hollow-cone dark-field transmission electron microscopy for

Hollow-cone dark-field transmission electron microscopy for grain size

and dislocation-density quantification of nanocrystalline materials

B. Yao1,2, H. Heinrich

2, 3, K. Barmak

4, K. R. Coffey

1,2, Kyu Cho

5, and Y.H. Sohn

1,2

1Advanced Materials Processing and Analysis Center, University of Central Florida, FL 32816, USA 2Department of Mechanical, Materials and Aerospace Engineering, University of Central Florida, Orlando, FL 32816,

USA 3Department of Physics, University of Central Florida, Orlando, FL 32816, USA 4Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA 5 U.S. Army Research Laboratory, Weapons and Materials Research Directorate, Aberdeen Proving Ground, MD 21005,

USA.

Methodologies based on hollow-cone dark-field (HCDF) transmission electron microscopy (TEM) are described to study

grain size and dislocation density in nanocrystalline materials. In order to accurately quantify the nanocrystalline grain

size, high quality TEM images are required where each grain has adequate contrast to its neighbor grains. Conventional

bright-field and dark-field TEM images often cannot provide this contrast for all the grains in the field of view. In this

study, HCDF-TEM imaging, with a slightly convergent beam, as an improved technique was found to be suitable to form

high contrast micrographs for nanocrystalline grain size quantification. A variant geometry of HCDF can also be used to

examine the dislocation structure. Compared to the conventional two-beam approach to form weak-beam dark field

(WBDF) TEM images for dislocation characterization, the method described in this study is readily applied to provide

high-quality WBDF-TEM images, and it is suitable to study both samples with coarse grains and those with ultrafine

nanocrystalline grains. Examples from both thin film and bulk nanostructured samples are provided.

Keywords hollow-cone dark field; nanocrystalline; grain size; dislocation density

1. Introduction

Nanocrystalline materials with a grain size below ~100 nm exhibit a variety of unique mechanical, chemical, and

physical properties that allow new applications and technologies [1-3]. For example, it has been reported that the yield

stress and microhardness of nanocrystalline materials can be 2-10 times higher than the corresponding coarse-grained

polycrystalline materials with the same chemical composition [4]. Electronic properties are also affected, as noted in

recent work wherein the scattering of conduction electrons at the grain boundaries of nanometric Cu films has been

shown to provide the most significant contribution to the resistivity increase in the films as thickness is reduced [5].

The interesting properties of nanocrystalline materials are due to their unique microstructure. The grain size and

density of dislocations are two of the most important microstructural parameters. The grain size is directly correlated to

the volume fraction of atoms at interfaces, which are responsible for many of the special properties of nanocrystalline

materials [6]. The small grain size of nanometric materials may also induce other nano-scale effects [7]. The

dislocation density strongly influences the strength and ductility of materials [8, 9]. It is known that most coarse-grain

metals or alloys can be hardened by increasing the dislocation density. Simultaneously, an increase in dislocation

density (e.g., through work hardening) typically results in a decrease of the ductility of materials, as it reduces the

capacity for further dislocation movement and accumulation [8]. The small size of nano-grains, however, may limit the

conventional operation (e.g., nucleation and growth) of dislocation sources, which results in a low ductility of

nanocrystalline materials. Recently, it has been reported that partial dislocations can be generated at grain boundaries

[9], which provides an approach to obtain nanocrystalline materials with both a high strength and plasticity.

Although many methods are available for grain size and dislocation density characterization, transmission electron

microscopy (TEM) is obviously one of the most powerful techniques. X-ray diffraction methods, based upon

diffraction line width, may provide an “average grain size” and dislocation density. However, such measurements are

sensitive to other factors that influence diffraction line width [10]. TEM-based grain size measurement techniques also

allow the distribution of the grain sizes, and the grain shapes to be determined. Similarly, the dislocation density and

distribution, the interaction of dislocations with precipitates or grain boundaries can be achieved from TEM

micrographs, given that these features are imaged appropriately.

