Advanced Transmission Electron Microscopy Lecture 2: Electron Holography by Jame
Davis, Judy, Operations Manager has reference to this Academic Journal, PHwiki organized this Journal Advanced Transmission Electron Microscopy Lecture 2: Electron Holography by James Loudon The Transmission Electron Microscope electron gun specimen (thinner than 200 nm) electromagnetic lens viewing screen Electron Holography When an electron wave passes through a specimen, its intensity in addition to phase change. An image records only the intensity in addition to not the phase. This is un as long as tunate as the phase contains valuable in as long as mation about the electric in addition to magnetic fields in the specimen. The term holography is used to describe an imaging technique which encodes the phase in as long as mation in an image. There are several methods to produce images which contain the phase in as long as mation in addition to the main ones will be covered in this lecture. specimen wavefronts x z y electron wave which went through the specimen electron wave which went through vacuum phase shift between the two rays
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Electron Holography Electron Holography A conventional image measures the intensity I(x,y) = y(x,y)2 = a2(x,y). The phase, f(x,y), is lost. How can we recover it Examples of E in addition to B-fields Measured using Electron Holography Semiconductor physics: built-in voltage across a p-n junction. Nanotechnology: Upper panel: remnant magnetic state in exchange-biased CoFe elements. Lower panel: micromagnetic simulation of the same elements. Field Emission: Electrostatic potential from a biased carbon nanotube. Geophysics: Exolved magnetite elements in the titanomagnetite system. Biophysics: Chains of magnetite crystals which grow in magnetotactic bacteria in addition to are used as long as navigation Refs: (a) Twitchett et al., J. Microscopy 214, 287, 2003. (b) Dunin-Borkowski R.E. et al., J. Appl. Phys. 90, 6, 2899, 2001. (c) Cumings J. et al., Phys. Rev. Lett. 88, 5, 056804, 2002., (d) Harrison R.J. et al., Proc. Nat. Acad. Sci. 99, 26, 16557, 2002. (e) Simpson E.T. et al., J. Phys. Conf. Ser. 17, 108, 2005.
Magnetic Imaging In normal operation, the main objective lens of the microscope applies a vertical field of 2T to the sample. This is obviously undesirable as long as magnetic imaging in addition to so the objective lens is usually turned off in addition to the diffraction lens which is lower down the column ( in addition to is normally used to produce diffraction patterns) is used as an objective lens. Some microscopes like the Cambridge CM300 in addition to Titan TEMs are equipped with a Lorentz lens which has a higher acceptance angle in addition to lower aberrations than the diffraction lens whilst still keeping the sample in a low field. With judicious fiddling, the specimen can be in a field of <5G. Obtaining In as long as mation from the Phase B z electron beam specimen f(0) 0 f(x) x S Constant determined by acceleration voltage Electrostatic potential Magnetic flux density Geometry as long as the integrals Note that the electrostatic potential can either come from specimen charging (not usually what is wanted) or from the mean inner potential, V0, which accounts as long as the fact that electrons travel faster through material than vacuum. Electron beam Electron accelerates Specimen Electron decelerates, returning to its original speed Electric field (or as long as ce or acceleration) z z Electric potential, V V0 Origin of the Mean-Inner Potential Atomic nuclei Electrostatic Contribution to Phase Shift Calculation the same as as long as a Potential Step This is the equation of simple harmonic motion This can be Taylor exp in addition to ed as E/e = 300kV, V0 ~ 10V which gives: The calculation SHOULD BE DONE RELATIVISTICALLY: this changes CE to Schrodinger equation (E is the energy of the electrons) Solution is with So the phase shift is or, if V is not constant m = electron mass, c = speed of light in a vacuum, l = electron wavelength in the vacuum. Magnetic Contribution to Phase Shift For small deflections, q = vx/v Based on Hirsch, Howie, Nicholson, Pashley, Whelan: Electron Microscopy of Thin Crystals q t B e- v In general as long as non-constant B SHOULD ALSO BE DONE RELATIVISTICALLY (but in fact all the relativistic bits cancel) F = ev×B = ma so a = evB/m in addition to vx=a×time To Reiterate: B z electron beam specimen f(0) 0 f(x) x S Constant determined by acceleration voltage Electrostatic potential Magnetic flux density Geometry as long as the integrals Phase Recovery Method 1: Off-Axis Electron Holography The electron biprism is a positively charged wire placed in the column to interfere electrons which went through vacuum with electrons which went through specimen. Note: many people (including me) use the term holography to refer to off-axis holography rather than a collective term as long as methods to recover the phase. How Does Off-Axis Holography Work + How Does Off-Axis Holography Work Separating the Phase To get the phase, we Fourier trans as long as m the intensity in addition to use Inverse Trans as long as m Original Image (called the hologram) Fourier trans as long as m Extract sideb in addition to in addition to put origin at centre wrapped phase vacuum glue 100 nm gives amplitude in addition to phase SrTiO3 SrRuO3 Extracting the Phase cont. Select sideb in addition to Inverse trans as long as m sideb in addition to The original wavefunction! The spatial resolution of the technique is determined by the size of the mask placed around the sideb in addition to . Minor difficulty: the image you recover is the real in addition to imaginary part of the wavefunction. To calculate the phase, you take the inverse tangent (actually arctan2) of the imaginary part upon the real part which gives the phase modulo 2p. So the phase image contains phase wraps which must be removed by adding 2p to selected areas of the image. This can be difficult if there are many phase wraps. Methods as long as Separating B in addition to V Constant determined by acceleration voltage Electrostatic potential Magnetic flux density In a magnetic sample, the phase will be a sum of electrostatic in addition to magnetic contributions. How