Solved with COMSOL Multiphysics 4.3a Magnetic Lens Introduction Scanning electron microscopes image samples by scanning with a high-energy beam of electrons. The subsequent electron interactions produce signals such as secondary and back-scattered electrons than contain information about the sample surface topography. Electromagnetic lenses are used to focus this electron beam down to a spot about 10 nm wide on the sample surface. Note: This model requires the Particle Tracing Module. Model Definition Particles (electrons) are released from near the bottom boundary of the simulation space and pass through a collimator. This collimator can typically be adjusted to remove stray electrons. A simple DC coil produces an axial magnetic field. This rotationally symmetric, inhomogeneous magnetic field results in non-axial electrons experiencing a radial force causing them to spiral about the axis. As they begin to spiral, they have a larger velocity component perpendicular to the mainly axial magnetic field, therefore the radius of their spiral/helical path decreases. Thus, a parallel beam of electrons entering the lens will converge to a point. If the region in which the magnetic field acts upon the electrons is sufficiently small, this coil acts as a ‘thin’ convex lens and the thin lens expression holds. MODEL EQUATIONS A simple model is set up to test the magnetic force within the Particle Tracing for ACDC interface. The equations solved are the equation of motion of a charged particle in a magnetic field (Lorentz force): d(mv) = q(v×B) (1) dt where q is the particle charge (C), v is the particle velocity (m/s) and B is the magnetic flux density (SI unit: T). The total work done on a particle by a magnetic field is zero. ©2012 COMSOL 1 | MAGNETIC LENS Solved with COMSOL Multiphysics 4.3a Results and Discussion The magnetic flux density is plotted in Figure 1. The strength of the lens depends upon the coil configuration and current. The lens within electron microscopes are generally very strong, in some cases focusing the electron beam within the lens itself. Figure 1: Plot of the magnetic flux density in the magnetic lens. Figure 2 plots the electron trajectories as they travel through the coil. The electrons are focused at a point along the z-axis. The focal length is given by: V f = K---- (2) 2 i where K is a constant based on the coil geometry and number of turns, V is the accelerating voltage and i is the coil current. The focal length increases with electron energy (i.e. V) because their high velocity means they spend less time experiencing a force due the magnetic field. However, as the current increases so does the magnetic 2 | MAGNETIC LENS ©2012 COMSOL Solved with COMSOL Multiphysics 4.3a field strength, therefore the electrons spiral in tighter paths bringing the focal length closer. Figure 2: Plot of the electron trajectories travelling through the magnetic lens. The ability to change the focal length of a lens is useful as it allows the focusing onto a surface in addition to adjusting the magnification. The effect of the focusing can be seen in Figure 3 which shows a Poincaré map of the particle position at three different snapshots in time. The sharpness of the cross-over can be improved using multiple lenses. ©2012 COMSOL 3 | MAGNETIC LENS Solved with COMSOL Multiphysics 4.3a Figure 3: Poincaré plot of the particle location in the xy-plane initially (red), at the focal point of the lens (blue) and at the last time step (black). Reference 1. M.J.Pritchard, Manipulation of Ultracold Atoms Using Magnetic and Optical Fields, PhD thesis, Durham University, September 2006, http://massey.dur.ac.uk/ resources/mjpritchard/thesis_pritchard.pdf. Model Library path: ACDC_Module/Particle_Tracing/magnetic_lens Modeling Instructions MODEL WIZARD 1 Go to the Model Wizard window. 2 Click Next. 3 In the Add physics tree, select AC/DC>Magnetic Fields (mf). 