Magnetic Lens

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Scanning electron microscopes take images of samples by scanning with a high-energy beam of electrons. The subsequent electron interactions produce signals such as secondary and back-scattered electrons that 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.

This application requires the AC/DC Module and 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 nonaxial 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 or helical path decreases. Thus, a parallel beam of electrons entering the lens converges 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 Charged Particle Tracing interface. The equations solved are the equation of motion of a charged particle in a magnetic field (Lorentz force):

where q (SI unit: C) is the particle charge, v (SI unit: m/s) is the particle velocity, and B (SI unit: T) is the magnetic flux density. The total work done on a particle by a magnetic field is zero.

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 lenses 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:

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 (that is, 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 field strength; therefore the electrons spiral in tighter paths bringing the focal length closer.

Figure 2: Plot of the electron trajectories traveling through the magnetic lens.

When charged particle beams are released, additional global variables are used to define beam properties such as the emittance and the Twiss parameters. These global variables can be used to characterize the shape of a beam and the transverse phase-space distribution of the beam particles. In Figure 3 the hyperemittance is plotted along the average beam trajectory as a color expression and as a tube radius expression. The nominal trajectory reaches its maximum thickness shortly after entering the lens and appears to be pinched off at the location where the beam is focused.

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 4, which shows a Poincaré map of the particle position at three different snapshots in time. The sharpness of the crossover can be improved using multiple lenses.

Figure 3: The nominal beam trajectory is plotted, with a color and thickness proportional to the 1-rms hyperemittance of the beam...

Figure 4: Poincaré maps 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).

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.

Particle_Tracing_Module/Charged_Particle_Tracing/magnetic_lens

Modeling Instructions

From the menu, choose .

In the window, click

.

1

In the window, click

.

2

In the tree, select .

3

4

5

In the tree, select .

6

Global Definitions

1

In the window, under click .

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In the window for , locate the section.

3

In the table, enter the following settings:


Name	Expression	Value	Description
Ic	0.32[A]	0.32 A	Coil current
Nc	1000	1000	Number of turns in coil

The coil geometry is constructed using cylinders, and it is available as a separate file in the Application Library. Insert the prepared geometry sequence from the file. You can read the instructions for creating the geometry in the appendix.

1

In the toolbar, click and choose .

2

Browse to the model’s Application Libraries folder and double-click the file magnetic_lens_geom_sequence.mph.

1

In the toolbar, click

to open the window.

2

Go to the window.

3

In the tree, select .

4

Click in the window toolbar.

5

In the tree, select .

6

Click in the window toolbar.

7

In the toolbar, click

to close the window.

Perfect vacuum (mat2)

Select Domain 1 only.

Magnetic Fields (mf)

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In the window, under right-click and choose the domain setting .

2

Select Domain 4 only.

3

In the window for , locate the section.

4

From the list, choose .

5

From the list, choose .

6

Locate the section. In the N text field, type Nc.

7

Locate the section. In the Icoil text 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 created for this purpose in previous steps.

Coil Geometry 1

1

In the window, click .

2

In the window for , locate the section.

3

.

4

Select Edges 22, 23, 57, and 82 only.

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In the toolbar, click

and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

Select Domains 2–5 only.

5

Locate the section. In the text field, type 0.5.

Free Triangular 1

1

In the toolbar, click

and choose .

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Select Boundary 30 only.

1

Right-click and choose .

Use a fine mesh on the surface where particles will be released.

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In the window for , locate the section.

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From the list, choose .

Free Tetrahedral 1

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In the toolbar, click

.

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In the window for , click

.

In the toolbar, click

.

1

In the window, expand the node, then click .

2

In the window for , locate the section.

3

Find the subsection. In the text field, type 0.

Streamline Multislice 1

1

In the window, click .

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In the window for , locate the section.

3

Find the subsection. In the text field, type 0.

4

button in the toolbar.

5

In the toolbar, click

. Compare the resulting image to Figure 1.

1

In the toolbar, click

to open the window.

2

Go to the window.

3

In the tree, select .

4

Find the subsection. In the table, clear the check box for .

5

Click in the window toolbar.

6

In the toolbar, click

to close the window.

1

In the toolbar, click

to open the window.

2

Go to the window.

3

Find the subsection. In the tree, select .

4

Find the subsection. In the table, clear the check box for .

5

Click in the window toolbar.

6

In the toolbar, click

to close the window.

In the window, collapse the node.

Charged Particle Tracing (cpt)

1

In the window, under click .

2

In the window for , locate the section.

3

.

4

Select Domain 1 only.

Particle Properties 1

1

In the window, under click .

2

In the window for , locate the section.

3

From the list, choose .

You need to provide the forces acting on the particles; in this case, the magnetic (Lorentz) force.

Magnetic Force 1

1

In the toolbar, click

and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

Locate the section. From the B list, choose .

Particle Beam 1

1

In the toolbar, click

and choose .

