The
evaluation of
grain orientation data
1.
Orientation Microscopy
The prime objective
of orientation microscopy is the acquisition of grain orientation data.
They are usually represented by three Euler angles or by the rotation
matrix which rotate the coordinate system of the specimen in the
coordinate system fixed to the particular crystal lattice.
A clear graphical
representation of the microstructure is obtained by constructing Crystal
Orientation Maps (COM), or in short Orientation Maps (OM)
[1]. Hereby the orientation parameters in the measured grid points are
assigned in an image to unique shades of three basic colors that are
red, green and blue. Orientation parameters in crystal orientation maps
may be two crystallographic directions (hkl)[uvw] for two sample
reference directions ("Miller maps"), the Euler angles (φ_{1},
Φ, φ_{2}) ("Euler maps"), or the Rodrigues vector R ("Rodrigues
maps"). Finally an orientation map of the sample is obtained which
illustrates the morphology of microstructure and the spatial
orientation distribution of the grains. If abutting points in a region
have the same or similar orientations as their neighbors, they are
identified as a grain, and the intersections of such regions are called
grain respectively phase boundaries. Grains with an orientation closely
in common are represented by similar colors. It is so possible to
quantitatively represent the orientations and misorientations in
materials on the submicron scale.
The
misorientations between neighboring raster points are calculated and,
by assuming specified threshold values, grain boundaries (GB) as well
as phase boundaries can be marked out in the COM. However, a grain
boundary in a planar sample section forms a closed perimeter line which
must not „leak“. A special „path finding“ algorithm along the grain
boundary segments is applied to fill missing spots where indexing may
have locally failed, and a (binary) grain boundary network of the
microstructure is obtained. The lines may be further skeletonized to
one pixel in width. Grain boundaries
are commonly marked by the Rodrigues vector, R, the axisangle
parameters of the misorientation, Δg, or the Σ
character according to the CSL model.
Other local properties or parameters can be represented graphically by
color maps of the microstructure in a similar way, such as dislocation
density, the predominant glide systems or twin systems in the grains,
the Schmid factor or the residual deformation energy in the individual
grains [2].
The
advantages of grainboundary COM over conventional light microscopical images of
grain boundary networks are for stereological evaluation:
+ The grainboundary COM is based quantitatively on measured crystallographic
orientations rather than on tricky methods of surface (GB) etching and
imaging.
+ A high contrast on a flat background is obtained over the whole
grainboundary COM due to a yes/no discrimination of intensity levels.
+ The data are available in a digital (binary) format on a regular
raster grid.
Therefore, standard methods of quantitative metallography can directly
be applied to skeletonized grainboundary COM. In many labs sophisticated
stereological programs are already available. 2D microstructure
parameters can so be determined such as the area fraction, planar size,
average grain size, shape and arrangement of grains, and their
statistical distributions. If the phases in the material have
sufficiently different lattice constants or differ in their element
composition such that the phases can be differentiated – with the aid
of simultaneous EDS analysis – in every pixel, quantitative
metallography can be extended to a phase discriminating stereology.
Under usual stereological assumptions, 2D stereology is often extended
to 3D stereology to calculate such parameters as volume fraction,
average grain volume, 3D shape and arrangement, contiguity of phases,
and their statistical distributions. Approaches are made to reveal the
true 3D microstructure on a grain specific scale by combining BKD with
insitu serial sectioning in a FIB&SEM instrument.
Since
a grain in the grainboundary COM is represented by the group
of pixels with similar grain orientations within the GB loop, its area
fraction is simply measured by counting the raster points which are
enclosed by the grain boundary perimeter line. This pointcounting
method is superior over the lineintercept method because it is not
affected by concave sections of the grain boundary line nor by
„islands“ formed by a second grain underneath which may shine through
the sample surface. A rapid but less precise alternative for estimating size
and shape of grains is the visual comparison of a grainboundary COM with standard
grain charts or reticules.
