, or a large area. This results in an image which can, to a first approximation corresponding to the predominance of the direct images, be considered as the projection of the defect distribution along the diffracted beam direction. The incident beam can also be restricted to a small width (typically 10–20 mm) in the scattering plane. In the same first approximation, this produces an image of those defects intercepted by the blade-shaped incident beam, hence the name ‘‘section’’ topograph. A better definition of the extended/restricted beam distinction is based on the width of the Borrmann triangle, wB ¼ 2t sin yB in the symmetrical Laue case, t being the crystal thickness. The extended beam, or projection topograph, situation corresponds to the case where the width w of the incoming beam in the scattering plane is such that wcwB . The restricted beam, or section topograph, case corresponds to w{wB . In the section topography, the position of the direct images within the image can be used to extract the depth of the related defects in the crystal, one edge of the image corresponding to the entrance face of the sample, the other to the exit surface. In very good crystals, with small or moderate absorption, the same interference effect that leads to the oscillations shown in Figure 5, produces interference fringes, called Kato’s fringes. These fringes are very sensitive to crystal distortion, and their modification is one of the first indications of a departure from the perfectcrystal situation (see Figure 11). The use of a polychromatic beam with a sufficiently wide bandwidth often offers advantages. Several reflections are recorded simultaneously, which is very helpful when characterizing defects or phenomena, and misoriented regions within a sample diffract at
p
k0
once, each region finding a wavelength l that satisfies the local Bragg condition. This ‘‘white beam’’ version of diffraction topography is identical to the wellknown Laue technique, except that the incident beam is broad. Each Bragg spot is now a topograph. This is an integrated wave diffraction image, because it results from the superposition of contributions from a range of wavelengths. It is mainly sensitive to the variations of lattice plane orientation dj, which produce a change in the diffracted beam direction, and not to Dd/d. ‘‘White beam’’ topography is widely used at synchrotron radiation facilities. The exposure time is often of the order of a fraction of a second, and it is possible to follow the evolution of any one of the Bragg spots in ‘‘real time,’’ using a CCD camera equipped with a scintillator. The incident beam on the sample can also be monochromatized. This ‘‘monochromatic wave’’ nevertheless displays a range of wavelengths (Dl/l) as well as an angular divergence o. Let oc be the diffraction width of the crystal (which reduces to oh in the case of a ‘‘perfect’’ crystal). If the divergence o (and/or (Dl/l)tan yB) 4 oc, the image is still an integrated wave topograph. If o (and (Dl/l)tan yB) { oh , the technique is called plane wave topography. It is not only suited to detect weak deformations (10 6–10 8 range), but also to obtain images of defects such as dislocations (see Figure 9) with much more details than in the case of direct images. Its use allows a quantitative analysis, because the local diffracted intensity is closely related to the effective misorientation dy. An example of the use of ‘‘plane wave’’ topography is shown in Figure 8: the seed of the investigated quartz crystal and the various growth sectors exhibit different impurity contents, which result in small variations of the lattice parameter (in the 10 5–10 7 range).
2 mm
Figure 8 Bragg case plane wave topographs of a quartz plate with nonhomogeneous distributions of impurity atoms in different growth % reflection, E ¼ 8:04778 keV, k p0 is the projection of the wave vector of sectors and in the seed plate (rectangle in the middle part), 4040 the incident wave on the crystal surface; the topographs were recorded at B40% of the maximum intensity on the low-angle side (left image) and B20% of the maximum intensity on the high-angle side (right image) of the rocking curve.
X-Ray Topography 347
h
(a)
(b) 400 µm
Figure 9 1 1 1 topographs of a Ge crystal recorded using the beam (35 keV) produced by an Si 1 1 1 monochromator: (a) top of the rocking curve, and (b) on the high-angle tail of the rocking curve (‘‘weak beam’’ image).
If an image is recorded at an angular position very far away (typically a few oc) from the center of the diffraction curve, the main contribution to Bragg diffraction originates from the distorted regions, and not from the perfect-crystal matrix. This allows reducing the ‘‘intrinsic’’ width of the images and is known, by analogy with electron microscopy, as weak beam topography (Figure 9).
