PROCESSING OF HRTEM IMAGE TO ANALYZE EDGE DISLOCATIONS
Federico M. Sciammarella1, Cesar A. Sciammarella2 and Luciano Lamberti2
1
Department of Mechanical, Materials and Aerospace Engineering
Thermal Processing Technology Center
Chicago (IL), 60616 – USA
2
Dipartimento di Ingegneria Meccanica e Gestionale
Politecnico di Bari
Bari, 70126 - ITALY
ABSTRACT
Mechanical behavior of materials at the atomic level has been an area of interest since Cauchy, with his pioneering work in
continuum mechanics. Today this interest is still there especially in the materials field where the development of new
semiconductor electronic devices and circuits based on silicon carbide that can function in high-temperature, high-power,
and/or high-radiation conditions are being made. These advancements allow for better performance and ultimately more
capacity. During the growth process of these SiC wafers mechanical defects can occur resulting in a defective component.
Without the advances made in electron microscopy and experimental methods in fringe analysis the study of these types of
defects would not be possible. This paper presents techniques developed to analyze displacements fields via electron
microscopy. Here we come full circle with Volterra’s pioneering work by analyzing strain fields in regions where the crystalline
structure is distorted.
Introduction
Since the early 19th century, efforts have been made to relate material properties to the interactions that occur at the atomic
level. These labors can be traced back to Cauchy with his attempt to relate continuum mechanics to the atomistic structure
that resulted in what is known today as the Cauchy–Born rule [1]. It is well known that macroscopic mechanical deformations
are the result of the interaction and organization of a very large quantity of atoms. The main objective of research in this topic
is to study the behavior of materials at the nanoscale to understand how and why materials deform and fail. In order for the
research to be feasible in this topic, it is necessary to develop models that contain atomistic information. These models must
include fine-scale features without introducing excessive computational demands.
To get information concerning the fine-scale structures has required enormous experimental developments. By combining Xrays with the use of the electron microscope details of the organization of matter have been investigated and the arrangement
of atoms in complex structures revealed. These breakthroughs in technology have made it possible to get a description of the
structure of matter from the geometric point of view. This information has enabled developments in the kinematics of
Continuum Mechanics. Basic concepts have been extended from the macroscopic field to the structure of crystalline matter. A
large amount of the work done has been connected with the application of the theory of Elasticity, relating the internal local
forces in the crystalline structure as defined by the stress tensor to the deformations defined by the strain tensor through
constitutive equations. Interestingly enough the concept of dislocation was introduced as a mathematical tool to discuss the
unicity of the displacement function in the theory of Elasticity, Volterra [2] in 1907, introduced the concept of dislocation. This
idea was further explored by Somigliana [3] and Love [4]. Volterra introduced basic types of dislocations that today are well
known in the mechanics of fracture. This abstract concept came back to the theory of the kinematic of solids as a result of the
need to explain discrepancies between simple models of the theoretical strength of a solid and experimentally measured
values (see, for instance, the work of Taylor [5], Orowan [6] and Polanyi [7]). Since then a whole discipline has come into
existence, the Mathematical Theory of Dislocations. This discipline analyzes the interactions of atoms in the crystalline
structure during the process of plastic deformation.
The development and understanding of atomistic and continuum models has greatly expanded beyond the classical disciplines
that we have mentioned and today there is a formidable array of tools available to formulate problems that involve the
kinematics and the dynamics of crystalline arrays. A good summary of this topic can be found in [8].
Returning to the subject of dislocations, X-ray analysis through the 3-D Fourier Transform, (FT), provided the experimental
evidence of the presence of dislocations in crystalline structures. The first images of edge dislocations in the crystalline
structure came in the 1960’s with the use of X-rays. This visualization was made through X-ray photogrammetry, using x-rays
recordings and an optical reconstruction. In the 1970’s the use of the electron microscope provided another tool to look at the
structure of crystalline structures and the actual images of dislocations were obtained with transmission electron microscopy
(TEM).
The visualization of dislocations and the images of defects in the crystalline structure have not lead to a quantitative analysis of
the displacement information contained in the images. This paper presents the application of methods of fringe pattern
analysis that have been developed through many years using techniques that provide strain and displacement information [910]. The image of a dislocation in a crystalline structure obtained by using high resolution electron microscopy (HRTEM) is
analyzed with the previously mentioned numerical tools on the basis of the theory of fringe dislocations [11].
A brief explanation of how the analyzed images are obtained via HRTEM is given to provide the reader a basic picture. The
images obtained in HRTEM are similar to that of a microscope, Fig. 1.
