TEM Imaging and Dynamical Scattering Duncan Alexander EPFL-CIME 1 Aspects of TEM imaging Objective lens focus Objective lens astigmatism Image delocalization Dynamical scattering 2-beam theory Thickness fringes Bend contours Double diffraction Contents 2
Objective lens focus A full ray diagram schematic of the objective lens will go from object plane to image plane and shows both focusing of the objective lens and diffraction pattern formation in the back focal plane see first TEM lectures and exercises today The image plane of the objective lens forms the object plane for the next lens in the series (i.e. the first intermediate lens); this is defined as the image plane. That is, the two lenses are coupled. At correct focus a point object is focused to a point in this image plane If we decrease objective lens strength the rays come to a point below this plane; this is called under focus. At the image plane there is an out of focus image which is then projected onto our detector This under focus image is basically equivalent to having correct objective focus but moving the sample down Figure from Williams & Carter Transmission Electron Microscopy 3 Objective lens focus Very important: when the sample is in focus there is minimum contrast (see phase contrast lectures) Quiz 1: which of these images is in focus? 1 2 3 Image 3 is in focus: no Fresnel fringe at edge of hole, no specular ( speckled ) contrast in the carbon film, therefore it has little contrast 4
Objective lens astigmatism When image is astigmatic different axes in the image plane have different focal points. This can be seen as different Fresnel fringes for different image directions Here seen for different objective foci 5 Which of these images of GaN nano-wires was taken with an objective aperture? 1 2 6
Image delocalization TEM image with no objective aperture. Image formed from direct beam and diffracted beams. Dark-field images from diffracted beams delocalize from bright-field image of direct beam. Gives shadow images that move with objective focus (draw ray diagrams for out of focus image). 7 Image delocalization Image of same nanowires but with objective aperture to make bright-field image. No diffracted beams => no shadow images. This is how you should take your TEM data! 8
Diffraction contrast on/off zone axis In bright-field imaging, zone axis condition => more scattering to diffracted beams Therefore intensity in direct beam goes down and bright-field image has strong contrast Example: GaN nanowire CBED SADP BF Off zone axis CBED SADP BF On zone axis 9 Dynamical scattering 10
TEM diffraction recap Reciprocal lattice + relrods => multi-beam scattering Excitation error s =>! deviation from Bragg condition hkl 0 11 Dynamical scattering " X-ray scattering is primarily kinematical: because interaction of X-ray with matter is weak, an X-ray is typically scattered at most one time SADP of Si on [1 1 0] zone axis " In contrast because of the coulomb interaction electrons have a strong interaction with matter " As a result they are often scattered many times on their path through a sample: dynamical scattering " Scattering and intensities in different diffracted beams are therefore not independent of each other, making them hard (impossible) to interpret 12
2-beam dynamical scattering " Consider Bragg condition combined with high probability of scattering 13 2-beam dynamical scattering " Consider Bragg condition combined with high probability of scattering!b 000 ghkl 14
2-beam dynamical scattering " Consider Bragg condition combined with high probability of scattering!b 000 ghkl 15 2-beam dynamical scattering " Consider Bragg condition combined with high probability of scattering!b 000 ghkl 16
2-beam dynamical scattering " Consider Bragg condition combined with high probability of scattering!b 000 ghkl 17 2-beam dynamical scattering " Consider Bragg condition combined with high probability of scattering!b 000 ghkl 18
2-beam dynamical scattering " Consider Bragg condition combined with high probability of scattering!b 000 ghkl 19 2-beam: Ig as function of excitation error s " For sample thickness t, excitation error s, from Howie-Whelan equations or Bloch wave theory the following can be derived: Intensity in the diffracted beam: Intensity in the direct beam: Extinction distance: 20
2-beam dynamical scattering " Plot Ig vs s for different t " Model using: #g= 100 nm #'('%)*'"# #'('+)*'"# $ % $ % envelope!"!"# $% &!"!"# $% & " As thickness t increases intensity curve modulates more quickly in s 21 2-beam dynamical scattering " Plot Ig vs s for different t " Model using: #g= 100 nm #'('%)*'"# #'('++*'"# $ % $ %!"!"# $% &!"!"# $% & " Only have maximum in Ig at s = 0 for: (integer n) 22
2-beam dynamical scattering " Plot Ig vs s for different t " Model using: #g= 100 nm #'('%)*'"# #'('+**'"# $ % $ %!"!"# $% &!"!"# $% & " When: (integer n) then: Ig = 0 at s = 0 23 Thickness fringes " For s = 0: " Ig and I0 modulate in t, and are in anti-phase with each other: " When thickness t = n#g (integer n) intensity in diffracted beam Ig = 0.! This is why #g is called the extinction distance 24
2-beam imaging of thickness fringes " Use cleaved wedge sample of Si to study thickness fringes " 90 wedge of Si of increasing thickness imaged in projection " Excite 2-beam Bragg condition (s = 0) and take bright-field and dark-field images " See bright and dark fringes from g and 000 reflections in anti-phase Bright field Dark field 25 2-beam imaging of thickness fringes " Extract normalised intensity vs thickness profiles from images " Compare to dynamical scattering simulation Simple model (no absorption): $ * $! 26
2-beam imaging of thickness fringes " Extract normalised intensity vs thickness profiles from images " Compare to dynamical scattering simulation: model with absorption works very well! Model with absorption: $ * $! 27 Thickness fringes as function of s " As excitation error s increased, fringes modulate faster in t " Demonstrate with g(3g) weak beam imaging condition with very large s " s ~0.1 nm 1 therefore dominates over #g term in: Dark field s = 0 Dark field s ~0.1 nm 1 28