Synchrotron Radiation Topography: Real-Time Investigations and Coherence-Related Effects Synchrotron radiation is now widely used for X-ray imaging. The typical exposure time to record a synchrotron radiation topograph is a fraction of a second. It is therefore possible to investigate evolving phenomena such as the movements of defects, boundaries during first-order phase transitions, or domains. Specific sample environments (cryostat, furnace, electric or magnetic field, strain, etc.) are added for each particular experiment. Image subtraction can help emphasize changes with respect to a reference state. It is possible to investigate quicker phenomena if they are periodic, through stroboscopic imaging. Choppers located on the beam path and synchronized with periodic magnetic or electric fields were used to investigate phenomena such as the motion of magnetic domains or the evolution of conduction channels, in the 10 2–10 3 s range. Stroboscopic experiments on bulk or surface acoustic waves, in the 10 8–10 9 s range, were performed using the pulsed structure of the synchrotron radiation source itself (Figure 10). The small source sizes and the large source-tosample distances, which are now common features of modern synchrotron radiation facilities, lead to highly coherent X-ray beams, which can be used for diffraction topography. In this case surface
12 µm
Figure 10 Stroboscopic topograph showing the propagation of surface acoustic waves – 12 mm wavelength – in an LiNbO3 crystal. Wave front distortions caused by the interaction of the waves with dislocations are visible. One dislocation region is framed. (Courtesy of E Zolotoyabko.)
inhomogeneities, or porosities, can be imaged in the diffracted beam even when they do not produce a strain field through their Fresnel diffraction image. Recording images at various sample-to-detector distances allow retrieving the phase jump involved in these images. The use of Bragg diffraction imaging with a coherent beam can even, in special cases, produce structural information at the atomic level, even though the spatial resolution of the images is on a much larger scale of micrometers. For instance, diffraction topographs of ferroelectric crystals, using a coherent beam, give access to a very elusive microstructural piece of information: how ferroelectric domains are connected across the domain wall and, in particular, how they match at the atomic level.
348 X-Ray Topography
Simulations of X-Ray Topographs
Conclusion
The information provided by the topographs allows a detailed quantitative analysis. This is performed through simulations. The simulation process requires introducing a deformation field with free parameters that will be refined by comparing the simulated image to the experimental one. Depending on how rapidly the distortions (expressed by the effective misorientation dyr) change locally within the crystal, this may be done using diffraction theories with various complexities. In the case of Figure 8, the Braggcase plane wave topographs of a quartz plate with nonhomogeneous distributions of impurity atoms, the local application of the dynamical theory for perfect crystals already provides very good results. In this case, the diffracted intensity can be approximated by the simple formula Ih ¼ IM RN ðyA dyÞ, where RN is the rocking curve of the setup, IM is a normalization factor, and yA is the Bragg angle in a crystal part assumed as the perfect reference one. This means that the contrast depends on the position of the ‘‘working point,’’ which is determined, through the parameter dy(r), by the local Bragg angle on the perfect reference–rocking curve. For faster varying deformations (Figure 11), a geometricoptical approximation may be chosen. The most general case requires a wave-optical approach, which is necessary to calculate, for instance, the contrast of a dislocation. The algorithms give very good results for the plane wave case.
X-ray diffraction topography is an invaluable tool to characterize single crystals and investigate many-crystal physics phenomena (creation of defects, domains, phase transitions, vibrations, etc.), whereas neutron topography remains a unique tool for the investigation of magnetic crystals, and more specially the antiferromagnetic ones. The scope of X-ray topography, when coupled with synchrotron radiation, extended to unexpected topics through the use of coherent beams or by in situ experiments, where the material, in an adequate sample environment device, is imaged while an external parameter (temperature, stress, etc.) is changed. Some of the applications, of course, require a better spatial resolution. A possibility of overcoming this limitation is to use an X-ray lens to magnify the image before the detector. Promising attempts are being made using parabolic compound refractive lenses, asymmetrically cut crystals, or bent mirrors (Kirkpatrick–Baez setup). See also: Crystal Growth, Bulk: Theory and Models; Dislocations; Scattering, Inelastic: X-Ray (Methods and Applications); X-Ray Absorption Spectroscopy; X-Ray Sources; X-Ray Standing Wave Techniques.