For orientation maps
using Miller indices as orientation parameters, a color triangle is
overlaid on the stereographic standard triangle of the crystal lattice
under consideration. In case of cubic crystal symmetry red is commonly
assigned to directions near the (001) corner, green to directions near
the (011), and blue to the directions near the (111) corner. Using this
legend, colored Miller maps provide an illuminating display of
the spatial and angular distribution of crystallographic directions
related to reference directions in the sample, for instance the sheet normal
direction, ND, and the rolling direction, RD, in rolled sheets. Two Miller
maps are required, one for the {hkl} insurface planes (i.e., in case
of cubic symmetry the <hkl> directions perpendicular on the
specimen surface), and one for the <uvw> in a reference direction
in the specimen surface, to fully represent the grain orientations in
the surface. When both Miller maps are considered, the spatial and
angular distribution of crystallographic orientations is clearly
visualized.
The unique colors, on
the other hand, can be interpreted in terms of orientation parameters
by comparison with a color legend. The full grain orientation may be
reconstructed from a set of two colored Miller maps in the following
way: From the symmetrically equivalent orientations only those have
been used for imaging whose Miller indices (hkl) of the insurface
planes fall in the standard triangle 001011111. One of the
permutations of the Miller indices, [uvw], of an orthogonal direction
is given in the standard triangle 001011111. For a full description
of crystal orientation the sign and sequence of the Miller indices
[uvw] is determined considering the condition of orthogonality between
(hkl) and [uvw]. If this condition is fulfilled for more than one of
the permutations of <uvw>, these solutions are symmetrically
equivalent. They cannot be discriminated physically from each other,
because the choice of the position of the axes of the elementary
lattice cell is arbitrary.
It is worth noting
that conventional light or scanning electron microscopy images may
display abutting grains by the same color or gray shade. Therefore,
Quantitative Metallography may fail when deriving grain size
distributions or other statistical parameters from microscopy images.
In crystal orientation maps, however, grains (and phases) are
discriminated unambiguously by indicating their crystal orientation and
lattice structure. Since crystal orientation maps are available in
digital form by their way of construction, the derivation of
statistical parameters (such as the distributions of grain size, length
of grain boundaries, grain size as a function of grain orientation, and
the fractions of Σ grain boundaries) is simply reduced to pixel
counting.
___________
[1] Download D.
Gerth and R.A. Schwarzer: Graphical representation of grain and hillock
orientations in annealed Al1%Si films. Textures and Microstructures 21
(1993) 177193
[2] Download
R.A. Schwarzer: Review Paper: Automated crystal lattice orientation
mapping using a Computercontrolled SEM. Micron 28 (1997)
249265
2. The
Microstructure Function
The meso and
microstructure of a polycrystal is described by the distributions of
size, shape, arrangement, orientation and defects of its constituent
grains and phases in threedimensional space. A mathematical
quantification of this state is given by the Microstructure
Function G [1] (also named Aggregate Function) [2, 3]:
The phases i, the
crystal orientations g and the lattice defects D in the volume elements
are specified at the places r = {x, y, z} in the specimen. g = {φ_{1},
Φ, φ_{2}}
are the orientation parameters, D = {d_{1}, d_{2},
..., d_{n}} the substructure parameters, and i = { i_{1},
i_{2}, ..., i_{n}} the phases which are characterized
by their crystal lattice and element composition.
In the simplest case,
that is a singlephase material without considering the grain
substructure, the microstructure function G(r) still depends on six
coordinates that are the three spatial and the three orientation
parameters. They make up a sixdimensional space which is almost beyond
human imagination. So a split was made in traditional materials science
by either emphasizing the morphology of microstructure or the density
of grain orientations. Two branches of science have developed almost
without mutual interaction:
• Stereology or Quantitative Metallography which is
based on microscopy techniques to describe the morphology and phases,
but with omission of the grain orientations.
• Crystal Texture Analysis which until recently was
based on polefigure measurement by diffraction methods without
considering the spatial coordinates.
It is a unique
feature of orientation microscopy in the SEM that this technique
enables the acquisition of the microstructure function at a high
spatial resolution with reasonably low effort: Scanning across the
specimen surface in a raster grid yields two spatial coordinates {x, y}
of the measured location. The third dimensional coordinate {z} can be
obtained in principle by serial sectioning. The grain orientation, g,
is the prime objective of orientation microscopy and readily available
along with {x, y}. Phases are described by their crystal lattice.
Additional information about the local element composition, e.g., from
a simultaneous EDS analysis, may be helpful in phase discrimination to
rule out less likely phases before performing the lattice check.