Thickness fringes in nanocrystal Even without choosing specific diffraction conditions, crystalline objects of varying thickness often show thickness fringes, depending on their diffraction condition Such fringes are regularly seen in bright-field images of strongly diffracting nanocrystals, such as this BaTiO3 powder: 29 Bend contours Thinned foils of crystalline samples are very often bent, e.g. from internal stress relief as the material is made thinner As the crystal bends, the local diffraction condition changes. For instance an (hkl) plane in the 2-beam condition will only be at s = 0 at the locations where the plane is at the exact Bragg condition relative to the incident e beam This leads to changes in contrast known as bend contours Bend contours: bright field Bend contours: dark field 30
Bend contour formation " Parallel incident e beam illuminating deformed sample " Orientation of crystal plane with respect to incident beam depends on location 31 Bend contour formation (dark-field) " Parallel incident e beam illuminating deformed sample " Orientation of crystal plane with respect to incident beam depends on location Deformed Ni 3 (Al,Ti) superalloy θ! 32
Bend contour formation (dark-field) " Parallel incident e beam illuminating deformed sample " Orientation of crystal plane with respect to incident beam depends on location " Bending of lattice equivalent to rocking Ewald sphere through the relrod!! DF image samples Ig vs s Deformed Ni 3 (Al,Ti) superalloy k D k I g 0 33 Bend contour formation (bright-field) " At first approximation see dark lines in bright-field image wherever a lattice plane comes into the exact Bragg condition! parallel pair of dark lines for each plane " Exact intensity profiles more complicated because of full dynamical scattering nature (e.g. modelled using Bloch wave theory) Bright-field image 34
Bend contour formation (bright-field) " At first approximation see dark lines in bright-field image wherever a lattice plane comes into the exact Bragg condition! parallel pair of dark lines for each plane " Exact intensity profiles more complicated because of full dynamical scattering nature (e.g. modelled using Bloch wave theory) Bright-field image θ! θ! 35 Bend contour formation (zone axis) " At low index ( important ) zone axis pairs of dark lines for each diffracting plane converge together " On zone axis exact intensity patterns more complex from full dynamical scattering " Can use this to help find zone axis in your sample! Deformed crystal on zone axis SADP from zone axis centre 36
Bend contours: consequence for imaging " For precise diffraction contrast (e.g. dark-field) imaging of crystal phases or strong beam imaging of defects, need very flat region of interest " If sample is distorted, condition only holds along bend contour and images are poor Dark-field images of $ -Ni3(Al,Ti) phase: Flat region of interest Bent region of interest 37 Dynamical scattering: double diffraction " Double diffraction is a type of dynamical scattering where the re-diffraction of one beam to another beam produces diffraction spots which are kinematically forbidden " Common example: silicon diffraction pattern on [1 1 0] zone axis: Kinematical simulation % % $"$"$ %!"!"# % $"$"$!"!"#!"!"! %% $"$"$ Experimental SADP 002 002 % $"$"$ 38
Dynamical scattering: double diffraction " Double diffraction is dynamical scattering where the re-diffraction of one beam to another beam produces extra diffraction spots which are kinematically forbidden " Common example: silicon diffraction pattern on [1 1 0] zone axis: Kinematical simulation Experimental SADP % % % $"$"$ $"$"$!"!"# % % % $"$"$!"!"! % $"$"$ 0 0 2!"!"# 0 0 2 0 0 2 " Example double diffraction event: 39 Ewald sphere/reciprocal lattice representation " Diffracted beam wave vector kd acts as direct beam for new scattering event " Centre additional reciprocal lattice, Ewald sphere at end of kd " Convolution of reciprocal lattice with itself 40
Ewald sphere/reciprocal lattice representation " Diffracted beam wave vector kd acts as direct beam for new scattering event " Centre additional reciprocal lattice, Ewald sphere at end of kd " Convolution of reciprocal lattice with itself 41 Ewald sphere/reciprocal lattice representation " Diffracted beam wave vector kd acts as direct beam for new scattering event " Centre additional reciprocal lattice, Ewald sphere at end of kd " Convolution of reciprocal lattice with itself 42
Ewald sphere/reciprocal lattice representation " Diffracted beam wave vector kd acts as direct beam for new scattering event " Centre additional reciprocal lattice, Ewald sphere at end of kd " Convolution of reciprocal lattice with itself 43 Double diffraction by second lattice " Re-diffraction by another crystal lattice can also occur if e -beam propagates through two different superposed lattices " Leads to formation of characteristic satellite spots Example: NiO reflections re-diffracted by Ni during in-situ NiO reduction Epitaxial relationship between the two FCC structures (NiO: a = 0.42 nm Ni: a = 0.37 nm) Images by Quentin Jeangros, EPFL/CSEM 44
Summary on imaging and dynamical effects Good TEM imaging requires correct use of objective defocus and astigmatism Imaging crystalline objects will often be subject to delocalisation (ghost images) use bright-field imaging with an objective aperture to cut these out! As the beam propagates across at TEM sample, we typically have dynamical scattering: multiple elastic scattering Intensities in different diffraction spots are then interdependent and therefore hard to interpret Dynamical scattering also leads to effects of thickness fringes, bend contours and double diffraction in your TEM data 45