PACS: 07.85; 61.10; 61.50.Cj; 61.72.y; 61.72.Dd; 61.72.Ff; 64.70.Kb; 75.60.Ch; 77.80.Dj Further Reading Authier A (2001) Dynamical Theory of X-Ray Diffraction. New York: Oxford University Press. Authier A, Lagomarsino S, and Tanner BK (1996) X-Ray and Neutron Dynamical Diffraction – Theory and Applications. New York: Plenum. Bowen DK and Tanner BK (1980) Characterization of Crystal Growth Defects by X-Ray Methods. New York: Plenum. Bowen DK and Tanner BK (1998) High Resolution X-Ray Diffractometry and Topography. London: Taylor and Francis.
Nomenclature b Cp Fh
100 µm
h
Figure 11 Measured (left) and calculated (right) 422 section topographs of a 565 mm thick silicon crystal with the edge of a 150 nm silicon oxide film, E ¼ 17:48 keV.
r0 u(r) Vc dy(r) yB l L0 oh
Burgers vector of a dislocation (m) polarization factor (X-ray diffraction) structure factor associated with the reciprocal vector h classical electron radius (2:8 1015 m) displacement vector (m) volume of the unit cell (m3) effective misorientation (rad) Bragg angle (rad) wavelength (m) extinction distance (dynamical theory) (m) intrinsic width of the diffraction curve (dynamical theory) (rad)
X-Ray Topography J Baruchel and J Ha¨rtwig, ESRF, Grenoble, France & 2005, Elsevier Ltd. All Rights Reserved.
V
Introduction A major use of X-rays, ever since they were discovered over a century ago, has been and still is the visualization of the inside of systems, opaque for other probes. Whatever the imaging process used, it implies an inhomogeneous response of the sample to the beam. Recently, X-ray imaging techniques have dramatically progressed, in connection with the improvements in sources (synchrotron radiation), detectors, and computer image processing. Among these new features, three-dimensional imaging with spatial resolution in the micrometer range (microtomography), the sensitivity enhancement associated with phase contrast imaging, and new X-ray microbeam-based imaging approaches such as fluorescence or diffraction microtomography are discussed below. This section is devoted to another way of obtaining X-ray images, based on Bragg diffraction: the ‘‘X-ray topographic’’ techniques (the ‘‘historical’’ name is retained: it is widely used, but misleading because it suggests that these techniques are surface techniques, whereas very often they are used to probe the bulk of single-crystal samples). They were mostly developed in the late 1950s and 1960s to visualize defects and distortions in crystals. Work performed on silicon made the routine production of large perfect crystals, as used by the microelectronics industry, possible. These techniques are nondestructive and sensitive to a large range of distortions (typically in the 10 3–10 8 range). Understanding the main contrast mechanisms implies knowing how crystal imperfections modify the diffraction behavior of the underlying perfect crystal. The dynamical theory of diffraction describes this behavior. The specific contributions of synchrotron radiation (and, briefly, neutrons) are outlined.
Principle of the Topographic Techniques X-ray topography is an imaging technique for single crystals based on Bragg diffraction. The local variations of the amplitude or direction of the diffracted beam provide information on crystal inhomogeneities such as defects, domains, or phases. This information is often averaged out, and lost, when using usual diffraction methods.
Detector C S
Figure 1 Principle of the topographic methods.