In this paper, methodologies based on hollow-cone dark field (HCDF) TEM imaging for grain size and dislocation

density characterization are described and demonstrated. The HCDF-TEM technique has been employed widely for

precipitates or secondary phase identification [11, 12]; orientation mapping of nano- and micro- grains [13-15]; and

phase quantification of materials with multiple phases [16]. In this paper, an introduction to a general HCDF-TEM

technique will be given first (section 2). The balance of the paper will describe the use of this technique to examine the

grain size (section 3) and dislocation density (section 4) of nanocrystalline materials. A comparison of the HCDF-TEM

technique with other TEM techniques will also be presented.

Microscopy: Science, Technology, Applications and Education A. Méndez-Vilas and J. Díaz (Eds.)

©FORMATEX 2010 1319

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2. Introduction to the HCDF-TEM technique

The HCDF-TEM imaging described in this work was conducted with a FEI/Tecnai F30 TEM/STEM microscope with a

field emission gun operating at 300 keV. In the HCDF-TEM mode, the incident electron beam is tilted at a fixed angle

with respect to the optical axis of the objective lens, and a rotation of the tilted beam forms a hollow-cone illumination.

A ray diagram of the HCDF-TEM is shown in Fig. 1(a). On the viewing screen of the microscope, the diffraction

pattern is deflected to allow a portion of a selected diffraction ring (e.g., 111 or 200) to coincide with a centered

objective aperture, as is normally done in conventional centered dark-field imaging. HCDF-TEM images are formed as

the position of the diffraction pattern over the objective aperture is circumferentially scanned as the electron beam is

rotated at the fixed angle to the optical axis. The geometry of the rotation of the diffraction pattern over the aperture is

illustrated in Fig. 1(b). The beam rotation can be conducted automatically via the microscope control software, or

manually through a mechanical probe. As will be described in detail below, a continuous rotation of electron beam is

necessary in order to form high-quality HCDF-TEM images for grain size quantification, where the grain-to-grain

contrast of all grains in the field of view is concerned. For dislocation density characterization, however, the rotation is

performed manually, and the HCDF-TEM images are not formed using the whole diffraction ring(s), but only from the

centered diffraction vector g.

Fig. 1 (a). The ray diagram of HCDF-TEM technique, and (b) The diffraction plane configuration for HCDF TEM imaging [17].

As reviewed previously, the HCDF-TEM techniques can be employed for a variety of applciations. The specific set-

up and operation procudures may vary significantly in different cases. This paper will focus on the application of

HCDF-TEM to grain size and dislocation density characterization. Detailed descriptions of the techniques will be given

in following sections.

3. Grain size quantification of nanocrystalline materilas

TEM micrographs ideal for grains size quantification should have most (and ideally all) grains in the field of view

having an adequately high contrast between adjacent grains such that the grain boundary network is clearly identifiable.

The idea of using the HCDF technique to obtain high quality micrographs is to maximize the diffraction contrast for all

imaged grains. Guided by this principle, the set-up of the HCDF imaging is modified as below. First of all, a

continuous rotation of a slightly-convergent beam is used to form the image. A high rotation speed of electron beam

(10 rotations per second) and a relative long CCD integration time (about 1 second) are typically used to maximize the

number of intensity averaging scans over the annular region. A slightly-convergent beam for has been found to aid

practical operation. Generally, HCDF imaging is normally done using a parallel (non-divergent) electron beam.

However, the HCDF image formed using a parallel beam has a very low intensity and, thus, a long CCD integration

time is required to form the images. The low intensity and the long integration time make it difficult to acquire high

quality micrographs, because of sample drift and because of the inability to judge whether the image is in sharp focus.

In order to increase the intensity of dark-field images, the beam is converged to about 40 mm in diameter on the

viewing screen, which results in different semi-convergence angles at different magnifications. The beam convergence

de-focuses the diffraction pattern at the diffraction plane (or the plane of the objective aperture) and introduces some

uncertainty regarding which diffracted electrons are contributing to the HCDF images. Nevertheless, the imaging

electrons are limited by the size of the objective aperture. The aperture size can then be optimized to provide the best

image for grain identification. As in the normal practice for HCDF imaging, the electron beam tilt angle is adjusted (in

our case, typically 0.3˚~0.4˚) so that the direct beam is just out of the objective aperture and does not contribute to the

micrographs. It should be noted that even with this slightly convergent illumination, high quality HCDF micrographs

formed with several diffraction rings still require a much longer CCD integration time (approximately a factor of ten

increase) compared to that used for BF images.

