Charged Particle Optics Radovan Vašina Application of charged particle optics Charged particle optics is/was used in the following areas: - Cathode ray tubes television, oscilloscopes, radars – obsolete - Electron and ion microscopes wavelength reduction to enhance resolution - Electron and ion lithographes - Particle accelerators - Plasma coating, microwave magnetron Vybrané partie z elektronové mikroskopie Comparison: Charged Particle and Light Optics Feature Charged Particle Optics Analogy in Light Optics Optical elements Electrostatic and/or magnetic field Glass and transparent materials Optical axis must be Vacuum Transparent Lenses Variable focus Fixed focus Focusing Changing field strength Moving lens or object Exchange lens Deflection and scanning Electrostatic and/or magnetic Mechanical Aberrations Not correctable at round lenses Easily correctable Wavelength ~ 2 – 200 pm (electrons) ~ 200 – 1000 nm Depth of focus High Low Magnification 2x – 1000 000x 1x – 2000x Maximum resolution 1 nm – 0.1 nm 500 nm Vybrané partie z elektronové mikroskopie Overview of optical elements Optical element Charged particle optics Analogy in Light Optics Sources Hairpin, Schottky emitter, CFEG Arc lamp, laser, LEDs etc. „Round“ lenses Magnetostatic, electrostatic lens Convergent, divergent lenses Apertures Round, annular, arrays Many types Deflectors Magnetostatic, electrostatic ~ Prisms, gratings, mirrors Multipoles Magnetostatic, electrostatic ~ Cylindric lenses Mirrors Electrostatic Concave, convex mirrors Grids Electrostatic Immersion lens Immersion lenses Immersion objectives Exotic Wien filter, RF cavities Vybrané partie z elektronové mikroskopie Creation of a charged particle optics device Mechanical design Calculation of the fields Calculations of the beam trajectories and optical properties Optimization iterations to get the performance Tolerancing Models Controlling software Vybrané partie z elektronové mikroskopie Basic facts Electron charge e=1.602x10-19C Electron mass m0=9.109x10-31kg Proton/electron mass ratio 1826 Relativistic energy = Non-relativistic = Φ = Electron wavelength = ! = " #$ Non-relativistically % = . &' Device Energy [keV] Wavelength [m] TEM 50 ÷ 1000 5.46 ÷ 1.22 x 10-12 SEM 1 ÷ 30 38.6 ÷ 7.04 x 10-12 Vybrané partie z elektronové mikroskopie Lorentz force Force acting on a single particle in a electrostatic and magnetostatic fields ( () * = e + , × . Electrostatic field - conservative - Acts always on the particle - Changes the particle energy and velocity - Usage up to ~30kV - Coulomb interaction among particles Magnetostatic field – not conservative - Acts only on a moving particle - Does not change energy of the particle - Usage up to energies of GeV and more - EFe - B Fm v Vybrané partie z elektronové mikroskopie Relativistic effects Vybrané partie z elektronové mikroskopie 0.00E+00 5.00E+07 1.00E+08 1.50E+08 2.00E+08 2.50E+08 3.00E+08 3.50E+08 1.0E+00 1.0E+02 1.0E+04 1.0E+06 1.0E+08 velocity[m/s] particle energy [eV] electron proton 1 10 100 1000 1.0E+00 1.0E+02 1.0E+04 1.0E+06 1.0E+08massincrease[1] particle energy [eV] electron proton Non relativistic acceptable Relativistic correction needed The relativistic effects can be neglected at ions Usage of relativistic potential beneficial for systems with no electrostatic lenses Φ/ = Φ 1 + 0Φ Paraxial approximation Vybrané partie z elektronové mikroskopie Ez Er dl r 12 = 0 4 15, = 0 267 8 + 67 9 + : 9 :; 1< − 67 9 = 0 8 = − 7 2 : 9 :; In an analogous way .8 = − 7 2 :.9 :; Linearity of the field only near to the optical axis The paradigm of the charged particle optics Angles and lateral distances from the axis are small Energy width is small All higher orders in the potential and field series are neglected Determination of the field (in cylindrical CS) Vybrané partie z elektronové mikroskopie EOD J.Zlámal&B.Lencová MEBS E.Munro&J.Rouse FEM with linear elements FEM with quadratic elements Finite difference methods Simion FEM 3D Ansys, Comsol, CST