Figure 1 schematically represents the main idea of all the diffraction imaging methods. The single-crystal sample C is set to diffract a fraction of the incoming beam, according to Bragg’s law, 2d sin yB ¼ l. The diffracted beam is recorded on a position-sensitive detector (usually X-ray films, nuclear plates or, increasingly, CCD cameras equipped with a highresolution scintillator). If a small volume V within C behaves differently from the matrix, intensity variations (‘‘contrast’’) can be produced on an area S of the two-dimensional image produced by the diffracted beam. This contrast only occurs if the platelet-shaped crystal displays inhomogeneities. The inhomogeneities that can be visualized using X-ray diffraction topography include defects such as dislocations (Figure 2), twins, domain walls, inclusions, impurity distribution, and macroscopic deformations, such as bending or acoustic waves, present within the single-crystal sample. Figure 3 is an example of such a topographic image. It shows the image recorded using the 444 reflection from a flux-grown platelet-shaped galliumsubstituted yttrium iron garnet crystal (Y3Gax Fe5 xO12, with xB1, Ga–YIG) using the MoKa1 radiation from a standard X-ray generator. In addition to a distorted region on the upper corner, associated with the crystal fixation, various features are observable on this topograph: dislocations, growth striations and growth sectors, magnetic domains, dissolution bands, and the initial stage of a crack. These features are observed because they entail a distortion of the crystalline lattice. It is worth giving the typical value of these distortions, which include either a lattice rotation dy, or a lattice parameter variation Dd/d, or a combination of both. For the growth striations, which are the main observable features, this distortion is in the 10 4 range, whereas the variation of magnetostriction associated with the
X-Ray Topography 343
(Figure 1), two ‘‘wavefields’’ propagate within the crystal if the direction of the incident beam propagation vector lies within an angle oh around the exact Bragg position (oh the intrinsic width of the diffraction curve, often called the Darwin width, being of the order of 10 5–10 6 rad):
b
oh ¼ ð2l2 Cp jFh jr0 Þ=ðpVc sin 2yB Þ
Figure 2 Schematic drawing of a dislocation, characterized by its line direction (the border of the incomplete atomic plane in this case) and the Burgers vector b (in this case – ‘‘edge’’ dislocation – horizontal and perpendicular to the line direction).
Growth striations Growth sectors
h
Initial stage of crack
1 mm
½1
where Cp is the polarization factor (Cp ¼ cos 2yB for p polarization, i.e., for the electric field vector E in the scattering plane, and Cs ¼ 1 for s polarization, i.e., for E perpendicular to this plane), r0 is the ‘‘classical electron radius’’ (2:8 1015 m), Vc is the unit cell volume, and Fh is the structure factor corresponding to the Bragg reflection used. A wavefield corresponds to the superposition of one diffracted and one forward-diffracted wave, coupled because the perfect crystal is set for Bragg diffraction. These waves decouple outside the crystal, producing two beams, parallel to the incident and diffracted directions, respectively (Figure 4). In the case of a monochromatic spherical wave, wavefields propagate and interfere within the whole ‘‘Borrmann’’ fan or Borrmann triangle ABC (Figure 4). The characteristic coupling length between wavefields is L0 (extinction length, or Pendello¨sung period), which corresponds, in the case considered, to the distance necessary for most of the energy participating in the diffraction process to move from the transmitted to the diffracted direction, or vice versa:
Magnetic domains
L0 ¼ ðpVc cos yB Þ=ðlCp jFh jr0 Þ
½2
Dissolution bands Dislocations Figure 3 Topograph of a flux-grown Ga–YIG crystal.
magnetic domains, as well as the distortion between growth sectors, are in the 10 6 range. The distortion associated with dislocations is a function of the distance to the core; at the level of the image border, it is in the 10 5 range. Large contrast on the image can, therefore, correspond to rather weak distortions, showing that these techniques are very sensitive.
Typical values of L0 are in the 1–100 mm range. If the crystal is rotated in a monochromatic beam, or if the beam is polychromatic, the diffracted power is a function of this ‘‘variable’’ (angle, or wavelength), leading to the integrated reflectivity r, defined as the area under the diffraction curve. Figure 5 shows r for a nonabsorbing perfect crystal, as a function of t/L0, t being the thickness of the plateletshaped crystal, for both the transmission (‘‘Laue’’) and reflection (‘‘Bragg’’) cases. These curves show that *
Some Results of Dynamical Theory A few results of the dynamical theory, necessary to understand the contrast mechanisms, are given below. The discussion is restricted to symmetrical transmission geometry as shown in Figure 1. In the case of a monochromatic plane wave beam, and a perfect crystal set in Bragg diffraction position
*
*
r is proportional to t for t{L0 (‘‘kinematical’’ limit of the dynamical theory); r is nearly constant for tcL0, the saturation value being proportional to jFh j; and r displays an oscillatory behavior in the Laue case (the image produced by a low-absorption wedgeshaped crystal will, therefore, exhibit ‘‘equal thickness’’ fringes associated with the maxima and minima of the curve, and to the varying
344 X-Ray Topography
The beams diffracted by two neighboring regions I and II of the sample produce contrast on the topograph if A
2B
B
C
Figure 4 Propagation of wavefields within the ‘‘Borrmann’’ triangle ABC in the case of a perfect crystal.
