Electron and Ion Optics, Softcover reprint of the original 1st ed. 1988 Microdevices Series

The field of electron and ion optics is based on the analogy between geometrical light optics and the motion of charged particles in electromagnetic fields. The spectacular development of the electron microscope clearly shows the possibilities of image formation by charged particles of wavelength much shorter than that of visible light. As new applications such as particle accelerators, cathode ray tubes, mass and energy spectrometers, microwave tubes, scanning-type analytical instruments, heavy beam technologies, etc. emerged, the scope of particle beam optics has been exten­ ded to the formation of fine probes. The goal is to concentrate as many particles as possible in as small a volume as possible. Fabrication of microcircuits is a good example of the growing importance of this field. The current trend is towards increased circuit complexity and pattern density. Because of the diffraction limitation of processes using optical photons and the technological difficulties connected with x-ray processes, charged particle beams are becoming popular. With them it is possible to write directly on a wafer under computer control, without using a mask. Focused ion beams offer especially great possibilities in the submicron region. Therefore, electron and ion beam technologies will most probably playa very important role in the next twenty years or so.

1. Introductory Survey.- 1-1. Introduction.- 1-2. Electromagnetic Fields.- 1-2-1. Maxwell’s Equations.- 1-2-2. Static Fields.- 1-2-3. Stokes’s Theorem.- 1-3. Some Basic Classical Mechanics.- 1-3-1. Hamilton’s Principle; The Lagrangian Equations of Motion.- 1-3-2. The Maupertuis Principle.- 1-4. A Little Reminder of Geometrical Optics.- 1-4-1. Fermat’s Principle; The Index of Refraction.- 1-4-2. Axially Symmetric Lenses.- Summary.- 2. Motion of Charged Particles in Electric and Magnetic Fields.- 2-1. The Lagrangian.- 2-2. Conservation of Energy.- 2-2-1. Motion of Free Particles; Velocity versus Potential.- 2-3. The Equations of Motion.- 2-4. The Trajectory Equations.- 2-5. The Relativistic Potential.- 2-6. The Electron Optical Index of Refraction.- 2-7. Particles in Homogeneous Fields.- 2-7-1. The Parallel-Plate Capacitor.- 2-7-1-1. Electrostatic Deflection.- 2-7-1-2. A Simple Velocity Analyzer.- 2-7-2. Homogeneous Magnetic Field.- 2-7-2-1. Long Magnetic Lens.- 2-7-2-2. Magnetic Deflection.- 2-7-3. The Simultaneous Action of Homogeneous Electric and Magnetic Fields.- 2-7-3-1. Mass Analysis and Other Applications.- 2-8. Scaling Laws.- Summary.- 3. Determination of Electric and Magnetic Fields.- 3-1. Analytical Methods.- 3-1-1. Series Expansions of Potentials and Fields.- 3-1-1-1. Planar Fields.- 3-1-1-2. Axially Symmetric Fields.- 3-1-1-3. Multipole Fields.- 3-1-2. Analytical Calculation of Axially Symmetric Potential Fields.- 3-1-2-1. Separation of Variables.- 3-1-2-2. Difficulties of Analytical Calculations (Electrostatic Field of Two Equidiameter Cylinders).- 3-1-2-3. Field of a Circular Aperture.- 3-1-2-4. Rapid Evaluation of Fields Produced by Two or More Circular Apertures.