4 | MAGNETIC LENS ©2012 COMSOL Solved with COMSOL Multiphysics 4.3a 4 Click Add Selected. 5 Click Next. 6 Find the Studies subsection. In the tree, select Preset Studies>Stationary. 7 Click Finish. GLOBAL DEFINITIONS Parameters 1 In the Model Builder window, right-click Global Definitions and choose Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 In the table, enter the following settings: Name Expression Vaccel 500[V] Ic 0.32[A] velec sqrt((2*Vaccel*e_const)/(me_const)) Nc 1000 GEOMETRY 1 Build a simple coil geometry using cylinders. 1 In the Model Builder window, under Model 1 click Geometry 1. 2 In the Geometry settings window, locate the Units section. 3 From the Length unit list, choose mm. Cylinder 1 1 Right-click Model 1>Geometry 1 and choose Cylinder. 2 In the Cylinder settings window, locate the Size and Shape section. 3 In the Radius edit field, type 10. 4 In the Height edit field, type 2.5. Cylinder 2 1 In the Model Builder window, right-click Geometry 1 and choose Cylinder. 2 In the Cylinder settings window, locate the Size and Shape section. 3 In the Radius edit field, type 6. 4 In the Height edit field, type 2.5. 5 Click the Build Selected button. ©2012 COMSOL 5 | MAGNETIC LENS Solved with COMSOL Multiphysics 4.3a Cylinder 3 1 In the Model Builder window, under Model 1>Geometry 1 right-click Cylinder 1 and choose Duplicate. 2 In the Cylinder settings window, locate the Position section. 3 In the z edit field, type -7.5. Cylinder 4 1 In the Model Builder window, right-click Geometry 1 and choose Cylinder. 2 In the Cylinder settings window, locate the Size and Shape section. 3 In the Radius edit field, type 2. 4 In the Height edit field, type 2.5. 5 Locate the Position section. In the z edit field, type -7.5. Cylinder 5 1 In the Model Builder window, under Model 1>Geometry 1 right-click Cylinder 1 and choose Duplicate. 2 In the Cylinder settings window, locate the Position section. 3 In the z edit field, type -2.5. Cylinder 6 1 In the Model Builder window, right-click Geometry 1 and choose Cylinder. 2 In the Cylinder settings window, locate the Size and Shape section. 3 In the Radius edit field, type 3. 4 In the Height edit field, type 2.5. 5 Locate the Position section. In the z edit field, type -2.5. Cylinder 7 1 In the Model Builder window, under Model 1>Geometry 1 right-click Cylinder 1 and choose Duplicate. 2 In the Cylinder settings window, locate the Position section. 3 In the z edit field, type 2.5. Cylinder 8 1 In the Model Builder window, right-click Geometry 1 and choose Cylinder. 2 In the Cylinder settings window, locate the Size and Shape section. 3 In the Radius edit field, type 3. 4 In the Height edit field, type 2.5. 6 | MAGNETIC LENS ©2012 COMSOL Solved with COMSOL Multiphysics 4.3a 5 Locate the Position section. In the z edit field, type 2.5. Cylinder 9 1 Right-click Geometry 1 and choose Cylinder. 2 In the Cylinder settings window, locate the Size and Shape section. 3 In the Radius edit field, type 20. 4 In the Height edit field, type 50. 5 Locate the Position section. In the z edit field, type -15. Difference 1 1 Right-click Geometry 1 and choose Boolean Operations>Difference. 2 Select the objects cyl5, cyl7, cyl3, and cyl1 only. 3 In the Difference settings window, locate the Difference section. 4 Under Objects to subtract, click Activate Selection. 5 Select the objects cyl4, cyl8, cyl2, and cyl6 only. 6 Click the Build Selected button. 7 Click the Go to Default 3D View button on the Graphics toolbar. Work Plane 1 1 Right-click Geometry 1 and choose Work Plane. 2 Click the Wireframe Rendering button on the Graphics toolbar. 3 In the Work Plane settings window, locate the Work Plane section. 4 From the Plane type list, choose Face parallel. 5 On the object dif1, select Boundary 3 only. Plane Geometry Click the Zoom Extents button on the Graphics toolbar. Circle 1 1 In the Model Builder window, under Model 1>Geometry 1>Work Plane 1 right-click Plane Geometry and choose Circle. 