Kinematical
Dynamical (Bragg case)
Dynamical (Laue case)
t /Λ Figure 5 Reflecting power r of a nonabsorbing perfect crystal as a function of the reduced thickness t /L0 for the transmission (‘‘Laue’’) and reflection (‘‘Bragg’’) cases. The kinematical approximation limit is also shown.
thickness of the crystal). Its saturation value, for tcL0 , is half of the one corresponding to the Bragg case.
Effect of Imperfections: Contrast Mechanisms For a nonabsorbing plate-shaped crystal illuminated by a parallel and polychromatic beam (a good approximation for the simplest and presently the most usual topographic technique, white beam synchrotron radiation topography), within a very simplified approach, the direction and intensity of the locally diffracted intensity depends upon * *
*
the structure factor Fh ; the angle y formed by the considered lattice planes, and the incident beam; and Y, a parameter that incorporates the modifications introduced by the crystal inhomogeneities on the perfect-crystal dynamical theory results.
1. ðFh ÞI aðFh ÞII , where the differences can either reside in the modulus or the phase or the structure factor. This structure factor contrast is the main contrast mechanism in neutron diffraction topography of magnetic domains, the magnetic structure factor often being different for the various domains (Figure 6); 2. yI ayII , that is, regions I and II are misoriented – subgrains, domains, etc. – one with respect to the other: this is orientation contrast; Figure 7 shows, for instance, white or black thick lines, corresponding to the separation, or superposition, of the images of subgrains; 3. YI aYII , the two regions display extinction contrast. Imperfections modify the perfect-crystal behavior described by the dynamical theory. They can hide the interference fringes, or lead to locally enhanced or reduced diffracted intensity, depending on the absorption conditions. New images can occur through the decoupling of a wavefield and the creation of new wavefields at a defect. In the low-absorption case, the predominant contrast mechanism is the so-called direct-image process. Assume again (Figure 1) that a polychromatic, extended beam impinges on a crystal such that a small crystal volume V contains a defect (inclusion, dislocation, etc.) associated with a distortion field, which decreases with growing distance from the defect core. A wavelength range Dl=l ¼ oh =ðtan yB Þ (B10 4) participates in diffraction by the ‘‘perfect’’ matrix crystal, whereas regions around the defect are at the Bragg position for components of the incoming beam which are outside this spectral range. The defect thus leads to additional diffracted intensity on the detector. The regions that produce the direct image of a given defect are some distance away from the core of the defect. This distance depends not only on the nature of the defect, but also on the diffraction process itself. The lattice distortion acts on diffraction through an angle, the effective misorientation dy, which reflects the change in the departure from the Bragg angle that is associated with the existence of the defect: dyðrÞ ¼ ðl=sin 2yB Þ@ðh uðrÞÞ=@ðsh Þ
½3
where h is the undistorted reciprocal lattice vector, u(r) is the displacement vector, and @/@(sh) is the
X-Ray Topography 345
(a)
(b)
IV
1 mm
II
II
(1 0 1) (1 0 0) III IV
III III
(1 1 0) (0 1 1) (c)
(d)
Figure 6 The observation, by neutron diffraction topography of antiferromagnetic domains in NiO is an example of ‘‘structure factor contrast.’’ These domains are characterized by their propagation vector, which lies along one of the four equivalent 1 1 1 directions; each type of domain has a nonzero magnetic structure factor for the Bragg reflection corresponding to its propagation vector, and zero for the other three magnetic reflections: (a) 1,1, 1, (b) 1,1,1, (c) 1,1, 1, and (d) drawing of the domain configuration – type I is missing in the investigated crystal. These images were recorded – exposure times B10 h – using a 157Gd screen acting as n-b converter and X-ray film; ‘‘white’’ means more illuminated.