(b)

Specimen

e beam

Optical axis

View screen

(a)

Rotation trace


1320 ©FORMATEX 2010

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Three micrographs, a HCDF, a BF, and a conventional DF of the same field of view in a 40 nm-thick Cu film

annealed at 600˚C for 30 minutes are shown in Figs. 2 (a), (b), and (c), respectively. A description of the preparation of

the Cu films and TEM samples can be found elsewhere [20]. The BF-TEM image has a low grain-to-grain contrast for

most grains because of the uniformly high intensity of the transmitted beam. Although a few grains satisfying the

Bragg diffraction condition can be identified, the weak diffraction contrast for most grains prevents identification of all

but a few of the grain boundaries. Conventional DF-TEM images provide high contrast for the grains whose diffracted

electrons pass through the objective aperture, but provide lower grain-to-grain contrast than BF imaging for the majority

of the grains present. The HCDF-TEM technique described above uses one or more diffraction rings of polycrystalline

samples (depending on the size of the objective aperture) to form the image. Consequently, most grains in the field of

view can exhibit contrast relative to adjacent grains, as shown in Fig. 2(a), and this allows the boundaries between

grains to be identified. In some cases, the grain boundaries themselves also show some contrast due to a different

density, the absence of a long-range ordering of atomic arrangement, or the presence of impurities at the interface. The

appearance of grain boundary contrast is also very helpful to the tracing work.

Fig. 2. (a) A HCDF , (b) a BF, and (c) a conventional DF micrograph of the same field of view in a 40 nm-thick Cu film annealed at

600˚C for 30 minutes.

As reported elsewhere [20], the image quality of the HCDF-TEM micrographs can be influenced by many factors,

and the key points are summarized here. The HCDF imaging puts stringent requirements on the microscope alignment

and focusing in order to reveal the small (less than 50 nm) grains and grain boundaries clearly. The small extent of

under-focus (e.g., ~100 nm) may not degrade the quality of BF-TEM images significantly. However, the same extent of

under-focus may induce a significant image blurring for the HCDF-TEM imaging technique described here. This

influence is typically more evident when smaller grains are examined. The image blurring is understood in the TEM as

the defocus causing the image to shift in the beam tilt condition. A rotating tilted beam makes this effect even more

evident. In order to acquire high quality HCDF images, it is generally required to conduct all alignments for high-

resolution TEM as a first step, followed by the specific alignments for the HCDF technique (i.e., the dynamical conical

DF Pivot Point X, Y), and finally a very careful focusing. The size of the objective aperture, along with the beam tilt

angle, controls the angular range of diffracted electrons that form the images and strongly influences the quality of

micrographs. Generally, it is recommended to select three or four lower-index diffraction rings (e.g., from 111 to 222

reflections of fcc phase) to form the image. The HCDF TEM micrograph using a small aperture can be very sharp, but

some regions of the image show poor grain-to-grain contrast. On the other hand, using a large aperture provides strong

grain-to-grain contrast for almost all of the grains, but the image is blurred and cannot be focused clearly. For a

microscope operating at 300 keV, a beam tilt angle of 0.3˚~0.4˚ and an aperture in a diameter about 60 µm are

appropriate. Another important factor influencing the image quality is from the material being examined, and the

artefacts induced during the TEM sample preparation. Factors favouring a higher image quality include a high atomic

number (thus high scattering ability) of the material being imaged, a columnar grain structure, large-angle grain

boundaries, and the absence of texture, strain, and defects. Although various materials may yield different qualities of

HCDF images, it has been demonstrated that HCDF technique consistently provide images with much higher quality

than those by BF or DF techniques. The last factor to consider, and perhaps the most important one, is in regard to the

quality of TEM samples. Like most other TEM techniques, the ideal TEM sample for HCDF imaging should be as thin

as possible and free of artefacts. Unfortunately, most TEM sample preparation techniques, such as the electro-polishing

[18], ion-milling [18], and focused ion beam milling [19], may induce some defects. Recently, a back-etch technique

was able to provide TEM samples of thin films with a very clear surface [20, 21]. An example HCDF TEM image from

a sample prepared using this back-etch technique is shown in Fig. 3(a), which can be compared to an image of a sample

prepared by ion-milling (shown in Fig. 3 b) from the same film.