- 3-1-3. Analytical Calculation of Multipole Fields.- 3-1-3-1. Short Multipoles.- 3-1-3-2. Long Multipoles.- 3-1-3-3. Ideal Multipoles.- 3-1-3-4. The Method of Conformal Transformation.- 3-1-4. On the Role of Magnetic Materials.- 3-1-5. Analytical Calculation of Magnetic Fields Produced by Currents.- 3-1-5-1. The Biot-Savart Law.- 3-1-5-2. Field of a Straight Wire.- 3-1-5-3. Field of a Circular Loop.- 3.1.5.4. Field of a Thin Solenoid.- 3-1-5-5. Field of a Multilayer Coil.- 3-1-5-6. Field of a Pancake Coil.- 3-2. Measurement of Fields and Analog Methods.- 3-2-1. Measurement of Magnetic Fields.- 3-2-1-1. Electromagnetic Induction.- 3-2-1-2. Hall Effect.- 3-2-1-3. Permalloy and Bismuth Probes.- 3-2-1-4. Magnetic Resonance.- 3-2-2. Analog Methods.- 3-2-2-1. The Electrolytic Tank.- 3-2-2-2. The Resistor Network.- 3-2-2-3. Other Analog Methods.- 3-3. Numerical Methods.- 3-3-1. Accuracy.- 3-3-1-1. Errors Due to the Nature of the Problem.- 3-3-1-2. Errors Due to the Number Representation in the Computer.- 3-3-1-3. Errors Due to the Numerical Method.- 3-3-2. The Finite-Difference Method.- 3-3-2-1. Methods of Solution for Systems of Algebraic Equations.- 3-3-3. The Finite-Element Method.- 3-3-4. The Charge-Density (Integral) Method.- 3-3-5. Numerical Differentiation and Interpolation.- 3-3-5-1. Differentiation.- 3-3-5-2. Lagrange Interpolation.- 3-3-5-3. The Interpolating Pulse.- 3.3.5.4. The Cubic Spline.- Summary.- 4. Focusing With Axially Symmetric Fields.- 4-1. Busch’s Theorem.- 4-2. The General Trajectory Equation.- 4-3. The Paraxial Ray Equation.- 4-4. Image Formation by Paraxial Rays.- 4-5. The Helmholtz-Lagrange Formula.- 4-6. Cardinal Elements.- 4-6-1. Asymptotic Cardinal Elements.- 4-7. Electron and Ion Lenses.- 4-8. Systems of Lenses.- 4-8-1. The Transfer Matrix.- 4-8-2. Combination of Two Thick Lenses.- 4-9. The Thin-Lens Approximation.- 4-9-1. Combination of Thin Lenses.- 4-10. Examples of Paraxial Focusing.- 4-10-1. Paraxial Trajectories in Homogeneous Fields.- 4-10-1-1. Homogeneous Electrostatic Field.- 4-10-1-2. Skew Rays.- 4-10-1-3. Homogeneous Magnetic Field.- 4-10-2. The Single-Loop Magnetic Lens.- 4-10-3. Lens Systems.- 4-10-3-1. Telescopic System.- 4-10-3-2. Magnification of Lens Systems.- Summary.- 5. The Theory of Aberrations.- 5-1. The Method of Characteristic Functions.- 5-2. Geometrical Aberrations.- 5-2-1. Spherical Aberration.- 5-2-1-1. Zero and Infinite Magnifications.- 5-2-1-2. Alternative Forms of the Spherical Aberration Coefficient.- 5-2-1-3. Scherzer’s Theorem.- 5-2-1-4. The Disk of Minimum Confusion.- 5-2-2. Astigmatism.- 5-2-3. Curvature of Field.- 5-2-4. Distortion.- 5-2-5. Coma.- 5-2-6. Anisotropic Aberrations.- 5-2-6-1. Anisotropic Astigmatism.- 5-2-6-2. Anisotropic Distortion.- 5-2-6-3. Anisotropic Coma.- 5-2-7. On the Relative Importance of the Different Geometrical Aberrations.- 5-3. Chromatic Aberration.- 5-3-1. Axial Chromatic Aberration.- 5-3-1-1. Zero and Infinite Magnifications.- 5-3-1-2. The Upper Limit of the Axial Chromatic Aberration.- 5-3-2. Chromatic Aberration of Magnification.