A few concepts in TEM and STEM explained

Transcription

1 A few concepts in TEM and STEM explained Martin Ek November 23, Introduction This is a collection of short, qualitative explanations of key concepts in TEM and STEM. Most of them are beyond what you need to know in the undergraduate courses, but it can be very useful to have some understanding of the underlying concept rather than to just accept them. They are also quite difficult to explain on the fly during the labs. For TEM: Section 2 on Fresnel fringes. We use these to gauge the defocus in TEM images, but what are they? Section 3 on diffraction. In order to understand what is happening when we tilt crystals in the TEM we need to go a little beyond Bragg s law. Section 4 on high resolution imaging. Here the importance of knowing the imaging conditions is discussed briefly. For STEM: Section 5 on the Ronchigram. Sometimes described as looking like a boiling brain, but the wierd name hides a not-to-difficult concept. 1

2 Underfocus Focus verfocus Figure 1: Fresnel fringes at different defocus for an indentation in a glass plate (the image is formed by light rather than electrons). Photons passing through the indentation will experience a phase change compared to the ones passing through the full glass plate. In a similar fashion there is a phase difference between electrons passing through a carbon film compared to ones passing through vacuum. 2 Fresnel fringes Fresnel fringes show up along edges when we look at something out of focus using a coherent illumination. This is the case for the edge between the carbon film and vacuum, your sample and the carbon film (or vacuum), or aperture edges in the TEM to give a few examples. What you need to know is that the appearence of fringes indicate that the object you are looking at is out of focus: dark fringes indicate overfocus and you need to reduce the strength of the objective lens (or lower the sample), bright fringes indicate underfocus and you need to increase the strength of the objective lens (or raise the sample). Figure 1 shows Fresnel fringes in an underfocused, in focus, and overfocused light image of an indentation in a glass plate. 1 This section will discuss why these fringes appear, in some detail for the simpler case of an opaque object (like an aperture) and very briefly for the much more difficult case of a transparent phase object (like the carbon film). Figure 2 shows the imaging geometry for an out-of-focus object. 2 There is a fixed image plane (which we observe), an object plane from which the image is formed, and the actual object. If we set the strength of the objective lens just right the object plane and the actual object will coincide, and we will form an in-focus image. In this case an opaque object should give a sharp transition from bright (in the vacuum where the electrons pass unimpeded) to dark (where the electrons hit the object and are excluded from the image). If the object is instead a transparent phase object, which only changes the phase of the electron wave and not its amplitude, the in-focus image would be completely uniform as we can only observe the amplitude. To get the electron wave in the object plane (which is what we will image) we need 1 From Experimental High-Resolution Electron Microscopy by John C. H. Spence. 2 Most of this material is adapted from 2

3 Actual object bject plane bjective lens Image plane (fixed) Fringes! Δf Figure 2: Image formation of a defocused object. The object plane which we image in the fixed image plane is below the actual object (overfocused). To get the electron wave at the object plane we propagate the wave from the actual object a distance f. to propagate the electron wave from the object a distance f (which is the defocus). To illustrate the concept lets consider what happens if we image vacuum as in figure 3 (a): in this case the object electron wave has uniform amplitude and phase. To see what the electron wave is in a certain point furhter down the microscope column (remember figure 2) we draw paths to this point from every part of the object. The wave in this point is then the sum of the object wave with a phase change (represented by the angle of the arrows) proportional to the distance from the object wave to the point. The parts of the object directly above the point will get roughly the same phase and add, but the areas further away in the object will contribute less as they rapidly become out of phase. Adding these arrows from the entire object produces the (Cornu) spiral shown in figure 3. We then get the total amplitude by connecting the two end-points of the spiral. As this is the same for every point in the propagated wave we have uniform intensity. If we now insert an opaque edge in the object some parts of the object electron wave will no longer contribute to the propagated wave. Think of this as shielding some of the paths we drew in (a) so that the arrows from these parts no longer contribute to the sum. This truncates the spiral as shown in (b) and (c). If the truncation removes primarily the parts that had the opposite phase of the straight path this actually increases the amplitude as shown in (c). f course it is also possible to remove mainy in-phase paths and thereby decrease the amplitude, as shown in (b). This produces the oscillating behaviour which far away from the edge will converge to the vacuum intensity. These are the Fresnel fringes! 3 3 For a lecture on this, see: vega.org.uk/video/subseries/8 3