(a) (b) (c)



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Fig. 3. The HCDF TEM images from TEM samples prepared with (a) the back-etch technique, and (b) conventional low angle ion-

milling method. The image from the ion-milled sample is typically less sharp and clear owing to ion-induced artefacts, which cause

non-uniform contrast within each grain. The tiny dots as circled in Fig. (b) are not observed in Fig. (a).

The quantification of grain size from the TEM micrographs is very critical. For some materials, the quantitative

microstructural analysis of TEM micrographs to extract grain size information can be conducted through automated

programs that identify and trace the grain boundaries [22]. For many other materials, however, the very complex

contrast present as a result of defects, strain and thickness contours, or artefacts introduced during sample preparation,

may preclude automated quantification and require instead manual tracing of the boundaries [5]. An example will be

used here to describe the method of the manual tracing of grain boundaries and to quantify the grain size. First of all, a

series of images tilted at different angles (e.g., -1.5°, 0°, 1.5°) of the same field of view are acquired, as shown in Fig.

4(a), (b), and (c) respectively. These images were then enlarged and printed out on high-quality papers. By

overlapping a transparency film on the printed images, the grain boundaries can be traced manually on the transparency,

and the traced network of grain boundaries in this example was shown in Fig. 4(d). The traced images were then

scanned to form digital images, which can be further processed and quantified conveniently by programs (e.g., Image J,

Q Materials). Fig. 4(e) shows the processed image of Fig. 4(d).

100 nm100 nm

100 nm100 nm

100 nm100 nm

Fig. 4. HCDF TEM images of the same field of view, but with slightly different sample tilt angles: (a) -1.5º; (b) 0º; and (c) 1.5º.

Figures (d) and (e) show the traced grain boundary network and the processed digital image for grain size quantification.

(a) (b) (c)

(d) (e)

(a) (b)



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4. Dislocation density quantification of nanocrystalline materilas

A variety of TEM techniques have been used to study dislocation structures [23], such as BF, weak-beam dark-field

(WBDF), and high-resolution TEM (HR-TEM). The HR-TEM image has a limited field of view, and furthermore, it

can only reveal dislocations on the imaged lattice planes. The BF-TEM imaging can show the strain contrast induced

by the dislocations. Unfortunately, the dislocations exhibited by the BF images typically have an inherent low contrast

(due to the intense background), and a low spatial resolution [23]. Consequently, it is not considered as a preferred

technique for dislocation examination, especially for samples with a high dislocation density.

The WBDF-TEM imaging has been considered to be one of the most suitable techniques to study dislocations and many

other defects. The basic idea of the WBDF technique is simple: using a large value of excitation error (s) for a selected

diffraction vector (g) gives a small effective excitation distance (ξeff) and hence a narrow image of most defects, since

the width of a dislocation is related to ξeff /3 [24]. The conventional approach to form WBDF-TEM images is based on

a two-beam condition, and the procedures to obtain this condition is:

1) An adjustment of the specimen orientation to obtain a two-beam diffraction condition where only the direct beam

and a single diffraction relfection (diffraction vector, g) are excited. Thus the diffraction pattern consists primarily of

two bright spots.

2) A tilt of the electron beam to bring the primary excited reflection, g, onto the optical axis and excit the higher

order 3g reflection. This is called g (3 g) condition since the 3 g vector is now on the Ewald sphere and g vector is

repositioned away from Ewald sphere by a displacement referred to as the excitation error.

3) Finally, the objective aperture is inserted to select the centered reflection, g, to obtain the WBDF-TEM image.

Fig. 5 shows an illustration of the geometry for the g (3 g) condition. The intensity of the g spot in the diffraction

pattern will be quite weak, hence the terminology of weak-beam dark field imaging.