2 In the Circle settings window, locate the Size and Shape section. 3 In the Radius edit field, type 2. 4 Click the Build Selected button. Last, create a circular edge to be used in the Multi-Turn Coil Domain feature as a reference edge. ©2012 COMSOL 7 | MAGNETIC LENS Solved with COMSOL Multiphysics 4.3a Work Plane 2 1 In the Model Builder window, right-click Geometry 1 and choose Work Plane. 2 In the Work Plane settings window, locate the Work Plane section. 3 From the Plane type list, choose Face parallel. 4 On the object dif1, select Boundary 13 only. Circle 1 1 In the Model Builder window, under Model 1>Geometry 1>Work Plane 2 right-click Plane Geometry and choose Circle. 2 In the Circle settings window, locate the Object Type section. 3 From the Type list, choose Curve. 4 Locate the Size and Shape section. In the Radius edit field, type 8. 5 Click the Build All button. MATERIALS Add materials for the air domain and metal collimator and coil. Material 1 1 In the Model Builder window, under Model 1 right-click Materials and choose Material. 2 In the Material settings window, locate the Material Contents section. 8 | MAGNETIC LENS ©2012 COMSOL Solved with COMSOL Multiphysics 4.3a 3 In the table, enter the following settings: Property Name Value Electrical conductivity sigma 6e7 Relative permittivity epsilonr 1 Relative permeability mur 1 Material 2 1 In the Model Builder window, right-click Materials and choose Material. 2 Select Domain 1 only. 3 In the Material settings window, locate the Material Contents section. 4 In the table, enter the following settings: Property Name Value Electrical conductivity sigma 0 Relative permittivity epsilonr 1 Relative permeability mur 1 MAGNETIC FIELDS In the Model Builder window, expand the Model 1>Magnetic Fields node. Multi-Turn Coil Domain 1 1 Right-click Magnetic Fields and choose Multi-Turn Coil Domain. 2 Select Domain 4 only. 3 In the Multi-Turn Coil Domain settings window, locate the Coil Type section. 4 From the list, choose Circular. 5 Locate the Multi-Turn Coil Domain section. In the N edit field, type Nc. 6 In the Icoil edit field, type Ic. Specify the reference edges to be used in the calculation of the current path for the circular coil. To obtain the best results, the selected edges should have a radius close to the average coil radius. In this case, select the edges intentionally created in previous steps. Reference Edge 1 1 Right-click Model 1>Magnetic Fields>Multi-Turn Coil Domain 1 and choose Edges>Reference Edge. 2 In the Reference Edge settings window, locate the Edge Selection section. ©2012 COMSOL 9 | MAGNETIC LENS Solved with COMSOL Multiphysics 4.3a 3 Click Clear Selection. 4 Select Edges 22, 23, 57, and 82 only. MESH 1 Scale 1 1 In the Model Builder window, under Model 1 right-click Mesh 1 and choose Scale. 2 In the Scale settings window, locate the Geometric Entity Selection section. 3 From the Geometric entity level list, choose Domain. 4 Select Domains 2–5 only. 5 Locate the Scale section. In the Element size scale edit field, type 0.5. Free Triangular 1 1 In the Model Builder window, right-click Mesh 1 and choose More Operations>Free Triangular. 2 Select Boundary 30 only. Size 1 1 Right-click Model 1>Mesh 1>Free Triangular 1 and choose Size. 2 In the Size settings window, locate the Element Size section. 3 From the Predefined list, choose Extremely fine. 4 Click to expand the Element Size Parameters section. Locate the Element Size section. Click the Custom button. 5 Locate the Element Size Parameters section. Select the Maximum element size check box. 6 In the associated edit field, type 0.4. Free Tetrahedral 1 1 In the Model Builder window, right-click Mesh 1 and choose Free Tetrahedral. 2 In the Settings window, click Build All. STUDY 1 1 In the Model Builder window, expand the Study 1 node. 2 Right-click Study 1 and choose Compute. 10 | MAGNETIC LENS ©2012 COMSOL