4 (a) No edge (b) A distance away from the edge (c) Close to the edge Total amplitude Image intensity Dark fringe Vacuum Bright fringe bject 0 Figure 3: Illustation of how a summation of the contribution of all points in the object yield the wave at a certain distance below the object: with a plane wave object without an edge(a), an object with an opaque edge some distance away (b), and close to the opaque edge (c). In the last case we actually get higher intensity in the image than if we had just imaged vacuum since the out-of-phase contributions are removed. In the above description you might have noticed that the direction of propagation of the object wave will make no difference: the fringes will be identical in over- and underfocus. This is the case for fringes from an opaque edge in the object. An edge from a phase object on the other hand changes the phase of the electron wave, which can either add or subtract to the phase change of the propagation. Figure 4 illustrates how this effects the propagated wave. This simplified picture shows that close to the edge in the underfocused case there are more in-phase contributions which will result in a higher amplitude (bright fringe). Conversely, in the overfocus case these contributions are more out-of-phase and reduce the amplitude, which roughly explains the different appearence of the fringes around your sample with defocus in the TEM. The Fresnel fringes themselves can be analysed to give information about the sample, but mostly they are used as a simple gauge of defocus. What you need to remember from this section is only that if the outermost fringe is dark the image is overfocused (reduce objective lens strength or lower sample), if instead it is bright the image is underfocused (increase objective lens strength or raise sample). However at the exact focus, phase objects are invisible. Since your sample likely (hopefully) 4

5 verfocus Underfocus bject Figure 4: Propagation of an electron wave up (underfocus) and down (overfocus) from a phase object. Notice the effect on the phase of the blue arrow which originates from the region with the phase object. is a phase object we usually record the images with a slight underfocus, but his is discussed in more detail in section 4. 3 Diffraction in TEM First, let s start with Bragg s law as illustrated in figure 5 (a). 4 In ordet to get intensity at spot G the rays reaching this point should be in phase, that is the path difference (marked with orange) should be a multiple of the wavelength λ. This requirement describes Bragg s law: 2d sin θ B = nλ If we instead look at this in reciprocal space as in (b) we have an incomming wavevector k I with the same direction as before, but with a length of 1/λ. The diffracted wave-vector k D has the same direction as the diffracted beam, with a length also equal to 1/λ. The scattering-vector K is then the difference between the two and has a length: K = k D k I = 2 1 λ sin θ According to this model we have diffraction if K corresponds to an actual reciprocal disctance between planes in the crystal, that is if K = 1/d. If you insert this definition of K into the formula above it produces Bragg s law. To be a bit more precise we should say that in order to get diffraction K = g where g is the reciprocal 4 Material adapted from ch. 11 and 12 in Williams & Carter 5

6 lattice vector G, which is the reciprocal of the vector between the planes in the crystal. In figure 5 notice how g has to fall on a circle (sphere in 3D) with radius k I = 1/lambda in order to result in diffraction. This is the Ewald sphere and it shows where the Bragg conditions are fulfilled. (a) Real space (b) Reciprocal space θ B d k I = k D = 1 λ θ k D k I 2θ B G K G Figure 5: (a) illustration of Bragg s law. When the (orange) path difference between the two rays is a multiple of the wavelength there is constructive interference and we see a diffraction spot at G. Some rays will always go straight through and form, the orgigin of the diffraction pattern. (b) shows the exact same diffraction geometry from a reciprocal perspective. Notice how the difference between the incomming and diffracted wave always end up on a circle independent of the angle θ: this is the Ewald sphere. We can now have a look at an actual crystal containing many planes and construct a reciprocal lattice which shows the location of all possible G. In figure 6 (a) a very simple real space lattice of (green) atoms is shown from the side with the electrons comming from the top (imagine it extending infinitely to the left and right). The corresponding reciprocal lattice with the origin and a series of G are shown in (b) (imagine it extending infinitely in all directions). Since λ nm the Ewald sphere is very large compared to g as typical plane spacings are on the order of 0.2 nm. As you can see the sphere only intersects (which it always will) and not any of the G shown, although it comes close enough for diffraction to occur. For small crystals the Bragg condition does not need to perfectly satisfied in order to result in diffraction. This can be justified by looking at Bragg s law which is derived from only two planes: if the angle is close to, but not exactly equal to, the Bragg angle the two rays will be almost in phase and therefore produce some intensity. Rather than describing discrete diffraction spots, Bragg s law gives a smooth variation with 6

