Physical Principles of Electron Microscopy. 2. Electron Optics

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1 Physical Principles of Electron Microscopy 2. Electron Optics Ray Egerton University of Alberta and National Institute of Nanotechnology Edmonton, Canada

2 Properties of an ideal image (Maxwell s rules) 1. For each point in the object there is an equivalent point in the image. Image defects: defocus (first-order focusing) aberrations (point à disk of confusion) 2. The object pattern and image pattern are similar. Defect: distortion (barrel, pincushion, spiral) A triangle imaged with magnification and inversion, showing ray paths. Points A,B,C are equivalent to a,b,c and the triangles are similar. M(r) increase M(r) decrease f(r) non-zero 3. If object is planar and perpendicular to the optic axis, so is the image. Defect: curvature of field

3 Imaging in Light Optics is based on refraction Snell s law: n 1 sinθ 1 = n 2 sinθ 2 Refraction by a glass prism For small angles, deviation α ~ (n-1)φ Focusing by a convex lens à (think of it as a prism whose angle increases with distance from the optic axis)

4 Fermat s Principle of least time (ray & wave optics) Total time T for travel between O and I is a minimum (otherwise the phases of alternative paths are almost random, so the net amplitude becomes zero). T = 4f/c t/c + n t /c path A = (2/c)[(2f) 2 + r 2 ] 1/2 path B Making these two times equal gives a formula for the focal length f, which can be compared with the Lensmakers Formula: path A t r path B 1/f = (n-1) [2/R 2 - (1-1/n)t/R 2 )] R is the radius of curvature for both lens surfaces and r 2 = Rt - t 2 /4 from geometry. 2f 2f

5 Ray diagrams with a few special ray paths: Thin-lens ray diagram (geometric optics) showing first-order focusing A ray parallel to the optic axis defines the focal length of the lens Thin-lens equations: 1/u + 1/v = 1/f (real image: u and v are positive) M = x i /x 0 = v/u (usually take M > 0 for a real image)

6 Lens aberrations (higher-order focusing) Axial aberrations: Spherical aberration: f depends on distance from optic axis Chromatic aberration: f depends on radiation wavelength (dispersion) Both of these give a circular disk of confusion. Axial astigmatism: f varies with azimuthal angle φ. Gives elliptical line foci from a non-round lens. Off-axis aberrations (less important in electron optics): Coma: each object point à comet-like effect in the image Third-order astigmatism à line foci, even for a round lens

7 Electron optics Ray bending cannot use solid lenses (too much scattering) 1. Electrons are charged particles (electrostatic deflection) Electrostatic force: F = (-e) E Used for beam deflection and focusing (axial symmetry) 2. Moving electron is equivalent to an electrical current, so there is a magnetic (Lorentz) force: F = -e (v x B) More effective when the electron speed v is high (limit is speed of light) Also used for deflection and focusing.

8 Electrostatic (einzel) lens Used in CRT display tube (TV, oscilloscope), ion optics (e.g. FIB machine, v is lower than for electrons) Advantages: lightweight, low power, no image rotation, self-compensating for voltage fluctuations Disadvantages: high voltage (insulation, surface breakdown), higher aberrations, no immersion lens for conducting specimen

9 Magnetic lens (usually electromagnetic) 1. Uniform field: point-to-point focusing > but cannot focus a parallel beam. 2. Localised field (axial symmetry) : B 1 rev. coil carrying direct current rotation through angle φ

10 Electron trajectories bell-shaped towards axis Div(B) = 0 à B r = -(r/2)(db z /dz) F φ = - e (v z B r ) + e (B z v r ) produces angular speed v φ, then F r = - e (v φ B z ) gives radial force towards axis F z = (v φ B r ) ß net effect = 0

11 Comments on magnetic focusing No force in direction of electron travel, so electron speed v remains constant at all times. Kinetic energy (KE) always stays the same. Rotation effect is ignored in e-ray diagram (plot of r versus z). Radial field B r causes force F r towards axis (focusing) and is present only where db z /dz is non-zero (e.g. in fringing field at the entrance and exit of a solenoid). So a long uniform field (as in a solenoid) is unnecessary. For efficiency, keep the field as short as possible, using polepieces.

12 Practical lenses Soft magnetic material surrounds windings except at the lens gap For mechanical stability, the electron column is vertical with O-ring seals between lenses. Cooling water ensures constant temperature and low thermal drift.

