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Advanced Methods in Transmission Electron Microscopy

Knut Müller, Kathar**in**a Gries: TEM Tutorial Riezlern 09/2008 evaluat**in**g distances **in** the image for example. What we understand under conventional TEM is the illum**in**ation of the specimen with a nearly plane wave. This wave will then propagate through the sample and to a first (but often very good) approximation, this wave suffers only a change **in** phase and not **in** amplitude. However, our wave is diffracted by our specimen -**in** this case the directions are given by Bragg's law, s**in**ce the specimen is crystall**in**e- and the task of the lens is to superimpose all diffracted beams aga**in** to form an image. But: Remember that the object exit wave shows ma**in**ly a phase modulation, and as we f**in**ally record **in**tensities, this **in**formation will not produce any contrast **in** the image. Nevertheless, we do actually see contrast **in** our image, and we will expla**in** why by consider**in**g lens errors **in** the follow**in**g. A characteristic quantity of any lens is the focal distance, where the focal plane is located. In this plane, we see the spectral decomposition (Fourier transform) of our object exit wave as shown **in** the diffraction pattern here. As our specimen is crystall**in**e, not every spatial frequency is present, so discrete Bragg diffraction spots show up. The most important lens errors spherical aberration and defocus can be modelled by apply**in**g an additional phase factor, which is the exponential term on the right side, to each **in**dividual po**in**t of the wave function **in** the focal plane. In the follow**in**g, we will **in**vestigate the contributions of spherical aberration and defocus, without go**in**g **in**to details of coherence effects, which are **in**cluded **in** the factor κ. One can show that phase contrast is largest if the s**in**e of χ is -1, for s**in**(χ) = 0 we will see no contrast. The contribution of spherical aberration is l**in**ear **in** the spherical aberration constant Cs and depends on the momentum transfer q **in** 4 th order. The so-called contrast transfer function is already shown here for a very low Cs value of 15µm, where it can clearly be seen that we can never image all spatial frequencies present **in** our specimen with the same contrast. An optimum curve would be -1 for every spatial frequency q. Moreover, spatial frequencies at nulls of the contrast transfer function may be present **in** the object, but they will not appear **in** the image. Due to this strange behaviour of the contrast transfer function, some spatial frequencies are more pronounced, others appear weaker, and some are 4/13

Knut Müller, Kathar**in**a Gries: TEM Tutorial Riezlern 09/2008 miss**in**g at all, so that the result is a distortion of the image. Now let us have a look at how the contrast transfer function depends on Cs. The follow**in**g series starts with Cs = 0, thus the function vanishes for all q, mean**in**g that we see no phase contrast us**in**g a perfect lens. With **in**creas**in**g Cs, the oscillation changes its rate and shape, so that at one time, we see that frequencies correspond**in**g to GaAs (400) -**in**dicated by the red circle- are transferred with good contrast, whereas the respective **in**formation is miss**in**g if the red circle co**in**cides with a null. One additional th**in**g which is worth to be mentioned here is the so-called **in**formation limit of the microscope. Obviously, the contrast transfer function is envelopped by a function that falls to zero at a certa**in** spatial frequency. It means that frequencies larger than this limit are cut by the imag**in**g process, and the correspond**in**g distances can never be resolved with the respective optical system. Let us now **in**clude the effect of a defocus Δf. As a first example, we use a Cs-value of 1mm which is typical for good microscopes without aberration corrector. We see aga**in** that the shape of the function s**in**(χ) is strongly **in**fluenced by the defocus sett**in**g. Now the movie starts at Δf = -200nm and will **in**crease up to Δf = 200nm. For certa**in** defocus values, we see that passbands develop, that is, the contrast transfer function is approximately -1 over a comparably wide frequency range. The defocus for which a passband develops for the lowest momentum transfer is called the Scherzer defocus, which can be seen top right. If the defocus **in**creases further, the contrast transfer gets worse for most frequencies, and also the **in**formation limit decreases drastically. 5/13 1 0 . 8 0 . 6 0 . 4 0 . 2 0 - 0 . 2 - 0 . 4 - 0 . 6 - 0 . 8 1 0 . 8 0 . 6 0 . 4 0 . 2 0 - 0 . 2 - 0 . 4 - 0 . 6 1 0 . 8 0 . 6 0 . 4 0 . 2 0 - 0 . 2 - 0 . 4 - 0 . 6 - 0 . 8 - 1 1 0 . 8 0 . 6 0 . 4 0 . 2 0 - 0 . 2 - 0 . 4 - 0 . 6 - 0 . 8 - 1 C S = 0 μ m - . . . 1 0 0 5 1 1 5 2 2 5 3 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 Δ f = 2 n m 1 0 x 1 0 C S = 4 5 0 μ m - . . . . 0 8 0 0 5 1 1 5 2 2 5 3 1 0 x 1 0 Δ f = - 1 3 6 n m 1 0 x 1 0 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 1 0 x 1 0 - . . . . 0 8 0 0 5 1 1 5 2 2 5 3 1 0 . 8 0 . 6 0 . 4 0 . 2 0 - 0 . 2 - 0 . 4 - 0 . 6 - 0 . 8 - 1 1 0 . 8 0 . 6 0 . 4 0 . 2 0 - 0 . 2 - 0 . 4 - 0 . 6 - 0 . 8 - . . . 1 0 0 5 1 1 5 2 2 5 3 1 0 . 8 0 . 6 0 . 4 0 . 2 0 - 0 . 2 - 0 . 4 - 0 . 6 C S = 1 9 5 μ m C S = 9 9 0 μ m Δ f = 2 0 0 n m 1 0 x 1 0 1 0 x 1 0 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 1 0 x 1 0

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