can you separate B in addition to V Method 1: If the specimen has a uni as long as m thickness (t) in addition to composition, the electrostatic term will just be constant: any changes in the phase will be the result of B only. Separating B in addition to V (cont.) If B is confined to the sample in addition to constant throughout the sample thickness, we can get the component of B normal to the electron beam explicitly as Method 2: If the sample can be heated above its Curie point so that it is no longer magnetic we have The magnetic contribution to the phase is then the difference of these two. See: Loudon J.C. et al. Nature 420, 797, 2002. Separating B in addition to V (cont.) Method 3: If the magnetisation of the sample can be reversed by tilting the specimen in addition to applying a B-field (usually done using the objective lens which can apply a vertical field of up to 2T), we have: The magnetic contribution to the phase is then: This, of course, relies on being able to exactly reverse the magnetisation. See R.E. Dunin-Borkowski et al., Microscopy Research in addition to Technique, 64, 390, 2004 in addition to refs. therein. Separating B in addition to V (cont.) Method 4: If the magnet is hard so that it tends to stay in a fixed magnetic state, holograms can be taken then the sample removed from the microscope in addition to turned upside down when holograms are taken, remarkably, the magnetic contribution to the phase is reversed but the electrostatic contribution remains the same. See R.E. Dunin-Borkowski et al., Microscopy Research in addition to Technique, 64, 390, 2004 in addition to refs. therein. v Electron beam F = -ev × B Turn over v Thin Thick Thin Thick F = -ev × B Phase Recovery Method 2: Out-of-Focus Imaging This technique is also known as Fresnel imaging or in-line holography. Unlike off-axis holography, where electrons which pass through the specimen are interfered with those which pass through vacuum, different regions of specimen are interfered by the simple method of taking an out-of-focus image. This is easier than off-axis holography as no biprism is required in addition to the specimen area of interest does not need to be close to the vacuum so the field of view can be much larger - the field of view achievable by electron holography is ~1 mm. The disadvantage is that getting the phase is difficult. It is good as long as a semi-quantitative overview of the specimen. Example: a specimen with three magnetic domains. Intensity Displacement In-focus image (blank) Out-of-focus image The distance telling you how far out of focus you are is called the defect-of-focus or defocus, Df in addition to is usually measured in mm. Method 2: Out-of-Focus Imaging (a) Magnetic domain walls in a magnetic thin film (of La0.7Ca0.3MnO3) at a defocus of 1.4 mm in addition to (b) a montage of images at different defoci (Df) (c) Magnetic domain walls in Nd2Fe14B. Taken from S. J. Lloyd et al., Phys. Rev. B 64, 172407, 2001 in addition to J. Microscopy, 207, 118, 2002. (c) The Transport of Intensity Equation There is a method of obtaining the phase using out-of-focus imaging. It requires two images equally disposed either side of focus in addition to an in-focus image. Combining Schrodingers equation with the condition as long as a steady electron current in addition to re-expressing the answer in terms of the intensity I in addition to phase f gives the Transport of Intensity Equation: This is a non-linear equation in addition to so difficult (but by no means impossible) to solve in the general case. If, however, the in-focus image has a constant intensity, I0 (this depends on the specimen), the equation simplifies to Poissons equation which can be solved by Fourier methods. Simplifying the Transport of Intensity Equation If the in-focus intensity is constant I0, we have: The Transport of Intensity Equation (TIE): Taking the 2D Fourier trans as long as m gives: (q is the Fourier space coordinate) So
Using the Transport of Intensity Equation x z z = Df z = -Df z = 0 To obtain the phase, take one image at positive defocus, another image at negative defocus in addition to subtract. Fourier trans as long as m, divide the answer by q2 in addition to multiply by all the constants. Inverse trans as long as m in addition to you have the phase. How Well Does TIE Work Flat, circularly magnetised permalloy elements (J.C. Loudon, P. Chen et al. in preparation.) Note: phase images are often displayed as the cosine of the phase: this gives a contour map where there is a phase shift of 2p between adjacent dark lines. The contour maps also resemble magnetic field lines. Phase Recovery Method 3: Foucault or Phase Plate Imaging When a electron is deflected by a magnetic field, the scattering angle is This is much smaller than the scattering angles as long as Bragg scattering from a crystal which are several mrad. The effect of a magnetic field is to shift the diffraction pattern.
Phase Recovery Method 3(a): Foucault Imaging If several magnetic domains are present, the spots in the diffraction pattern are split. If an aperture is used to block one of the split beams, only one set of domains appear bright. This as long as m of dark-field imaging is called Foucault imaging. Phase Recovery Method 3(b): Phase Plate Imaging Instead of blocking one set of beams, a thin sheet of carbon can be used to induce a phase shift in one of the split beams. For magnetic imaging a phase shift of p should be used. The optimal phase shift depends on the object. For weak phase objects, p/2 is best. After some maths, it can be shown that the resulting image should have black field lines with a phase shift of 2p between each. Carbon sheet giving p phase shift. Image has field lines Method 3: Phase Plate Imaging Phase plate image of circular permalloy elements. cos(phase) recovered using TIE. J.C. Loudon, P. Chen et al. in preparation. This method was suggested by A.B. Johnson in addition to J.N. Chapman (J. Microscopy, 179, 119, 1995) as long as visualising magnetic fields in addition to it is very rarely used. It is also not very clear how to obtain the phase itself from a phase-plate image. Phase plates are more commonly used to enhance the contrast from biological specimens.
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