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Advanced Methods in Transmission Electron Microscopy

Advanced Methods in Transmission Electron Microscopy

Knut Müller,

Knut Müller, Katharina Gries: TEM Tutorial Riezlern 09/2008 evaluating distances in the image for example. What we understand under conventional TEM is the illumination 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, since the specimen is crystalline- and the task of the lens is to superimpose all diffracted beams again to form an image. But: Remember that the object exit wave shows mainly a phase modulation, and as we finally record intensities, this information will not produce any contrast in the image. Nevertheless, we do actually see contrast in our image, and we will explain why by considering lens errors in the following. 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 crystalline, 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 applying an additional phase factor, which is the exponential term on the right side, to each individual point of the wave function in the focal plane. In the following, we will investigate the contributions of spherical aberration and defocus, without going into details of coherence effects, which are included in the factor κ. One can show that phase contrast is largest if the sine of χ is -1, for sin(χ) = 0 we will see no contrast. The contribution of spherical aberration is linear 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, Katharina Gries: TEM Tutorial Riezlern 09/2008 missing 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 following series starts with Cs = 0, thus the function vanishes for all q, meaning that we see no phase contrast using a perfect lens. With increasing Cs, the oscillation changes its rate and shape, so that at one time, we see that frequencies corresponding to GaAs (400) -indicated by the red circle- are transferred with good contrast, whereas the respective information is missing if the red circle coincides with a null. One additional thing which is worth to be mentioned here is the so-called information limit of the microscope. Obviously, the contrast transfer function is envelopped by a function that falls to zero at a certain spatial frequency. It means that frequencies larger than this limit are cut by the imaging process, and the corresponding distances can never be resolved with the respective optical system. Let us now include 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 again that the shape of the function sin(χ) is strongly influenced by the defocus setting. Now the movie starts at Δf = -200nm and will increase up to Δf = 200nm. For certain 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 increases further, the contrast transfer gets worse for most frequencies, and also the information 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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