Boundary element methods Symmetry n of the field with optical elements: 2n multipoles n = 0 Round lenses - focusing n = 1 Dipoles – deflectors n = 2 Quadrupoles - stigmators n = 3 Hexapoles - correctors n = 4 Octopoles – universal % ∈ 1,4 n > 4 Multipoles - correctors Symmetry of the fields and OEs Vybrané partie z elektronové mikroskopie Axisymmetrical fields (n=0) Vybrané partie z elektronové mikroskopie Magnetic field of a round lens Electrostatic field of a round unipotential lens Axial field Bz Axial field Ez Axial fields Total electrostatic field can summed up as a weighted sum of the contribution of particular electrodes Assigning the voltages to the respective electrodes - Magnetostatic field - Axial field - Deflector magnetostatic field - Axial field Assigning the currents to the respective coils calculated with excitation 1A*Number of turns 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -20 -15 -10 -5 0 Potential[V] z[mm] 0 0.02 0.04 0.06 0.08 0.1 0.12 -60 -40 -20 0 Bz[T] z[mm] Vybrané partie z elektronové mikroskopie Equation of motion Vybrané partie z elektronové mikroskopie Time dependent Relativistically corrected potential for electrons Time independent A, = q + , × . C 1 1D , 1 − E = F + , × . Φ/ = Φ 1 + 0Φ 0 = 2 E G = 2 1 1; Φ/ 1 + H′ + J′ H′ = − 1 2 + 0Φ 1 + HK + J′ Φ/ L + G .M − J′.9 1 1; Φ/ 1 + H′ + J′ J′ = − 1 2 + 0Φ 1 + H′ + J′ Φ/ M + G −.L + H′.9 Integration methods Vybrané partie z elektronové mikroskopie The initial value problem JN = O D, J J DC = JC Runge Kutta Bulirsch Stoer - Richardson extrapolation ℎ → 0 - Using rational functions for extrapolation - Error function of the integrator dependent on ℎ Interpolation Vybrané partie z elektronové mikroskopie A general problem: The value of the field known only at discreete points (FEM not BEM). The integration routine needs to obtain the accurate field values at a general place. Interpolation of the axial fields: paraxial approximation - Cubic or quintic splines: cubic or quintic polynomials with continuous 2nd or 4th derivative - Fourier Bessel series - Hermite series Interpolation Vybrané partie z elektronové mikroskopie Interpolation of general data: - Bicubic spline - ZRP method – interpolation using Laplacian base functions (author J.Chmelík) Using a local coordinate system H = ; − ;R, J = 7 − 7S Φ H, J = ∑ UVWV H, JX VY where WV H, J fulfill Laplace equation. The coeficients are fitted from the 4-8 nearest neighbor points with Singular Value Decomposition method. Weight of the points depends on the distance from the element center Analogy with light optics Basic phenomena of light optics valid in the charged particle optics Thin lens approximation Z = S + [ But they are thick lenses two main planes in case of immersion lenses two focal lenghts direct magnification \ = ] ^ angular magnification \_ = _] _^ a b f Vybrané partie z elektronové mikroskopie Ordinary differential equation of the second order 2 independent solutions for the initial conditions 7S ;` = 0, 7′S ;` = 1 ⇒ 7S ;V = 0, 7′S ;V = \S 7[ ;` = 1, 7′[ ;` = 0 ⇒ 7[ ;V = \ 7′[ ;V = − 1 OV Relationship between direct and angular magnification Φ/ / ;V Φ/ / ;` \\S = 1 Solution of the equation of motion Vybrané partie z elektronové mikroskopie rb zo zi ra rb(zi) Electrostatic lens: how it focuses 1. Intensity Er changes quickly, change of the radial velocity 2. Intensity Ez is constant, change of axial velocity 3. Smaller intensity -Er change of the radial velocity Axial symmetry of the fringe field causes linear focusing effect 1. 2. 3. Vybrané partie z elektronové mikroskopie Electrostatic lenses Vybrané partie z elektronové mikroskopie 1 electrode lenses: 2 electrode lenses: 3 electrode lenses: 4 and more aperture lens accelerating, decelerating unipotential zoom lenses Magnetostatic lens: how it focuses 1. 2. 3. 