Image Formation in TEM

Image formation in TEM is best described by Abbe’s theory and is displayed in Fig- ure 4.1. In this illustration the system is simplified to a single lens system due to the fact that the resolution of a TEM is mainly determined by the objective lens. A real TEM system contains further intermediate and projection lenses. The theory of image formation processes in TEM are widely covered in the literature, good examples can be found in the works of Williams and Carter [12], Zhang [118] and Reimer and Kohl [119]. The surface of the specimen is illuminated by a parallel or near parallel electron beam. The lattices of the specimen then diffract the electron beam, forming beams traveling in multiple directions. The electron-surface interactions are governed by quantum mechan- ical diffraction theory beyond the scope of this thesis, the resultant effect however, is that the electron waves undergo phase and amplitude changes. The scattered electrons are then collected by an objective lens which is used to form an image, a Fourier trans- form of the sample is produced at the focal point of the lens. Depending on the method of imaging needed; the objective aperture can be adjusted in size to select different ar- eas of the projected Fourier spectrum. For example, a narrow aperture only allows the unscattered and narrowly deviated beam through, and is known as bright field imaging (BF). Conversely, to get dark field (DF) imaging, only highly deviated electrons are taken into account, excluding the central spot. The commonly used phase imaging is performed when a large aperture is used and high resolution can be obtained.

The electron wave is forward scattered through the specimen with a transmitted wave- function ψ(R). Propagation through the objective lens then results in the beams being focussed on the back-focal plane, the electron wave here would be represented by its Fourier transform, Ψ(K). K represents the frequency vector perpendicular to the optical axis and is sometimes considered in terms of the scattering angle, θ (K = θ/λ; λ is the wavelength of electrons) and R represents the real vector. The electrons that have been scattered at an angle θ will undergo a phase shift due to spherical and chromatic

Figure 4.1: Abbe’s theory of image formation demonstrated in a one lens TEM. Figure adapted from Wang [120]

aberrations of the lens. A function A(K) (lens function) can be multiplied to Ψ(K) to allow for this effect.

The electron wave at the image plane is then given by taking another Fourier transform, =[Ψ(K)A(K)]. In effect, the image amplitudes seen in the image plane are based on the inverse Fourier transform of Ψ(K) with the added function A(K) accounted for, i.e. the image contrast in TEM micrographs results from the scattering of electrons within the sample, and from the transfer properties of the optical system itself.

So we have seen that the electron image is a result of the inteference between the scattered beams at different angles, and that the interference pattern is affected by the phase modulation caused by the aberration of the objective lens. The image is calculated according to equation 4.5,

I(R) = |ψ(R) ⊗ =[A(K)]|2. (4.5)

By considering the weak-phase object approximation (WPOA) [12], where only thin samples are considered and weak scattering is assumed, it is possible to obtain an an- alytical expression for the image intensity. Using the WPOA the electron wave upon

exiting the sample is given by,

ψ(R) = exp[−iσVp(R)] ≈ 1 − iσVp(R), (4.6)

where,

σ = π

λE (4.7)

Within equation 4.7, σ represents the interaction constant, E is the electron energy and Vp is the projected potential. This is obtained by integrating the sample potential along

the optical axis (z direction),

Vp(R) =

Z

V (R, z)dz (4.8)

If the spherical aberration and defocus is taken into account, then the lens function is,

A(K) = H(K)exp[iχ(K)], (4.9) where, χ(K) = πλ∆z|K|2+1 2λ 3_C s|K|4, (4.10)

here, ∆z is the defocus, Csthe coefficient of spherical aberration and λ is the wavelength

of the electrons. The objective aperture is represented as a top-hat function, H(K) and the phase shift is χ(K). Using equations 4.9 and 4.6 an expression is obtained for the intensity to the first order:

I(R) ≈ 1 + 2σVp(R) ⊗ =[H(K) sin χ(K)] (4.11)

It is shown then that a convolution between the projected potential and an impulse response from the instrument is responsible for the contrast in a weak-phase object. In

Fourier space, the spatial frequencies of 2σVp(R) will therefore be multiplied by H(K)sin

χ(K), which is known as the transfer function, T(K). For a more detailed description of the transfer function see reference [12].

This transfer function is applicable if the sample is a weak-phase object, the function then describes the contrast level in the TEM micrograph. For example, when the T(K) is negative, there is positive phase contrast which results in atoms appearing dark against a bright background. When T(K) is positive, the reverse happens and bright atoms are seen against a dark background and when T(K)=0, there will be no detail in the image for that value of K. This is due to the subtraction or addition of amplitudes given to the forward scattered beam by the phase shift function [12], which has been shown to complicate the transfer function through its dependance on defocus, electron wavelength and the Cs of the lens. It is also oscillatory with K (see equation 4.9). Optimisation of

the transfer function occurred in 1949, when Scherzer balanced the effect of spherical aberration against a particular negative value of defocus [121]. This is known as the Scherzer defocus (∆ zsch),

∆zsch= −1.2(Csλ)

1

2 (4.12)

In summary, image formation in conventional TEM proceeds through the process of taking the Fourier transform of the forward scattered electron wave at the exit face of the sample, multiplying it by the lens transfer function and taking the inverse Fourier transform of the amplitude in the back focal plane. The image produced is made more complex by modifications made by the lens system, but contrast is achieved through coherent interference of the scattered waves from the sample. The difficulty in quan- titative interpretation of TEM micrographs arrises due to the fact that only exit wave intensities and not phase information can be recorded. TEM can however provide de- tailed structural images of clusters, and a more quantitative approach can be made by selecting different types of scattered electron from the sample. Such a method is called scanning TEM (STEM) and is discussed in the next section.