In practical operation, however, it is very challenging to utilize this WBDF technique to study dislocations in

ultrafine nanocrystalline materials (e.g., grain size less than 100 nm in diameter). The technique requires that a

diffraction pattern from a single grain be obtained and that the grain be aligned to meet the two-beam condition. This is

not possible for many nanostructured materials of interest. For example, the nanocrystalline Al phase in a trimodal Al

metal matrix composite has a grain size typically ranged from 30 to 100 nm [25]. At this grain size, it is practically

impossible (limited by the size of selected-area aperture of the microscope) to obtain a selected area diffraction pattern

(SADP) from an individual grain due to the overlap of grains and inclusion of surrounding grains in the selcted area.

Convergent-beam electron diffraction (CBED) provides a possible solution for some materials, however, the input of

high energy from the convergent beam into a small grain may easily change its defect structure [23]. Even when the

diffraction pattern from an individual nanoscale grain can be obtained, the alignment of the nano-grain required to

obtain a two-beam condition is much more challenging. A small mechanical drift during the sample tilt can easily

change the field of view of the sample, especially when a convergent electron beam and high magnifications are used.

Unfortunately, such a drift cannot be monitored since the microscopy is conducted in the diffraction mode to search for

the two-beam condition.

The HCDF-TEM technique provides a more effective approach to examine the dislocation structures of nano- and

micro- grains. Based on the above description and Fig. 5, two necessities are required to obtain WBDF-TEM images

showing the dislocation structure. First, the selected reflection g is on the optical axis. Second, the g vector has a

relatively large positive excitation error. It can be understood from the geometry of the HCDF-TEM technique (as

shown in Fig. 1b) that the first requirement, the diffraction vector g on the optical axis, is inherently satisfied, when the

electron beam tilt angle is adjusted so that the centered objective aperture coincides with the selected diffraction ring g.

Once such a beam tilt angle is obtained, changes in sample orientation to find grains with their g vectors having an

appropriate excitation error do not change this condition.

The procedure to obtain grain orientations with an appropriate positive excitation error, s, for a selected g vectors is

straightforward. The basic principle is that the magnitude of the excitation error can be judged from the intensity of

illuminated grains. For example, if the g vector is very close to the Ewald sphere (s close to 0), the corresponding grain

Fig. 5. An illustration of the geometry of the two-

beam condition g(3g) for WBDF-TEM images to

reveal the dislocation structure.



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is highly illuminated in the dark field image. On the other hand, if the vector is far away from the sphere (with a very

large s), the grain is dark in the micrograph. In theory, the grains with a moderately large s should have an intermediate

intensity, but practically, it is difficult to identify these grains simply by their intensities. Our approach is to proceed as

follows:

1) A dark field image is obtained by using a reflection (g vector) of interest. A brightly illuminated grain in the

dark field image is chosen for dislocation imagine. The highly illuminated grain can be obtained by manually rotating

the diffraction pattern and/or by tilting the samples.

2)The sample is then tilted about both axis to maximize the brightness of the chosen grain in the dark field image.

This establishes that the g vector for the grain of interst is on the Ewald sphere.

3)Whenever highly illuminated grains are obtained (i.e., s close to 0), a small tilt of the specimen can increase the

excitation error, either in a positive or negative direction, where a rotation of electron beam may be necessary when the

diffraction spot is out of the objective aperture. The image can be recorded when the grain intensity is reduced, but the

intensity may not be reduced to the extent that the defect structure is not observable. High-quality micrographs of the

dislocation structure can only be obtained if the excitation error is in a positive direction. The sign of the excitation

error can be judged by comparing the corresponding image qualities when samples are tilted in two different directions.

The image with the positive excitation error will provide dislocation contrast and can be used for measurements of

dislocation density.