7 the highest intensity at the Bragg angle (where the rays are completely in-phase) to 0 intensity (where the rays are completely out-of-phase). nly for an infinite lattice do we get actual point-like spots. In this sense the diffraction laws are relaxed when there are only a few planes present, which is the case in at least one direction for a thin TEM sample. (a) Real space (b) Reciprocal space k I 51 b a 10 - b* a* Figure 6: Illustration of a real space lattice of atoms (a) viewed from the side, and the corresponding reciprocal lattice (b) with lattice points corresponding to planes. The crystal in (a) should be imagined to stretch infinitely to the left and right (and also into and out of the paper), while the reciprocal lattice extends infinitely in all direction. The large Ewald sphere in (b) shows for which planes the diffraction conditions (almost) fulfil the Bragg condition. Looking closer at figure 6 (b) we can see that the incident wave-vector k I is in this case comming down perpendicular to the reciprocal lattice-vector a. This means that the normals of these planes are perpendicular, and the planes themselves parallel, to the incomming electrons. Again this means that none of theses planes satisfy the Bragg condition, but since the Bragg angle is very small (since λ is small, and the Ewald sphere large) and the crystal is thin we still get diffraction. Since these G are in the same plane as and perpendicular to k I they are in the so called zero-order Laue zone (ZLZ). Further out where the curvature of the Ewald sphere brings it closer to G one layer up along b we also get diffraction, and these are in the first-order Laue zone (FLZ). Even further away from the sphere will intersect a layer 2b up, and so on. The key thing to remember is that often in an electron diffraction pattern we have a set of spots around which are formed from planes parallel to the electron beam, surrounded by rings of spots corresponding to where the Ewald sphere intersects the higher layers in the reciprocal lattice. How are the concepts of reciprocal lattices and Ewald spheres used practically at the TEM? For imaging we often want to view the sample from a certain direction 7

8 (a) Perpendicular (b) Tilted Side view k I k I ZLZ Top view FLZ Figure 7: Illustration of the formation of a diffraction pattern with a crystal having planes parallel to the viewing direction (a) and slightly tilted (b). The topmost images show how the Ewald sphere intersects the reciprocal lattice from from a side view. The bottom images illustrate the same from a top view, which also correspond to the appearence of the diffraction pattern on the screen. nly the layers along 2b are shown for clarity. and the diffraction pattern allows us to see the sample orientation. In high resolution TEM in particular we want the atoms to line-up into columns, which is the same as saying that we want to have many planes of atoms parallel to the electron beam. Figure 7 illustrates the appearence of a diffraction pattern as the crystal is tilted. The first case corresponds to the same conditions as in figure 6 and we can see a series of spots from the parallel planes in the ZLZ surrounded by a ring of FLZ spots (which might not be visible for crystals with small unit cells). nly the different b layers are shown in order to remove some clutter from the illustration: just assume that there are plenty of possible G in each layer which can be excited where the Ewald sphere intersects. When the crystal is tilted the diffraction pattern will no longer be symmetric arround, but will have more intensity in one direction. Also, the rings of spots outside the ZLZ will get a crescent like appearence. rienting a crystal into a good direction for high resolution imaging involves recognising these patterns and trying to make them round and symmetrical around by tilting. If we look at a crystal with a b then a b since a = 1/ a. From figure 6 we can see that looking perpendicular to a we expect many spots in the ZLZ (since there are many G to interesect) and a long spacing between the ZLZ and the FLZ (since the distance between the layers in b is large). If we instead look perpendicular to b we should expect fewer spots with larger spacings in the ZLZ singe the planes parallel to the beam have small spacings. n the other hand 8

9 now the short a deetermines where the higher order zones appear and as a result they will be closer to. 4 High resolution imaging in TEM For high resolution imaging in TEM the change in phase of the electron wave as it passes through a very thin sample is used. If the sample is properly oriented so that we view along colums of atoms (i.e. parallel to planes of atoms) there will be a phase difference between the electron which have experienced the attractive force of the atomic nuclei, and the electrons passing between the atoms. The phase of the electron wave therefor contains information about the location of the atomic columns, and potentially some information about the chemistry as heavier nuclei will have a stronger effect than a lighter ones. In the same way a thicker specimen will give a different phase shift compared to a thin one. It is not possible to directly detect the phase however, and we need to transfer the phase information into amplitude variations in the electron wave hitting the detector. We do this by allowing the objective lens to add a further phase shift to the parts of the electron wave that have been scattered by the atomic columns, thereby generating an interference pattern between these electrons and the unaffected electrons which have passed straight through the sample. The phase shifts caused by the objective lens are mainly determined by three parameters: the electron wavelength (related to the accelerating voltage) which determines to what angle the atomic columns scatter the electrons, the spherical aberration (C s ) which is an unavoidable lens imperfection for round magnetic lenses, and the defocus ( f) which we can control by changing the strength of the objective lens. In practice we use defocus to compensate for the spherical aberration by recording images with a slight underfocus in a way that allows as much phase information as possible to be reliably transfered into amplitude variations. To show the effect of electron wavelength (and spherical aberration), crystal thickness, and defocus simulated images of NaNb are shown in figure 8, 9 and 10. NaNb has a flat unit cell with angles of almost 90 and for the simulations and the polyhedra-models we are looking down the short axis. In these axamples we will mainly see Nb atoms (in the centre of the octahedra), but sometimes also Na (situated in the channels between the octahedra and illustrated by a purple dot) and (on the corners of the octahedra). There are many different Nb-Nb spacing in this structure as there are both octahedra sharing only corners and octahedra sharing either two or four edges. This gives plenty of detail in the images which can be affected by the microscope parameters. 9