h 1 mm
differentiation along the reflected beam direction. It was shown that the width of the direct image corresponds to the contribution of regions such that dy is of the order of oh. This can be understood when considering that the regions where the distortion is ooh are within the diffraction range of the perfect crystal, and that those where the distortion is coh have a negligible contribution for the considered Bragg diffraction spot. This leads to ‘‘intrinsic’’ image widths, for an edge dislocation such as the one schematized in Figure 2, BL0 j . b . hj=4 ranging in the 1–100 mm scale, and therefore implies a limitation on the density of defects that can be resolved on a topograph (B104–105 cm 2 for dislocations). This also implies that direct images are only seen in crystals that are thick enough. Indeed, the dynamical theory predicts a width oh for thin ðtoL0 =pÞ crystals inversely proportional to the thickness. If the crystal is thin, only very distorted regions with a strongly reduced volume – and consequently reduced diffracted intensity – can contribute to the direct image, which cannot therefore be observable. The effective misorientation dy allows, in addition, determining on which reflections a given defect can produce an image. Equation [3] shows that dy ¼ 0 if u is perpendicular to h. The defect is not visible on the corresponding topographs. A dislocation, for instance, is usually not visible when h . b ¼ 0, b being the Burgers vector. The invisibility criteria are powerful ways of characterizing the defects. The effective misorientation can also be written as a function of the local variation in the rotation dj of the concerned lattice planes and the local relative variation in the lattice parameter, Dd/d, as dy ¼ ðDd=dÞtan yB 7dj
½4
When taking absorption into account, the situation can change drastically. Direct images do not occur in the high-absorption case (mt410), and are replaced by dips in the intensity corresponding to the disruption of a dynamical-theory-related effect, the anomalous transmission (also called the Borrmann effect). Both types of images (‘‘additional’’ intensity and ‘‘dips’’) are simultaneously present on the topographs, as well as interference fringes (‘‘dynamical’’ and ‘‘intermediary’’ images) for samples with intermediate absorption (mtB225).
Diffraction Topographic Techniques Figure 7 White beam topograph of an Fe–3%Si crystal, E E25 keV, showing orientation contrast associated with subgrains. A fir-tree magnetic domain pattern is also visible.
As already discussed, Bragg diffraction imaging may be used either in transmission or in reflection. In addition, the beam can be extended or restricted, and the image can be an integrated or a nonintegrated one.
346 X-Ray Topography
Figure 1 shows an extended, divergent, and/or nonmonochromatic beam illuminating the whole sample, or a large area. This results in an image which can, to a first approximation corresponding to the predominance of the direct images, be considered as the projection of the defect distribution along the diffracted beam direction. The incident beam can also be restricted to a small width (typically 10–20 mm) in the scattering plane. In the same first approximation, this produces an image of those defects intercepted by the blade-shaped incident beam, hence the name ‘‘section’’ topograph. A better definition of the extended/restricted beam distinction is based on the width of the Borrmann triangle, wB ¼ 2t sin yB in the symmetrical Laue case, t being the crystal thickness. The extended beam, or projection topograph, situation corresponds to the case where the width w of the incoming beam in the scattering plane is such that wcwB . The restricted beam, or section topograph, case corresponds to w{wB . In the section topography, the position of the direct images within the image can be used to extract the depth of the related defects in the crystal, one edge of the image corresponding to the entrance face of the sample, the other to the exit surface. In very good crystals, with small or moderate absorption, the same interference effect that leads to the oscillations shown in Figure 5, produces interference fringes, called Kato’s fringes. These fringes are very sensitive to crystal distortion, and their modification is one of the first indications of a departure from the perfectcrystal situation (see Figure 11). The use of a polychromatic beam with a sufficiently wide bandwidth often offers advantages. Several reflections are recorded simultaneously, which is very helpful when characterizing defects or phenomena, and misoriented regions within a sample diffract at
p
k0