Fig. 6 illustrates the geometry between the electron beam, g vector, and the axis of sample tilt. With an arbitrary

angle between the sample tilt axis and the g vector (i.e., the direction normal to the diffraction plane), a tilt of the

sample may induce two effects. One is to change the relative angle between the diffraction plane (or g vector) and the

optical axis on the vertical plane in Fig. 6, which effectively alter the sign and magnitude of excitation error s. The

change of excitation error generally induces a smooth change of the illumination intensity of grains. The other effect is

to change the orientation of g vector on the horizontal (or X-Y) plane, which causes the shift of diffraction spot around

the corresponding diffraction ring. In HCDF-TEM technique, it is possible that the originally centered diffraction spot

will move out of the objective aperture when the sample is tilted, which causes an abrupt change of illumination

intensity, and a sharp contrast in the field of view can be observed. In this case, a slight adjustment of the rotation of

the electron beam can easily bring the diffraction spot beck to the objective aperture under the hollow-cone TEM mode.

It should be noted that all the procedures mentioned above are conducted under the image mode and judged by the

illumination intensity in HCDF images, therefore, the small mechanical drift in the conventional two-beam approach is

not a practical problem in the this technique.

Compared to the conventional two-beam approach for WBDF-TEM images, the HCDF method presented in this

paper is more readily employed, especially when ultrafine nanocrystalline grains are examined. Figs. 7(a) and (b) show

the WBDF images of a small nanocrystallie Al material and a coarse grain Al, respectively, acquired using the HCDF

technique described above. The sample used is a trimodal Al metal matrix composite, and the preparation of

composites and TEM samples can be found elsewhere [20].

The dislocation density can be quantified in two different approaches from TEM micrographs. The first one is based

on the dislocation length (in three dimensions, or 3D) per unit volume, which is also named as volume dislocation

density. As the TEM micrographs are two-dimensional (2D) projected image, a factor should be included to transform

the 2D length to 3D. Further, the thickness of the sample or the grains should be measured in order to determine the

grain volume, which typically is not readily achieved. The other approach, also named as areal dislocation density, can

be employed more conveniently, which is defined as the number of dislocations per unit area. The area of grains or

interested regions can be measured conveniently. The number of dislocations is counted as half of the total interception

point of dislocations with the two surface planes of the TEM samples. The point of the interception of dislocations with

the perimeter of grains is not included in the sum. In both approaches, the invisibility criterion (i.e., dislocations

invisible when g.b=0) is also a concern. For example, when g vector of Al {111} is used to form the micrographs, and

b is typically in <110> directions, only half of the dislocations can be revealed.

Fig. 6. The geometry relations between g-vector

(lying in or very close to X-Y plane when

excitation error is close to zero), optical axis of

microscope, and alpha tilt axis.



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100 nm100 nm

Fig. 7. The WBDF micrographs showing the dislocation structure in (a) nanocrystalline Al phase and (b) coarse grain Al phase. The

dislocations in nanocrystalline Al phase are formed through the cryomilling, while those in coarse grain Al phase are produced by the

thermo-mechanical deformation process.

5. Conclusions

Methodologies based on HCDF-TEM are described to study grain size and dislocation density in nanocrystalline

materials. It has been demonstrated that HCDF-TEM imaging, with a slightly convergent beam, as an improved

technique was found to be suitable to form high contrast micrographs for nanocrystalline grain size quantification.

Factors which may influence the quality of HCD-TEM micrographs are also reviewed briefly. A variant geometry of

HCDF can also be used to examine dislocation structure. Compared to the conventional two-beam approach to form

WBDF-TEM images for dislocation characterization, the method described in this study is more readily applied to

provide high-quality WBDF-TEM images, and it is suitable to study both samples with coarse grains and those with

ultrafine nanocrystalline grains. Examples from both thin film and bulk nanostructured samples are presented.

Acknowledgements The grain size quantification work was supported by the Semiconductor Research Corporation, Task

1292.008. Partial support from the MRSEC program of the NSF under DMR-0520425 is also acknowledged. The dislocation study

was sponsored by U.S. Army Research Laboratory (ARL) and was accomplished under Cooperative Agreement W911NF-07-2-

0073. The views and conclusions made in this document are those of the authors and should not be interpreted as representing the

official policies, either expressed or implied, of ARL or the US Government. The US Government is authorized to reproduce and

distribute reprints for Government purposes notwithstanding any copyright notation hereon. The use of the Materials

Characterization Facility of the Advanced Materials Processing and Analysis Center at University of Central Florida is gratefully

acknowledged.

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