Pattern quality is a (qualitative) measure of local dislocation density
and lattice strain. To enable a comprehensive orientationstereological
interpretation of the microstructure, pattern quality, confidence
index, and the concentrations of several elements (if measured
simultaneously by EDS) are stored along with the grain orientation data
for every point (x, y).
With the availability
of automated EBSD the route is now open to a comprehensive "Orientation
Stereology" by merging both aspects of microstructure [1].
Depending on the
application at issue, several special functions may be derived from the
universal microstructure function G(r) which are either sections
through the sixdimensional space or integrals of it. Texture
descriptors are the Orientation Density Function (ODF), f(g),
pole figures, the MisOrientation Distribution Function (MODF),
orientation correlation functions, and texture fields. Of particular
interest for applications in industry are tensorial materials
properties that can be calculated from orientation data if the
anisotropy of the property is known for the single crystalline material
[4].
______________
[1] H.J. Bunge and R.A. Schwarzer: Orientation stereology  A new
branch in texture research. Adv. Engin. Materials 3 (2001) 2539
[2] H.J.
Bunge: Texture, microsturcture and properties of polycrystalline
materials. In: R.K. Ray and A.K. Singh (eds.): Texture in Materials
Research, Oxford & IBH Publishing Co., New Delhi 1999, 344
(ISBN 8120413121)
[3] H.J.
Bunge: Texture and structure of polycrystals. In: R.L. Snyder, J. Fiala
and H.J. Bunge (eds): Defect and Microstructure Analysis by
Diffraction, Oxford University Press, New York 1999, 405
531 (ISBN 0198501897)
[4] H.J.
Bunge: Texture analysis in materials science – Mathematical
methods. Butterworths London 1982 (ISBN 0408106425)
Paper back reprint:
CuvillierVerlag, Goettingen 1993 ( ISBN
3928815814))
3. Pole
Figures and ODF
The Orientation
Density Function (ODF) and pole figures can be constructed directly
from a data set of individual grain orientations by associating the
orientations or the volume fractions of grains having these
orientations with points or discrete cells of finite size in the Euler
orientation space, on pole figures (stereographic projection) or on
inverse pole figures (standard triangle of the crystal symmetry group).
With increasing number of data points such discrete representations are
difficult to survey. So they are often normalized, smoothed, filtered
and converted to equal density line representations, or the densities
are scaled to color intervals in the plots of ODF sections, pole
figures or inverse pole figures. This method of discrete representation
appears easy and straightforward, but has drawbacks even if the Euler
plots and Pole Figure Plots are smoothed and normalized to equal
densities. First, the distortions of the Euler space for small Φ angles are usually not
taken into account (nor the distortions of the stereographic
projection) since for simplicity the Euler space is subdivided into
cells of constant angular increments. Second, the number of measured
orientations is often not high enough to guarantee statistical
relevance.
An elegant
approach to smooth and to condense the data set into a continuous
representation is the calculation of the ODF, f(g), by series expansion
into generalized spherical harmonics (T functions) after Bunge [1]:
The expansion
coefficients (termed "C coefficients") are then expressed by
V_{s} stands
for the volume fraction of grain s, and K_{s}(l) for the
convolution kernel of the expansion. Common convolution kernels are a
Dirac δ function
in case of a large number S of orientations, or Gauss type
distributions in Euler space with a half width Ψ_{0s} at 1/e
maximum of the Gauss peak at the orientation point g_{s}
[2, 3]
The C coefficients
are a highly concise and convenient description of crystal texture.
They enable the calculation of normal and inverse pole figures,
important elastic and plastic materials properties, and tensorial
properties in general [1]. ODFs calculated from EBSD and from Xray
pole figure measurements usually agree very well if the data have been
acquired on the same specimen area.
N.B.: In Xray
diffraction, the reflections and hence pole figures are indexed
according to their diffracting lattice planes. By tradition, inverse
pole figures referred to the specimen surface are also indexed
according to the insurface lattice planes rather than to the
crystallographic directions perpendicular to the specimen surface. This
inconsequent definition is of no relevance in case of cubic crystal
symmetry, but has to be kept in mind when studying materials of lower
symmetry.
______________
[1] H.J. Bunge: Texture
analysis in materials science – Mathematical methods.