1. Magnetic induction Br changes quickly, a change of the tangential velocity 2. Magnetic induction Bz is constant, change of radial velocity 3. Magnetic induction Br changes quickly, the tangential velocity stops Axial symmetry of the fringe field causes linear focusing effect Vybrané partie z elektronové mikroskopie Properties of magnetostatic lenses Vybrané partie z elektronové mikroskopie First order properties O = c + 1 d 10 e 1 − e e = fg 300 Φ/ 1 = 1 + 1 2 Excitation parameter HE = Xh '/ scaling laws → trajectories for same HE same Rotation Axial aberrations UiV ∝ k l m Zn ik( ( U V ∝ 1 + \ f d2 d1 s Coil o = 2 p .9 ; Φ/ 1; Properties of deflectors Vybrané partie z elektronové mikroskopie Electrostatic (weak) perpendicular field: (parabolic trajectory) Derivative of the movement (L (9 = qrs ' Deflection does not depend mass and charge – usage at focusing elements ion devices FIB, SIMS Magnetostatic (weak) perpendicular field: (circular trajectory) Derivative of the movement (M (9 = s t = ur "v w Deflection does depend on the mass and charge – filtering elements at ion devices Weaker dependency on the particle energy – usage of transversal fields at accelerators Deflectors A pair of deflectors can manipulate with the beam In the lateral position and angularly HEx!!&8 ≈ − z. ee {2cD HEs`|&8 ≈ z 1 + ee {2cD Pivot point is the position in the column where the beam rocks about Dist +PP -PP α α Pivot point Pivot point Vybrané partie z elektronové mikroskopie Quadrupole multipole with n=2 Forces on the charged particle Fx independent of y position and proportional to x position Fy independent of x position and proportional to y position Field/forces on the axis is/are zero Usage as stigmator Light optics analogy Properties of quadrupoles Vybrané partie z elektronové mikroskopie The final lens is not perfect: the focusing ability is varies along the azimuthal angle with a π period The stigmator shifts the position of the virtual object differently in the orthogonal planes, depending linearly in each plane How stigmation works 1 OL = } sin } • 1 OM = − } sinh } • Stigmator Final lens Ideal focus Y–Z plane X–Z plane Focusing Defocusing Object = crossover Vybrané partie z elektronové mikroskopie Multipoles octupole quadrupole 2 dipole 1 dipole 2 Superposition of all +U -U +U -U +U -U +U -U +Q1 +Q1 -Q1 -Q1 +Q1 +Q1 -Q1 -Q1 -Q2 +Q2 +Q2 -Q2 -Q2 +Q2 +Q2 -Q2 +D1 +D1 +D1 +D1 -D1 -D1 -D1 -D1 +D2 +D2 -D2 -D2 -D2 -D2 +D2 +D2 +U+Q1- Q2+D1 +D2 - U+Q1+ Q2+D1 +D2 +U- Q1+Q2 +D1-D2 -U-Q1- Q2+D1- D2 +U+Q1- Q2-D1- D2 - U+Q1+ Q2-D1- D2 +U- Q1+Q2- D1+D2 -U-Q1- Q2- D1+D2 quadrupole 1 Vybrané partie z elektronové mikroskopie Scaling with Accelerating Voltage • Equal trajectories if qƒ ' = E„%cD • Equal trajectories if uƒ '/ = E„%cD • Particle mass depends on Φ! Use relativistic corrected Φ/ Consequences: – Electrostatic field does not and magnetic field does split particles according mass and charge – Permanent magnet effect ∝ '/ – Charging effects ∝ ' – Geometrical misalignment effects do not depend on Φ – Magnetic saturation effects at high Φ Vybrané partie z elektronové mikroskopie Aberrations Vybrané partie z elektronové mikroskopie Aberrations cause blurring of an point source image -Diffraction aberration due to wave character of particles -Geometrical aberrations describing deviation from the linear behavior -Chromatic aberrations due to finite energy width of the beam -Parasitic aberrations … stemming from the imperfection in manufacturing bad alignment instabilities of HT and power supplies thermal drift of the column assemblies lack of homogeneity of the lens materials Axial aberrations Correction not possible in charged particle optics in such an easy way Scherzer’s theorem (1936) Electromagnetic lenses have unavoidable aberrations (spherical and chromatic) as long as the following conditions are fulfilled: - Lens fields are rotationally symmetric → mul[poles - The electromagne[c fields are sta[c → RF operation - There is no space charges → electron mirrors 1… = 0.18Uiz† 32 bfafCS +≈ 1‡ U‡ ∆ z fCC ≈ Vybrané partie z elektronové mikroskopie Aberrations Vybrané partie z elektronové mikroskopie