Image Formation
Although the paper is devoted to the analysis of the image of a dislocation obtained with HRTEM we will present a brief
explanation of how the images are obtained via HRTEM. This explanation will help the understanding of the image analysis
and at the same time provide some insights on the relationship between image formation and the data retrieval process. The
images obtained in HRTEM are similar to the images produced by a microscope illuminated by coherent light, Fig. 1. There is
an illuminating beam that is focused on an object, an aperture diaphragm, a focusing lens. Using Fourier optics terminology
the objective lens creates the diffraction pattern of the object in its back focal plane, and then through the diaphragm it
performs an inverse FT. The direct beams through the specimen interfere with the diffracted beams and produce an image of
the object. The incident wave front of the electron beam, ideally a plane wave front, interacts with the structure of the object.
Figure 1. Schematic of High Resolution Transmission Electron Microscope (HRTEM)
Information detection can be represented by the conventional mathematics of amplitude and phase modulation. The emerging
wave front E(r) eφ(r) contains information of the object structure. The lens has aberrations (astigmatism, spherical, chromatic),
and these aberrations are included in the information retrieved by the wave front. The information is captured as an intensity
distribution. This intensity distribution is a function of the microscope contrast transfer function. In order to get high resolution
the specimen thickness must be very small, in some cases on the order of 10 nm. The information resolution depends on the
instrument properties and in the coherence of the electron beam, and is degraded by the presence of the aberrations. The
microscope image interpretation is helped by the use of specialized software to introduce corrections. Hence the resolution
that can be obtained depends greatly on the analysis of the contrast transfer function. Since unit cell distances are on the
order of 10-1 nanometers in order to get strain information resolution must be on the order of 500 pm. Only recently have
technological advances made it feasible to get high quality HRTEM images and only the central portion of the beam can be
utilized to make the observations. Outside of this region the image is highly distorted by spherical aberration. Hence to obtain
quality HRTEM images requires the use of recent high quality transmission electron microscopes.
Image Analysis
The image analyzed in this paper was obtained from Ref. [12]. The reason for the selection is the extensive information that
enables us to perform the investigation. The basic unit of a hexagonal polytype of a SiC is shown in Fig. 2 where a1=a2=a3.
Figure 2. (0001) plane marked in grey and [0001] direction marked blue, [112 0] direction marked purple [1100]
direction marked pink.
Figure 3. Dislocation present due to the (1120) plane slipping in [1100] direction adding extra atoms on top of (0001).
Figure 3 shows optical micrograph of the etch pit bands of Potassium Hydroxide (KOH) etched Si (0001) face of a 4H SiC
wafer with etch pit bands along <11-20> directions. These bands are evidence of threading edge dislocations. One can see
the picture that is analyzed. Black dots in the picture of bottom left indicate the rows of extra atoms that cause the dislocation
in the crystal lattice. The bright dots indicate the position of the atoms in the crystal and the dark array of lines shows the interatomic spaces. The array of atoms is observed from the top and one assumes that the array positions repeat through the
thickness of the thin slice of the crystal that is being subjected to observation.
Analysis of the Deformations
Figure 4 shows the atomic arrangement in the (0001) with the extra-planes causing the deformation of the crystal indicated by
the red dots. The green dots show the hexagonal structure, for reference the crystallographic axes of the HCP are also shown
in this figure.
The bright dots are the positions of the atoms and the dark areas are the inter-atomic spaces. We can imagine this array to
repeat vertically in depth and that we have a normal view. In reality the impinging electrons are not in plane wave-fronts. The
electrons travel in wave packets that have slightly different plane orientations and slightly different wavelengths. Upon arrival to
the surface of a metal, some of the electrons will experience reflection, other will be scattered, and some will pass through.
The different proportions depend on the atomic structure of the metal. A more advanced view of the process of the image
formation is that the impinging electron packets will be absorbed by the electron layers on the surface of the thin foil prepared
for the HRTEM. This process will generate a complex system of electromagnetic waves that will cause an electron to be
emitted at the other end of the assumed trajectory of the incoming electron, in the outgoing face of the foil. The emitted
electron will have a memory of the passage of the wave through the thin foil and at the end the image lens will reproduce an
image of the structure array with all the imperfections arising from the different sources of noise present in the system. These
imperfections are the cause of the difference between the ideal mathematical model of the microscope and the actual physical
realization of the microscope. The optical equivalent would be a 2-D array of dots that it is illuminated by a quasimonochromatic plane wave-front parallel to the plane of the array and is imaged on a plane by a lens system. The lens system
produces an FT of the object in the focal plane, and depending on the aperture of the system, it will let some of the spatial
frequencies go through to form an image at the image plane. If the array is distorted the analysis of the geometry of the
distorted array (see [9] and references given) will provide the continuum mechanics rendering of the kinematics of the
deformation field, a second order tensor field. If the field is continuous the displacement field will be a continuous single valued
function [11].