once, each region finding a wavelength l that satisfies the local Bragg condition. This ‘‘white beam’’ version of diffraction topography is identical to the wellknown Laue technique, except that the incident beam is broad. Each Bragg spot is now a topograph. This is an integrated wave diffraction image, because it results from the superposition of contributions from a range of wavelengths. It is mainly sensitive to the variations of lattice plane orientation dj, which produce a change in the diffracted beam direction, and not to Dd/d. ‘‘White beam’’ topography is widely used at synchrotron radiation facilities. The exposure time is often of the order of a fraction of a second, and it is possible to follow the evolution of any one of the Bragg spots in ‘‘real time,’’ using a CCD camera equipped with a scintillator. The incident beam on the sample can also be monochromatized. This ‘‘monochromatic wave’’ nevertheless displays a range of wavelengths (Dl/l) as well as an angular divergence o. Let oc be the diffraction width of the crystal (which reduces to oh in the case of a ‘‘perfect’’ crystal). If the divergence o (and/or (Dl/l)tan yB) 4 oc, the image is still an integrated wave topograph. If o (and (Dl/l)tan yB) { oh , the technique is called plane wave topography. It is not only suited to detect weak deformations (10 6–10 8 range), but also to obtain images of defects such as dislocations (see Figure 9) with much more details than in the case of direct images. Its use allows a quantitative analysis, because the local diffracted intensity is closely related to the effective misorientation dy. An example of the use of ‘‘plane wave’’ topography is shown in Figure 8: the seed of the investigated quartz crystal and the various growth sectors exhibit different impurity contents, which result in small variations of the lattice parameter (in the 10 5–10 7 range).
2 mm
Figure 8 Bragg case plane wave topographs of a quartz plate with nonhomogeneous distributions of impurity atoms in different growth % reflection, E ¼ 8:04778 keV, k p0 is the projection of the wave vector of sectors and in the seed plate (rectangle in the middle part), 4040 the incident wave on the crystal surface; the topographs were recorded at B40% of the maximum intensity on the low-angle side (left image) and B20% of the maximum intensity on the high-angle side (right image) of the rocking curve.
X-Ray Topography 347
h
(a)
(b) 400 µm
Figure 9 1 1 1 topographs of a Ge crystal recorded using the beam (35 keV) produced by an Si 1 1 1 monochromator: (a) top of the rocking curve, and (b) on the high-angle tail of the rocking curve (‘‘weak beam’’ image).
If an image is recorded at an angular position very far away (typically a few oc) from the center of the diffraction curve, the main contribution to Bragg diffraction originates from the distorted regions, and not from the perfect-crystal matrix. This allows reducing the ‘‘intrinsic’’ width of the images and is known, by analogy with electron microscopy, as weak beam topography (Figure 9).
Synchrotron Radiation Topography: Real-Time Investigations and Coherence-Related Effects Synchrotron radiation is now widely used for X-ray imaging. The typical exposure time to record a synchrotron radiation topograph is a fraction of a second. It is therefore possible to investigate evolving phenomena such as the movements of defects, boundaries during first-order phase transitions, or domains. Specific sample environments (cryostat, furnace, electric or magnetic field, strain, etc.) are added for each particular experiment. Image subtraction can help emphasize changes with respect to a reference state. It is possible to investigate quicker phenomena if they are periodic, through stroboscopic imaging. Choppers located on the beam path and synchronized with periodic magnetic or electric fields were used to investigate phenomena such as the motion of magnetic domains or the evolution of conduction channels, in the 10 2–10 3 s range. Stroboscopic experiments on bulk or surface acoustic waves, in the 10 8–10 9 s range, were performed using the pulsed structure of the synchrotron radiation source itself (Figure 10). The small source sizes and the large source-tosample distances, which are now common features of modern synchrotron radiation facilities, lead to highly coherent X-ray beams, which can be used for diffraction topography. In this case surface
12 µm
Figure 10 Stroboscopic topograph showing the propagation of surface acoustic waves – 12 mm wavelength – in an LiNbO3 crystal. Wave front distortions caused by the interaction of the waves with dislocations are visible. One dislocation region is framed. (Courtesy of E Zolotoyabko.)
inhomogeneities, or porosities, can be imaged in the diffracted beam even when they do not produce a strain field through their Fresnel diffraction image. Recording images at various sample-to-detector distances allow retrieving the phase jump involved in these images. The use of Bragg diffraction imaging with a coherent beam can even, in special cases, produce structural information at the atomic level, even though the spatial resolution of the images is on a much larger scale of micrometers. For instance, diffraction topographs of ferroelectric crystals, using a coherent beam, give access to a very elusive microstructural piece of information: how ferroelectric domains are connected across the domain wall and, in particular, how they match at the atomic level.