Deviation of the real wavefront from the ideal one ),( φθχ Ideal wavefront real wavefront Aberration function With the length dimension [m, μm, …] Dependent on the radial angle o and azimuthal angle ‰ Precision needed λ/4 ~ 1pm Relative precision ~ deviation of the wavefront / diameter of the wavefront in the objective …i.e. 1pm/100μm = 10-12/10-4 = 10-8 Comparable with modern astronomical telescopes Aberrations Vybrané partie z elektronové mikroskopie ( ) ( ) ( ){ } ( )∑∑ ++= ++ n m n bmn n amn nmCmC 1/sincos, 1 ,, 1 ,, φθφθφθχ Representation of geometrical aberration function Křivánek, Delby & Lupini Where θ …semiangle ∈ 0 ÷ ~50 mrad Φ …azimuthal angle ∈ 0 ÷ 2π n …order ∈ 0 ÷ 7 m …multiplicity ∈ 0 ÷ 6 For odd n even m = 0, 2, 4 …n+1 e.g. C1,0; C1,2; C3,0;C5,2 For even n odd m = 1, 3, 5 …n+1 e.g. C0,1; C2,1;C2,3;C4,5 For m = 0 no azimuthal dependence is no term Cn,0,b e.g. C3,0 Wavefront errors Vybrané partie z elektronové mikroskopie Defocus C1,0 Astigmatism C1,2,a Axial coma C2,1,b 3-f Astigmatism C2,3,a Spherical aberr.C3,0 Spherical aberr.C5,0 Resolution at scanning probe systems Vybrané partie z elektronové mikroskopie Resolution depends on: - Spherical axial aberration - Chromatic axial aberration - Diffraction aberration - Spot size influence by the brightness - Coulomb interactions - Many other effects…noise, vibrations, stray magnetic field 0 5E-09 1E-08 1.5E-08 2E-08 2.5E-08 3E-08 0 20 40 60 80 D50[m] Aperture diameter [um] proportional to alphai D50 resulting D50 brightness D50 diffraction D50 spherical D50 chromatic αi EBA aperture angle αo Angular magnification of the final lens Image creation Full field systems Advantages: - traditional approach, - image instantaneously available - high resolution due to optics Disadvantages: - more complex, more expensive - pixelated detectors - complicated specimen preparation Scanning probe systems: Advantages: - simpler, cheaper, for any specimens - integral detectors for various signals Disadvantages: - image not present at one instant - resolution due to scanning Vybrané partie z elektronové mikroskopie State of the art SEM: general specifications Parameter Value(s) Landing energy 20 ÷ 30000 eV ~3 orders of magnitude Probe current 0.78 pA ÷ 410 nA ~6 orders of magnitude Maximum/minimum field of view 3 mm / 50 nm ~6 orders of magnitude Resolution <1 nm Beam current control Continuous Vacuum modes Hi Vac (~10-4Pa), Low Vac (10-50 Pa) Vybrané partie z elektronové mikroskopie Different modes Full frame, Line, Spot, Pattern Channeling contrast STEM Crossover Mirror Scanning electron microscope -4000V Detector +8kV Diagnostics detector Vybrané partie z elektronové mikroskopie Imaging Scanning electron microscope • Electron or ion source • Lenses • Mechanical alignment • Alignment deflectors • Scanning deflectors linear device • Stigmators • Apertures • Magnetic shielding • Vacuum 10-4 to 10-8 Pa • Differential pumping • Mechanical stiffness, vibration and acoustic resistance • Drifts • Detectors • Stable sources • Alignments • Control model and SW Vybrané partie z elektronové mikroskopie Ion Columns • Various sources and species: Ga, In, Li, He, Ar • Electrostatic elements • Coulomb interactions crucial for resolution • Sputtering of apertures • Higher sensitivity on vacuum – beam sputtering • Micromachining and Gas-assisted etching • Destructive method • Volume sputtered is independent of the scan field Vybrané partie z elektronové mikroskopie Workflow for electronic industry 1. Select the region of interest 2. Dig out the material in the vicinity 3. Form the lamella, cut on one side 4. Approach the needle 5. Weld the needle and lamella, cut the lamella 6. Transport the lamella to special holder, weld it 7. Cut the needle, repolish the lamella to ~20nm 8. Transport to TEM Vybrané partie z elektronové mikroskopie Design issues • Colinearity and circularity of the optical elements better in sub-µm region • Assembling of heavy parts with micron precision – different systems