10 (a) 120 kv Cs= 2.2 mm (b) 200 kv Cs= 1.0 mm Libra 120, W-filament (c) 300 kv Cs= 0.6 mm JEL 3000F, FEG JEL 2010, LaB6 Figure 8: (a) This 120 kv TEM cannot resolve the edge sharing octahedra. (b) a 200 kv TEM with a LaB6 filament resolves all the Nb columns. (c) This microscope corresponds to the JEL 3000F here in Lund. All Nb atoms are resolved and even the columns start to appear as faint grey spots. (a) t= 1.2 (b) 3.5 (c) 6.5 nm Figure 9: Illustration of the effect of increasing thickness on images simulated for a 3000F. nly for a crystal 1.2 nm (about 3 unit cells) thick is the image directly interpretable in terms of dark atom positions and bright voids. For greater thicknesses the contrast starts to revert and the columns become more prominent. 10

11 (a) Δf= -30 (b) -42 (c) -54 nm Figure 10: The effect of varying the defocus around its optimal value of -42 nm (b) on the 3000F. The optimum value is determined by the accelerating voltage and the spherical aberration. These focus changes represent less than one click on the course focus knob. Notice how in (c) the contrast has reverted to show bright atoms. From this we can conclude that it is very important for the interpretation of high resolution TEM images to know the settings that were used to acquire them. It is always a good idea to try to match the images with simulations, but beware: simulated images tend to have much more contrast than experimental images so the comparison is not always straight forward. 11

12 5 The Ronchigram The Ronchigram might seem very complicated from the description in the STEM manual, but as you can see in the ray diagram in figure 11 the underlying principles are not too difficult. The most important concept here is that at exact focus only a single point on the sample is illuminated and as a consequence magnified to cover the entire image: the point is infinitely magnified. When the Ronchigram is defocused the image is more recogniseable as a simple projection of the sample. verfocused Focused Underfocused Sample Illumination Image Infinite magnification! Figure 11: The Ronchigram: a shadow image of an object formed by a convergent illumination. The illustrations show how the image would appear for different defoci with a perfect lens. In the previous illustration the illumination was focused perfectly to a single spot. This is not the case in a real microscope where we have a considerable spherical aberration, which causes high angle rays to be deflected more than rays which have a smaller angle to the optical axis. Each ray will therefore be focused to different points depending on their angle to the optical axis. The Ronchigram will no longer be either a projected shadow image or an infinitely magnified point, but something in between. A real Ronchigram of an amorphous carbon film is shown in figure 12 (a). The amorphous film has an even speckled ( dotty ) pattern which is imaged differently in different parts of the underfocused Ronchigram. The ray paths in get a bit more complicated with spherical aberration as seen in (b), but we can break them down into three cases as in figure 12 (c). The central part of the Ronchigram comes from rays with low angles which are underfocused. These form a normal shadow image in the centre (c1). The rays with higher angles are focused on the sample and form images with infinite magnification. Two rays from oposite sides of the optical axis, but with equal angles, will magnify a single point on the sample to the outer parts of the Ronchigram, which will stretch into a ring with 12

13 (a) Underfocused Ronchigram with Cs (c) The zones in the Ronchigram Underfocused image (Ring of) Infinite radial magnification 2 (b) Ray diagram (Ring of) Infinite angular magnification 3 Probe Amorphous carbon Image Figure 12: (a) Underfocused Ronchigram with spherical aberration, (Cs ). Three zones (1-3) show different types of images, which are formed according to (c). infinite angular magnification (c3). Two rays on the same side of the optical axis, but with slightly different angles, will coincide on a ring on the sample. In the image the points on this ring will be stretched radially to form an area with infinite radial magnification (c2).5 This explains what we see in the Ronchigram, but how is this useful to us? First of all the Ronchigram tells us the focus. In a focused Ronchigram the central shadow image dissapears and is replaced by an area of infinite magnification. This also helps us to select the aperture: ideally we want only rays passing through a single point on the sample to form our probe. This is not the case for the rays in the ring of infinte radial magnification, and we select an aperture which excludes these and only contains the central part of the Ronchigram. If there is astigmatism present the Ronchigram will no longer be round, but instead oval (as there is different defocus in different directions). Using the stigmator coils in the microscope we can compensate for this and correct the Ronchigram until it is round. 5 Adapted from slides by Alan J. Craven 13