348 X-Ray Topography
Simulations of X-Ray Topographs
Conclusion
The information provided by the topographs allows a detailed quantitative analysis. This is performed through simulations. The simulation process requires introducing a deformation field with free parameters that will be refined by comparing the simulated image to the experimental one. Depending on how rapidly the distortions (expressed by the effective misorientation dyr) change locally within the crystal, this may be done using diffraction theories with various complexities. In the case of Figure 8, the Braggcase plane wave topographs of a quartz plate with nonhomogeneous distributions of impurity atoms, the local application of the dynamical theory for perfect crystals already provides very good results. In this case, the diffracted intensity can be approximated by the simple formula Ih ¼ IM RN ðyA dyÞ, where RN is the rocking curve of the setup, IM is a normalization factor, and yA is the Bragg angle in a crystal part assumed as the perfect reference one. This means that the contrast depends on the position of the ‘‘working point,’’ which is determined, through the parameter dy(r), by the local Bragg angle on the perfect reference–rocking curve. For faster varying deformations (Figure 11), a geometricoptical approximation may be chosen. The most general case requires a wave-optical approach, which is necessary to calculate, for instance, the contrast of a dislocation. The algorithms give very good results for the plane wave case.
X-ray diffraction topography is an invaluable tool to characterize single crystals and investigate many-crystal physics phenomena (creation of defects, domains, phase transitions, vibrations, etc.), whereas neutron topography remains a unique tool for the investigation of magnetic crystals, and more specially the antiferromagnetic ones. The scope of X-ray topography, when coupled with synchrotron radiation, extended to unexpected topics through the use of coherent beams or by in situ experiments, where the material, in an adequate sample environment device, is imaged while an external parameter (temperature, stress, etc.) is changed. Some of the applications, of course, require a better spatial resolution. A possibility of overcoming this limitation is to use an X-ray lens to magnify the image before the detector. Promising attempts are being made using parabolic compound refractive lenses, asymmetrically cut crystals, or bent mirrors (Kirkpatrick–Baez setup). See also: Crystal Growth, Bulk: Theory and Models; Dislocations; Scattering, Inelastic: X-Ray (Methods and Applications); X-Ray Absorption Spectroscopy; X-Ray Sources; X-Ray Standing Wave Techniques.
PACS: 07.85; 61.10; 61.50.Cj; 61.72.y; 61.72.Dd; 61.72.Ff; 64.70.Kb; 75.60.Ch; 77.80.Dj Further Reading Authier A (2001) Dynamical Theory of X-Ray Diffraction. New York: Oxford University Press. Authier A, Lagomarsino S, and Tanner BK (1996) X-Ray and Neutron Dynamical Diffraction – Theory and Applications. New York: Plenum. Bowen DK and Tanner BK (1980) Characterization of Crystal Growth Defects by X-Ray Methods. New York: Plenum. Bowen DK and Tanner BK (1998) High Resolution X-Ray Diffractometry and Topography. London: Taylor and Francis.
Nomenclature b Cp Fh
100 µm
h
Figure 11 Measured (left) and calculated (right) 422 section topographs of a 565 mm thick silicon crystal with the edge of a 150 nm silicon oxide film, E ¼ 17:48 keV.
r0 u(r) Vc dy(r) yB l L0 oh
Burgers vector of a dislocation (m) polarization factor (X-ray diffraction) structure factor associated with the reciprocal vector h classical electron radius (2:8 1015 m) displacement vector (m) volume of the unit cell (m3) effective misorientation (rad) Bragg angle (rad) wavelength (m) extinction distance (dynamical theory) (m) intrinsic width of the diffraction curve (dynamical theory) (rad)