Chapter 2 Characterization Techniques for Carbon Nanotubes
2.4 Transmission Electron Microscopy and Electron Diffraction
Transmission electron microscopy is a powerful tool for microstructure and nanostructure characterization. It is a microscopy where fast electrons go through a very thin specimen, interacting with the atoms in the specimen as electrons pass through.
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Scattered electrons are focused to form a magnified image onto an imaging device such as a fluorescent screen, a photographic film, or a sensor such as a CCD camera. The physical principles behind the development of TEM should be attributed to the following two aspects: (1) the wave-like characteristics of electrons first postulated by Louis de Broglie in 1925 [17]; (2) the discovery of electron focusing lenses which use electromagnetic field to focus moving electrons in desired direction by Hans Busch in 1926 [18]. The first transmission electron microscope was built by Ernst Ruska and Max Knoll in 1932 [19]. Ernst Ruska shared the Nobel Prize in Physics with Gerald Binning and Hans Rohrer in 1986. The wavelength of an electron accelerated by a voltage between 100 kV to 400 kV is about two orders magnitude smaller than the size of an atom which has a diameter of about 0.1 nm. In principle, it is possible to resolve material structure well below the atomic level according to the Raylaigh criteria [20]. However, a TEM with this resolution limit is impossible to construct due to the imperfections of the magnetic lenses. Nowadays, a good TEM can achieve the resolution on the order of 0.1 nm. In contrast, electron diffraction is much less sensitive to the imperfections of magnetic lenses, but more dependent on the convergence of incident electrons [21].
Within this research, CNT images and electron diffraction patterns are taken on JEM 2010F FasTEM electron microscope (vender JEOL). All the diffraction patterns are taken by the nanobeam electron diffraction method with beam waist of about 80 nm.
2.4.1 Theory of Electron Imaging in TEM
A TEM produces an image in the following two-step Abbe principle: (1) the incident electrons are scattered by the specimen and form a diffraction pattern on the back focal plane of the objective lens; (2) the scattered electrons are recombined to form
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an image on the image plane. Two imaging mechanisms are usually used: amplitude contrast and phase contrast [22]. In the amplitude contrast mode, the image contrast is the result from the electrons scattered in different angles within a specimen. The areas of the specimen with higher mass or larger Coulomb potential will scatter more electrons toward large angular regions which are away from optical axis. If a small aperture is used, most electrons which are scattered in large angles can be excluded except the selected beam. As a result, the areas corresponding to the higher mass or strong atomic potential positions within the specimen will turn dark in the image. On the other hand, image contrast in the phase contrast mode comes from the phase difference of electrons caused by interactions between electrons and the Coulomb potential of the specimen. A large objective aperture is usually used to allow more scattered beams to pass through the objective lenses to form an image. It offers a much higher structural resolution and is usually called high resolution TEM compared to the amplitude contrast mode. But it requires that the specimen be thin.
Like the image formation in an optical microscope, the scattered electron waves in the diffraction plane will form a two-dimensional structure image projected from three- dimensional specimen on the image plane. However, in order to better interpret the electron imaging process, both non-linear imaging and dynamical electron diffraction effects need to be considered [23].
For phase contrast imaging, the interactions between the incident electrons and the specimen are usually weak and theories can be greatly simplified. In the phase-grating approximation, the wave function of the electron scattered by specimen can be described by [24]
25 ( , ) e px p( , )
o x y iV x y
, (2.4.1) where
/ ( U) is the relativistic interaction constant with U being the accelerating voltage applied on electrons, V x yp( , ) is the projected Coulomb potential of specimen in a plane perpendicular to the direction of the incident electron beam and is the wave length of the electrons. For a thin specimen constituted of light atoms, the weak phase object approximation can be applied and Eqn. 2.4.1 can be simplified to( , ) 1 ( , ) o x y i V x y
, (2.4.2)
where the unit 1 represents the transmitted wave which has no interaction with the specimen and the imaginary part corresponds to the scattered waves. The image wave on the image plane is a convolution between the object wave and a contrast transfer function
:
( ) ( ) ( )
i r o r T r
, (2.4.3)
where is the convolution operator. The convolution operation of two functions can be expressed as the product of their corresponding Fourier transforms in the reciprocal space:
( ) ( ) ( )
i q o q T q
. (2.4.4)
The contrast function in a TEM includes the information of aperture function, spherical aberration and imperfection of focusing of the objective lens and is given by [25]
( ) ( )exp[2 ( )]
T q a q
i q , (2.4.5) where a q( ) is the aperture function of the objective lens and4 2 1 1 ( ) 4 s 2 q C q fq , (2.4.6)
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where Cs and Δf are the spherical aberration coefficient and defocus of the objective lens, respectively. Using Eqn. 2.4.5 and 2.4.6, Eqn. 2.4.3 can then be expressed as
( , ) 1 ( , ) [ ( , )] ( , ) [ ( , )]
i x y V x yp Im T x y i V x yp Im T x y
. (2.4.7)
In weak phase object approximation, Vp 1, the image intensity can then be further simplified into
2 i
( , ) | Ψ( , )| 1 2 p( , ) [ ( , )]
I x y x y V x y Im T x y . (2.4.8) Eqn. 2.4.8 shows that only the imaginary part of the contrast transfer function contributes to the image intensity in the weak phase object approximation and linear imaging.
2.4.2 TEM Imaging of CNT
Because CNTs are formed by graphene layers and carbon atoms have a low atomic number ( ), the weak phase object approximation discussed in the last section can be used to interpret the TEM images of carbon nanotubes structure (an image actually corresponds to the projected Coulomb potential of the CNT [26]). A high resolution TEM equipped with a field emission electron gun can easily obtain structural images of a CNT with a resolution of about 0.2 nm. Thermal and mechanical vibrations, stage drift, and instabilities of the magnetic lenses will compromise the quality and the resolution of a TEM image.
Fig. 2.4.1 shows high resolution TEM images of a SWNT, a DWNT and a six- wall carbon nanotube. Hollow cylinder structure of the carbon nanotubes is seen clearly in the images. In the DWNT and the six-wall carbon nanotubes, two and six concentric cylinders can be identified in the figure. The two parallel darks lines run along the tube axis are the projected structure of the tube walls. The diameter of the nanotube can be
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measured from a line profile perpendicular to the tube axis. But the measurement needs to be very careful because the position and width of the dark line are very sensitive to the imaging condition such as the defocus of the objective lens. It also causes errors in the diameter measurement if the nanotube is not oriented within a horizontal plane, i.e. not perpendicular to incident beam. The error becomes more significant when measuring the smaller diameter tube due to the pronounced curvature [27].
Figure 2.4.1 TEM images of (a) SWNT, (b) DWNT, and (c) 6-wall carbon nanotube taken with JEM 2010F operated at 80 kV.