for mechanical alignment: kinematical mounting (cone, prism & plane), deformable string of balls + V grooves • Vacuum issues: • usage of a liner – thin non magnetic tube – excludes electrostatics • absence of a liner implies usage of vacuum sealed magnetostatic lenses • pressure gradient from the chamber up to the electron gun – differential pumping Electrostatic lenses • Electrostatic breakdown on the electrostatic lenses: vacuum 10kV/mm, surface much less – rounded shapes of the insulators: ruby balls, PEEK, Macor, ceramics Vybrané partie z elektronové mikroskopie Design issues • Saturation in the yokes of the magnetostatic lenses: design, high saturation soft magnetic materials: ARMCO iron (pure annealed Fe 2.15 T), permalloy (48% Ni and Fe) • Remanence – degaussing rigorous but lengthy normalization easier • Soft magnetic material turned before and after annealing • Mechanical strain destroys the ferromagnetic properties • Thermal load of the magnetostatic lenses due to the Joule heating water cooling ~ 1000 W – vibrations, drifts Vybrané partie z elektronové mikroskopie Design issues Design constraints results in the physical dimensions of the lenses Magnification in the full field view systems obtained by ratios of object and image distance → TEM length ~ 3 m : condenser + objec[ve + projec[ve Scanning probe system more length effective ~ 0.5 m: condenser + final lens Charging of the materials Almost all design materials are covered with thin semi or non conductive oxide layer Solution: covering with carbon Vybrané partie z elektronové mikroskopie Electrostatic versus Magnetic Electrostatic Magnetostatic Drivers High voltage High currents Environment Vacuum Non magnetic Column heating no yes Scaling with beam energy ~ HV ~ HV 1/2 Scaling with particle mass no ~ m 1/2 Speed Faster Slower Accuracy High Lower (hysteresis) Cost More expensive Less expensive Application area Ion optics Low energy electrons Fast systems (beam blanker, lithography) High energy electrons (TEM) Low cost systems Vybrané partie z elektronové mikroskopie Alignments Ideal system Real system Aligned real system Pairs of deflectors are mostly used for alignments. The position of the pivot point defines their usage. Deflections with different pivot points can be added thanks to linearity. Vybrané partie z elektronové mikroskopie Thank you for your attention Vybrané partie z elektronové mikroskopie Vybrané partie z elektronové mikroskopie Further reading - M. Lenc, B. Lencová: Optické prvky elektronových mikroskopů Metody analýzy povrchů 2. Elektronová mikroskopie a difrakce. (L. Eckertová, L. Frank ed.) Academia 1996 - B. Lencová, M. Lenc: Optika iontových svazků. Metody analýzy povrchů 3. (L Frank, J.Král, ed.), Academia 2002 - L. Reimer, Scaning Electron Microscopy -- Physics of Image Formation and Microanalysis, Springer-Verlag 1998 - J. Orloff, ed. Handbook of Charged Particle Optics, CRC Press, 2nd edition 2008 - P. W Hawkes, E. Kasper: Principles of electron optics, vol. 1-3. Academic 1988 and 1996 Determination of the field in 3D Vybrané partie z elektronové mikroskopie Finite differences method (FDM …Simion) Advantages: easy to program Disadvantage: not precise enough Finite elements method (FEM … MEBS, CST, Comsol + open source…) Advantages: covers any geometry, non-linearities, commercial packages Disadvantage: Precise field values only on the nods, rest interpolated, evaluation of the higher order derivatives restriction to the modeled space Boundary element method (BEM … Lorentz, open source ) Advantages: reduces dimensionality of the task by 1, precise field, any geometry, suitable for semi-infinite space Disadvantages: not suitable for non-linear material properties Transfer matrix method Vybrané partie z elektronové mikroskopie Drift space H H′ 1 = 1 • 0 0 1 0 0 0 1 H H′ 1 Lens H H′ 1 = 1 0 0 − 1 O 1 0 0 0 1 H H′ 1 Deflector H H′ 1 = 1 0 0 0 1 g .CG Φ/ 0 0 1 H H′ 1 O = c + 1 d 10 e 1 − e e = fg 300 Φ/ Thin lens approximation