Figure 4. Image analyzed showing crystallographic axis in the region of the dislocation. Extra-atomic planes
are indicated in red, the green dots outline the hexagonal structure outline.
If there are internal sources of displacement, then the displacement will be at a given region a multi-valued function and a
fringe dislocation as defined in [11] will be present. Within this framework of concepts and limitations arising form the
idealization of the 3-D array of atoms as a superposition of identical 2-D arrays stacked in the vertical direction we proceed to
the analysis of the kinematics of the deformation of the crystal. The analogy between the optical process and the electronic
process can not be over extended. In the optical process the photons follow the intensity reflectivity or transmisivity of the
surface that they are reflected from or they go through. In the electron microscopy case the intensity depends on the atomic
structure of the material that is involved in the imaging process. Atoms are shown as bright spots and the inter-atomic regions
appear increasingly dark. The image is a map of the electronic density distribution. In the point of view that electrons do no go
through the metal but are absorbed in the receiving end, setting electromagnetic waves inside the metal that finally cause an
electron to be emitted at the other end of the field, one would think that emissions will be larger where the electronic density is
greater, around the core of the atom and lesser in the periphery regions. The above arguments are required to give meaning to
the displacement field observations and the gradients.
Determination of the Displacement Field
Using the similitude of the optical process and the electronic process and restricting ourselves to a two dimensional analysis
as mentioned before, we have a two dimensional array of bright spots that we can consider a moiré pattern grid. To this grid
we can apply all the methodology described in [9]. To apply the moiré pattern analysis software it is necessary to start from a
reference array from which the kinematics of the deformed array will be computed. From the literature and from the analysis of
the pattern a value for the parameter a was selected.
Figure 5. FFT pattern of the crystal structure shown in Fig. 4 (a) and equivalent 120° strain rosette (b)
Lattice parameter a=0.3073 nm at 300 K
Figure 5 shows the FFT of the crystal structure and the axes of the frequency space. The diffraction pattern shown in the figure
corresponds to the structure lying inside a hexagon. In place of using the Cartesian orthogonal system of reference, we can
follow the symmetry of the crystal and use a hexagonal system. We can get from this FFT three systems of fringes at
o
orientation 120 to each other. We have the equivalent of a three elements rosette, each at 120° to each other. The
correspondence between rosette orientations and directions (orthogonal to fringe orientation) in the FFT space is also shown
in the figure. Calling a, b, c the axes of the rosette, principal strains can be expressed as a function of the strains measured
along the rosette axis:
a)
ec
c)
b)
eb
ea
Figure 6. Phases of the harmonics that modulate the structure of the crystal: a) -60°, b) +60° and c) 0° in the FFT space
The displacement field can be represented by the following vector equation,
(1)
The derivatives can be computed from the displacements
(2)
(3)
(4)
(5)
where θ is the rigid body rotation.
The components of the Almansi’s non-linear strain tensor referred to the Eulerian coordinates can be computed from
derivatives (2-6). It follows,
(6)
(7)
(8)
Finally the principal strains can be computed for the strain tensor. Figure 7 indicates where the point of maximum principal
strain is located. The location is indicated at point 1 and in the figure to the right the image has been enlarged. The blue dot
corresponds to the atom that is close to the extra layer of atoms causing the dislocation.
37.93°
Figure 7. Close up of the region of large deformations where the two extra-atomic planes converge.
The principal strains are: ε1=0.3593, ε2=−0.5886. The direction of ε1 is inclined by 37.93o with respect to direction ea.
Consequently, the direction of ε2 is inclined by 127.93o. The rigid body rotation is 1.6o .The principal direction of tension is
almost the direction [11-20] described in the work carried out in [12]. Amazingly it has been shown here that (as expected) the
direction of maximum tension is close to the gliding plane.
Figure 8. Displacement derivatives in regions surrounding fringe dislocations along rosette axes b and c
Enlarged views are also provided. In the case of small deformations and rotations the derivative
will be the strain.
Discussion
The fringe analysis provides important information that is being retrieved from the image of the structure produced by the
electron microscope. The presence of fringe dislocations in the fringe patterns indicates the place where the strains in the
array analyzed as a 2-D grid, reach a maximum. The points are located inside the atomic field and are positioned in the interatomic space. For example, for the strain at a point on Fig. 7, the Eulerian form of the Green’s tensor was utilized. Strains in
this region must be compressive in one direction because in this region extra atomic planes have been introduced during the
crystal growth process. The maximum negative strain takes place a little bit up and left with respect to point 1 shown in Fig.7
(distance is too small to be shown in the figure).