- 5-3-3. Anisotropic Chromatic Aberration.- 5-3-4. Magnetic Chromatic Aberration.- 5-4. Asymptotic Aberrations.- 5-4-1. The Dependence of the Asymptotic Aberration Coefficients on the Magnification.- 5-4-1-1. Polynomial Expression for the Asymptotic Spherical Aberration Coefficient.- 5-4-1-2. Polynomial Expression for the Asymptotic Axial Chromatic Aberration Coefficient.- 5-4-2. Aberrations of Thin Lenses.- 5-4-2-1. Spherical Aberration.- 5-4-2-2. Axial Chromatic Aberration.- 5-5. Aberrations of Lens Combinations.- 5-5-1. Addition of Spherical Aberrations.- 5-5-2. Addition of Axial Chromatic Aberrations.- 5-6. Other Sources of Aberrations and Aberration Correction.- 5-6-1. Diffraction.- 5-6-2. Space Charge and Surface Charges.- 5-6-3. High-Frequency Fields.- 5-6-4. Lack of Axial Symmetry.- 5-6-5. Other Methods of Correction.- 5-6-5-1. Coaxial Lenses.- 5-6-5-2. Symmetric Trajectories.- 5-6-5-3. Position of the Limiting Aperture.- 5-6-5-4. Digital Image Processing.- 5-6-6. Synthesis.- 5-6-7. On the Measurement of Aberrations.- 5-6-8. Brightness.- 5-7. Simultaneous Action of Different Aberrations.- 5-7-1. Negligibly Small Sources.- 5-7-2. Finite Sources.- 5-7-2-1. Negligible Chromatic Aberration.- 5-7-2-2. Negligible Spherical Aberration.- 5-7-3. Aberration Mixing for Lens Combinations.- 5-7-4. Figures of Merit.- Summary.- 6. Numerical Techniques for Ray Tracing and Calculation of Aberrations.- 6-1. Analytical Models.- 6-2. Numerical Ray Tracing.- 6-2-1. The Runge-Kutta Method.- 6-2-2. Multistep Methods.- 6-2-2-1. Numerov’s Method.- 6-2-3. Additional Remarks on Accuracy.- 6-3. Numerical Calculation of Aberration Integrals.- 6-3-1. Trapezoidal Integration.- 6-3-2. Simpson’s Rule.- 6-3-3. Romberg Integration and the Gaussian Quadrature.- Summary.- 7. Electrostatic Lenses.- 7-1. General Properties and Relationships.- 7-2. Electrostatic Lens Models.- 7-2-1. Analytical Models.- 7-2-2. The Piecewise Linear Model.- 7-2-3. The Piecewise Quadratic Model.- 7-2-4. The Spline Model.- 7-3. Two-Electrode Immersion Lenses.- 7-3-1. Geometrically Symmetric Lenses.- 7-3-1-1. A Linear Model.- 7-3-1-2. An Analytical Model.- 7-3-1-3. The Two-Cylinder Lens.- 7-3-1-4. The Double-Aperture Lens.- 7-3-1-5. Polynomial Lenses.- 7-3-2. Asymmetric Lenses.- 7-3-2-1. Analytical Models.- 7-3-2-2. The Asymmetric Two-Cylinder Lens.- 7-3-2-3. A Hybrid Lens.- 7-4. Unipotential Lenses.- 7-4-1. Symmetric Lenses.- 7-4-1-1. A Piecewise Linear Model.- 7-4-1-2. A Piecewise Quadratic Model.- 7-4-1-3. An Analytical Model.- 7-4-1-4. The Three-Cylinder Lens.- 7-4-1-5. The Triple-Aperture Lens.- 7-4-1-6. Other Types of Symmetric Lenses.- 7-4-2. Asymmetric Lenses.- 7-5. Three-Electrode Immersion Lenses.- 7-5-1. Geometrically Symmetric Lenses.- 7-5-1-1. The Three-Cylinder Lens.- 7-5-1-2. Other Types of Geometrically Symmetric Lenses.- 7-5-2. Asymmetric Lenses.- 7-6. Multielectrode Lenses.- 7-6-1. Four-Electrode Lenses.- 7-6-2. Lenses with Five or More Electrodes.- 7-6-3. Spline Lenses.- 7-7. Comparison of Different Electrostatic Lenses.- 7-8. Lenses Immersed in Fields.- 7-8-1. The Exponential Model.- 7-8-2. The Single-Aperture Lens.