13 Focusing power of a magnetic lens in the thin-lens approximation (a << f ): electron KE For Lorentzian field B z = B 0 /(1 + z ß much lower for ions Reversing the lens current reverses the image rotation but does not change the focusing power

14 Defects in electron lenses: Spherical aberration mid-plane of thin lens Focus change Δf increases with distance x from axis. Axial symmetry implies Δf = c 2 x 2 + c 4 x 4 + But x = f 1 tanα ~ f α and r s = Δf tanα ~ Δf α so r s ~ c 2 (fα) 2 α = C s α 3 where C s is the coefficient of spherical aberration For a broad beam (radius x), r s is the radius of the disk of confusion. Paraxial focus is at F (Gaussian image plane) Screen could be moved to the left to display a smaller disk of least confusion.

15 Practical cases of spherical aberration demagnification e.g. SEM objective magnification (electron rays reversed) magnification (Rayleigh criterion)

16 Chromatic aberration concerns the kinetic energy E 0 of the electrons 1/f = (π/16)(ab 02 )[e 2 /(8mE 0 )] à f = c E 0, Δf = c ΔE 0 = (f/e 0 ) ΔE 0 If E 0 decreases, f decreases à focus closer to lens (dashed rays below) object plane image plane M = v/u = tanα / tanβ ~ α/β r i = Δv tanβ ~ β Δv, equivalent to r c ~ β Δv/M ~ α Δv/M 2 1/u + 1/v = 1/f à Δv = (v 2 /f 2 ) Δf and M>>1 à u ~ f, v ~ Mf So Δv = M 2 Δf and r c ~ α (Μ 2 Δf)/M 2 = α Δf = α (f/e 0 ) ΔE 0 More generally, r c = C c α (ΔE 0 /E 0 ) where C c = f approximately Rayleigh criterion

17 Calculations for an objective lens with a = 1.8 mm and E 0 = 200keV using thin-lens approximation and more accurate theory (Glaeser, 1952) 1. C s and C c are roughly equal to f 2. Small r s and r c require a strong lens (small focal length ~ 1 or 2 mm) Causes of chromatic aberration: 1. Spread of kinetic energy from electron source 2. Fluctuations in accelerating voltage (drift and ripple) 3. Energy losses in specimen, due to inelastic scattering

18 Axial astigmatism arises from different 1/f in the x-z and y-z planes à two elliptical line foci It is a parasitic aberration, due to imperfection in lens-polepiece material or machining Solution: use a weak quadrupole lens (called a stigmator)

19 Electrostatic (a) and magnetic (b) stigmators viewed along the optic axis Need to adjust amplitude and rotation (mechanically or electrically) while sweeping through focus. In light optics, a weak cylindrical lens is used to correct astigmatism of the eye

20 Distortion R = Mr + C d r 3 C d > 0 gives pincushion distortion, C d < 0 gives barrel distortion. Related to spherical aberration, C s > 0 à C d > 0. Arises mainly from final (projector) lens in TEM. Distortion is more visible at low magnification. Important in high-resolution (lattice) imaging of crystals but does not lead to loss of resolution. Curvature of field is unimportant in TEM because the depth of focus is large

21 Correction of lens aberrations Scherzer (1936) proved that any electron-optical system will suffer from spherical and chromatic aberration if all of the following are true: 1. The optical system has rotational (axial) symmetry 2. The system produces a real image of the object 3. The focusing fields are time-independent 4. No charge is present on the electron-optical axis 3. is violated in particle accelerators in order to correct chromatic aberration and is being pursued in femtosecond electron optics. [also allows correction by an electrostatic mirror, where v is reversed] 4. has been investigated in the TEM by using foils or mesh, one problem being hydrocarbon contamination 2. seems a necessary requirement (intermediate virtual images don t help) 1. has been the main target of development, using weak multipole lenses

22 Multipoles Dipole is used for beam deflection (scan coils) E r Weak quadrupole used to correct axial astigmatism Strong quadrupole lenses are used in particle accelerators Sextupoles & octupoles are used for aberration correction in the TEM and SEM Diagram by A. Bleloch & Q. Ramasse in Aberration-Corrected Analytical Electron Microscopy (Wiley, 2011). E=constant Multipoles E r 3 E r 2 Dipole is used for beam deflection (scan coils) Weak quadrupole used to correct axial astigmatism Strong quadrupole lenses are used in particle

23 Quadrupole-Octupole corrector (Scherzer 1947, Krivanek 1994, Nion STEM) y-cor. 4-fold stig. cor. x-cor. Diagram by A. Bleloch & Q. Ramasse in Aberration-Corrected Analytical Electron Microscopy (Wiley, 2011).

24 Aberration-corrected STEM (Nion Company, Seattle)

25 Hexapole (sextupole) corrector (Hawkes 1965, Rose 1981, Haider 1998, CEOS TEM/STEM corrector) transfer lenses long Hex1 Hex2 rotated 60deg Diagram by A. Bleloch & Q. Ramasse in Aberration-Corrected Analytical Electron Microscopy (Wiley, 2011).

26 CEOS sextupole corrector