Butterworths, London 1982 (ISBN 0408106425)
Paper back reprint:
CuvillierVerlag Goettingen 1993 ( ISBN
3928815814))
[2] F. Wagner: Texture determination by
individual orientation measurements. In: Experimental Techniques of
Texture Analysis, H.J. Bunge (ed.). DGMInformationsGes., Oberursel
1986, 115123 (ISBN 3883551015)
[3] J. Pospiech: The smoothing of the
orientation distribution density introduced by calculation methods.
Textures and Microstructures 2627 (1996) 8391
4.
Gentle sample preparation
The
preparation of a flat and clean sample surface without unwanted
deformation is not always an easy task. Owing to the low information
depth of EBSD and the steep sample tilt, a thin foreign surface layer
can lead to diffuse patterns. A finish by ion polishing (Ar+ at about 1
keV and flat angle of incidence) may be advisable. Ga from a FIBing
treatment, however, tend to segregate at grain boundaries in Al and Mg
samples.
Chemicalmechanical
polishing (CMP) is often a good choice, in particular if supported by
placing the sample at positive potential of some 10 V during mechanical
polishing with a suitable electrolyte (ECMP =
electrochemicalmechanical polishing).
__________
Download
R.A. Schwarzer: The preparation of Mg, Cd and Zn samples for
crystal orientation mapping with BKD in an SEM.
Microscopy Today 15/March (2007) 40, 42
0001
polefigures of a crossrolled AM20 magnesium sheet.
Initial thickness of the hot extruded material
was 7 mm.
It was
clockwise crossrolled in steps of 90° with 200 mm/s in 17 stitches
down to 0.95 mm. The rolls were kept at 100 °C. Between the rolling
steps, the samples have been annealed (recrystallized) at 420 °C for 20
min. (1.72.5 wt% Al, min. 0.2 wt% Mn, max.
0.2 wt% Zn, balance Mg)

5.
Electromigration failures in copper interconnect lines

About
1 µm thick copper layers have been electroplated on trenchstructured
SiO2 /Si substrates by the damascene technique. The trenches were
1.0 µm wide, 0.5 µm deep and PVD coated with 50 nm Ta as a barrier
layer to protect the silicon semiconductor from diffused copper. In
addition, a PVD copper seed layer has been deposited at 50 °C. After
the electroplating process the wafers were annealed for 10 min. at 120
°C. Finally the continuous copper layer has been removed by
chemicalmechanical polishing (CMP) down to the damascene trenches to
achieve narrow interconnect lines with sharp edges.
In the
crystal orientation maps the grains are clearly discriminated from each
other. Some twin grains are seen. It is worth mentioning that
individual grains are hardly recognized in SEM micrographs, since
orientation contrast with backscatter and secondary electrons is low. A
fragmented central seam of grain boundaries has formed.

The ODF and
pole figures have been calculated with the series expansion method from
the measured crystal orientations, separately for those grains which
fill an interconnect line at full width ("bamboo grains") and for those
grains which form the central seam. For both types of grain, a strong
<111> and a weak <115> fiber texture have been found. For
the bamboo grains, the fiber axes are aligned in specimen normal
direction, i.e. from the bottom to the top of the trenches, whereas for
the seamforming grains the fiber axes are directed perpendicular to
the sidewalls.
Since
<111> is the direction of fast grain growth and <115> is
the twin component to <111> in copper, texture analysis has
revealed competing columnar grain growth from the bottom and from
the sidewalls of the trenches. A variety of general grain
boundaries is so formed. By additives to the electrolytic bath and
appropriate electroplating conditions, crystal growth from the
sidewalls can be suppressed in favor of grain growth from the bottom.
Seamless interconnect lines are then obtained.
___________
A. Huot, A.H. Fischer, A. von Glasow and R.A. Schwarzer: Quantitative
texture analysis of Cu damascene interconnects.
In: O. Kraft, E. Arzt, C.A. Volkert, P. Ho and H. Okabayashi (Eds.):
Proc. 5th Intern. Workshop on StressInduced Phenomena in
Metallization, MPI Stuttgart 1999. AIP Conference Proceedings 491,
Melville N.Y. 1999, 261264 (ISBN13 9781563969041)