- 7-8-3. Cathode Lenses, Electron and Ion Sources.- 7-8-3-1. Thermionic Guns.- 7-8-3-2. Field-Emission Guns.- 7-8-3-3. Ion Sources.- Summary.- 8. Magnetic Lenses.- 8-1. General Properties and Relationships.- 8-2. Long Lenses.- 8-2-1. Homogeneous Magnetic Fields.- 8-2-2. Linear Magnetic Fields.- 8-2-3. Long Lenses with Low Spherical Aberration.- 8-3. Magnetic Lens Models.- 8-3-1. The Rectangular Model.- 8-3-2. The Step-Function Model.- 8-3-3. The Piecewise Linear Model.- 8-3-4. The Spline Model.- 8-3-5. Glaser’s Bell-Shaped Model.- 8-3-5-1. Generalization of the Bell-Shaped Model.- 8-3-6. The Grivet-Lenz Model.- 8-3-7. Other Models.- 8-4. Short Lenses.- 8-4-1. Conventional Lenses.- 8-4-2. Unconventional Lenses.- 8-4-2-1. Superconducting Lenses.- 8-4-2-2. Reduction of the Coil Size by Other Means.- 8-4-2-3. Rotation-Free Miniature Lenses.- 8-4-2-4. Iron-Free Magnetic Lenses.- 8-4-2-5. Single Pole-Piece Lenses.- Summary.- 9. Computer-Aided Optimization and Synthesis of Electron and Ion Lenses.- 9-1. Is Aberrationless Electron/Ion Optics Possible?.- 9-1-1. The Lower Limit of the Axial Chromatic Aberration of Magnetic Lenses.- 9-2. Optimization: Synthesis versus Analysis.- 9-3. Early Attempts of Synthesis.- 9-4. Calculus of Variations.- 9-4-1. The Lower Limits of the Spherical and Axial Chromatic Aberration Coefficients.- 9-5. Dynamic Programming.- 9-6. Optimal Control Procedure.- 9-7. Analytical Functions.- 9-8. Reconstruction of Electrodes and Pole Pieces from the Optimized Axial Field Distributions.- 9-9. Polynomial and Spline Lenses.- 9-9-1. Polynomial Lenses.- 9-9-2. Spline Lenses.- 9-9-2-1. Two-Interval Spline Lenses.- 9-10. The Synthesis Procedure.- 9-10-1. Application: Unconventional Electrostatic Lenses.- 9-11. Artificial Intelligence Techniques.- Summary.- 10. Multipole Lenses.- 10-1. The Fields of Multipole Lenses.- 10-2. The Paraxial Ray Equations.- 10-3. Image Formation by Paraxial Rays.- 10-4. Systems of Quadrupoles.- 10-4-1. Transfer Matrices.- 10-4-2. Thin-Lens Representation.- 10-4-3. Doublets.- 10-4-4. Triplets.- 10-4-5. Multiplets.- 10-4-5-1. Beam Matching.- 10-5. Aberrations of Multipole Lenses.- 10-5-1. Geometrical Aberrations.- 10-5-2. Correction of Aberrations by Means of Multipoles.- 10-5-3. Chromatic Aberration.- 10-5-3-1. The Achromatic Quadrupole Lens.- Summary.- 11. Beam Deflection.- 11-1. Deflection for Scanning.- 11-1-1. Electrostatic Deflection Fields.- 11-1-2. Magnetic Deflection Fields.- 11-1-3. Stigmatic Imaging with Small Deflection.- 11-1-4. Deflection Aberrations.- 11-2. Electrostatic and Magnetic Prisms.- 11-2-1. Electrostatic Prisms.- 11-2-2. Magnetic Prisms.- 11-3. New Symmetries-New Possibilities.- Summary.- 12. High-Intensity Beams.- 12-1. Space-Charge Optics.- 12-1-1. Space-Charge Forces.- 12-1-1-1. The Electrostatic Force.- 12-1-1-2. The Magnetic Force.- 12-1-2. Beam Spreading.- 12-1-3. Production of High-Intensity Beams.- 12-1-3-1. Space-Charge Flow.- 12-1-3-2. The Pierce Gun.- 12-1-4. Maintenance of High-Intensity Beams.- 12-1-4-1. Focusing by Homogeneous Magnetic Fields.- 12-1-4-2. Periodic Focusing.- 12-2. The Boersch Effect.- Summary.- References.