This appears to be a very meaningful result since the core region of the atom should behave as a very rigid sphere. Looking
at the overall field, (although Fig.8 does not directly provide the principal strains that must be calculated using the Almansi’s
non linear tensor) the plotted derivatives show the typical strain distribution that is obtained from the analysis of a Volterra’s
type of edge dislocation. For the first time it is possible to have an idea of the magnitude of the deformation caused by
dislocations generated during the growth stage of a 4HSiC crystal in the region where the cooling rates have altered the
regularity of the crystalline array. In reality, there is not just a single dislocation but there are two dislocations which are
interacting with each other as it is shown in Fig. 6.
Moiré method provides the Eulerian components of the strain tensor, that is, the strains at the location in the deformed
configuration. It was also observed that at point “c” (X-coordinate is 4.271 nm) of Fig. 7 the lattice constant is very close to the
value for the perfect crystal (0.3073 nm). This indicates the passage from compression to tension in the array, a sort of neutral
region.
The presented analysis is the result of applying the same software Holo Moiré Strain AnalyzerTM [13] (HOLOSTRAIN) that has
been utilized for many years in the macroscopic and microscopic analysis of strain and has yielded excellent results for
deformations in the nanometric range. This software is based on the Continuum Mechanics approach to the analysis of the
deformation of the continuum. It is a geometrical approach based in the accurate measurement of the position of reference
signal that is embedded in the analyzed surface. The quantities measured are distances and the variation of distances is
transformed into angular variables by introducing the concept of phase of the signal. The accuracy of the results depends on
the accuracy of the reference signal and the accuracy with which the reference signal and the modulated signal can be
measured. The Continuum Mechanics requires continuum functions of the C3 type. This means that the function that provides
displacement must be a holomorphic function. In this case the presence of a dislocation indicates the fact that there is a
source of displacements inside the crystal. The ability of recovering a displacement function depends on satisfying the
Nyquist’s condition. If this condition is satisfied it is possible to recover the function with a degree of accuracy dictated by the
number of harmonics that is possible to capture. Of course this concept is independent of the relative size involved whether
measuring large objects or features in the nanometric range. As always, the accuracy of results depends on the signal to noise
ratio. If there is a good signal and low enough noise it is possible to recover the information. What can be said in this particular
case, although it may seem strange that strains at the level of fraction of nanometers were obtained, is that there is a
description of the displacement field on a purely geometric basis without considering the physical properties of the medium
involved.
A different topic, of course, is the relationship of the elastic solution to the actual atomic array. Volterra’s solution is applied
generally to cylinders that model the region under analysis. The core of this cylinder is hollow: this hollow region represents the
region where the elastic analysis does not apply. What occurs in this region depends on the atomic structure of the particular
material being considered. What was observed in the analysis of the distorted array, because of the technique applied, is
independent of the particular material considered. The only point to analyze is whether the classical differential geometry
approach to Continuum Mechanics is good enough to model the events taking place in the small region that is being
considered in this study.
Remarkably, no analytical solution is currently available for the case analyzed here although the problem of interactions
between edge dislocations in crystals was solved many years ago. It is our hope to encourage a further dialogue and or
investigation into this matter.
References
[1] Born, M. and Huang, K., Dynamical Theory of Crystal Lattices. Oxford University Press, 1954
[2] Volterra, V., Ann Ec. Norm. 24, 401, 1907.
[3] Somigliana, C., Atti Reale Accademia dei Lincei, 23, 1914, 463
[4] Love, A.E.H. A treatise on the mathematical theory of elasticity, Cambridge, 1927.
[5] Taylor, G.I. Proceedings of the Royal Society. A145, 362, 1934.
[6] Orowan, E., Z. Phys, 89, 634, 1934.
[7] Polanyii, M.Z., Z. Phys, 89, 660, 1934.
[8] Liu, W.L., Karpov, E.G., Zhang, S. and Park, H.S. “An Introduction to Computational Nano-Mechanics and Materials”,
Computer Methods in Applied Mechanics and Engineering, 193, 1529-1578, 2004.
[9] Sciammarella, C.A. “Overview of optical techniques that measure displacements”, Experimental Mechanics, 43, 1-19, 2003
[10] Sciammarella, C.A. and Sciammarella, F.M. “Measuring mechanical properties of materials in the micron range using
electronic holographic moiré”, Optical Engineering, 42, 1215-1222, 2003.
[11] Sciammarella, C.A. and Sciammarella, F.M. “Properties of isotethic lines”. In Fringe 2005. The 5th International Workshop
on Automatic Processing of Fringe Patterns, 54-64, Springer Verlag, Stuttgart (Germany), September 2005.
[12] Skowronski, M. Sources of threading edge dislocations. Available on line at:
http://neon.mems.cmu.edu/skowronski/Silicon%20carbide/%20unit/%20cell.htm
[13] General Stress Optics Inc., Holo-Moiré Strain Analyzer software HoloStrain™ Version 2.0, (2004). www.stressoptics.com