2

Select Boundary 30 only.

3

In the window for , locate the section.

4

In the N text field, type 10000.

5

Locate the section. In the εrms text field, type 0.1[um].

6

Locate the section. In the E text field, type 0.5[keV].

Step 1: Time Dependent

1

In the window, under click .

2

In the window for , click to expand the section.

3

Find the subsection. From the list, choose .

4

From the list, choose .

5

From the list, choose .

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Locate the section. Click

.

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In the dialog box, choose from the list.

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In the text field, type 5e-9.

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In the text field, type 50.

10

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In the toolbar, click

.

Particle Trajectories (cpt)

In the window, expand the node.

Particle Trajectories 1

1

In the window, expand the node, then click .

2

In the window for , locate the section.

3

Find the subsection. From the list, choose .

4

Find the subsection. From the list, choose .

Color Expression 1

1

In the window, click .

2

In the window for , locate the section.

3

In the text field, type sqrt(cpt.Ftx^2+cpt.Fty^2+cpt.Ftz^2).

4

Locate the section. Click

.

5

In the dialog box, select in the tree.

6

7

In the toolbar, click

.

8

button in the toolbar. Compare the resulting image to Figure 2.

Now observe the beam hyperemittance along the nominal beam trajectory.

Average Beam Position and Hyperemittance

1

In the window, under click .

2

In the window for , type Average Beam Position and Hyperemittance in the text field.

Point Trajectories 1

1

In the window, expand the node, then click .

2

In the window for , locate the section.

3

Find the subsection. From the list, choose .

4

button. From the menu, choose .

5

Select the check box. In the associated text field, type 4E10.

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From the list, choose .

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In the toolbar, click

. Compare the resulting image to Figure 3.

Now construct a to visualize the radial distribution of particles initially, at the focal point, and at the exit of the modeling domain.

1

In the toolbar, click

.

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In the window for , locate the section.

3

From the list, choose .

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In the text field, type -6.

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Locate the section. From the list, choose .

1

Right-click and choose .

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In the window for , locate the section.

3

In the text field, type 7.

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Right-click and choose .

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In the window for , locate the section.

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In the text field, type 34.

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In the toolbar, click

.

2

In the window for , type Poincaré Maps in the text field.

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Click to expand the section. From the list, choose .

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In the text area, type Poincaré Map.

Poincaré Map 1

1

In the toolbar, click

and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

Locate the section. From the list, choose .

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Locate the section. From the list, choose .

6

button in the toolbar.

Poincaré Map 2

1

Right-click and choose .

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In the window for , locate the section.

3

From the list, choose .

4

Locate the section. From the list, choose .

Poincaré Map 3

1

Right-click and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

Locate the section. From the list, choose .

5

In the toolbar, click

.

6

button in the toolbar. Compare the resulting image to Figure 4.

Appendix A - Geometry Instructions

In the toolbar, click

and choose .

1

In the window for , locate the section.

2

From the list, choose .

Cylinder 1 (cyl1)

1

In the toolbar, click

.

2

In the window for , locate the section.

3

In the text field, type 10.

4

In the text field, type 2.5.

Cylinder 2 (cyl2)

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In the toolbar, click

.

2

In the window for , locate the section.

3

In the text field, type 6.

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In the text field, type 2.5.

5

.

Cylinder 3 (cyl3)

1

In the window, under right-click and choose .

2

In the window for , locate the section.

3

In the text field, type -7.5.

Cylinder 4 (cyl4)

1

In the toolbar, click

.

2

In the window for , locate the section.

3

In the text field, type 2.

4

In the text field, type 2.5.

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Locate the section. In the text field, type -7.5.

Cylinder 5 (cyl5)

1

Right-click and choose .

2

In the window for , locate the section.

3

In the text field, type -2.5.

Cylinder 6 (cyl6)

1

In the toolbar, click

.

2

In the window for , locate the section.

3

In the text field, type 3.

4

In the text field, type 2.5.

5

Locate the section. In the text field, type -2.5.

Cylinder 7 (cyl7)

1

Right-click and choose .

2

In the window for , locate the section.

3

In the text field, type 2.5.

Cylinder 8 (cyl8)

1

In the toolbar, click

.

2

In the window for , locate the section.

3

In the text field, type 3.

4

In the text field, type 2.5.

5

Locate the section. In the text field, type 2.5.

Cylinder 9 (cyl9)

1

In the toolbar, click

.

2

In the window for , locate the section.

3

In the text field, type 20.

4

In the text field, type 50.

5

Locate the section. In the text field, type -15.

Difference 1 (dif1)

1

In the toolbar, click

and choose .

2

Select the objects , , , and only.

3

In the window for , locate the section.

4

Find the subsection. Click to select the

toggle button.

5

Select the objects , , , and only.

6

.

7

button in the toolbar.

Work Plane 1 (wp1)

1

In the toolbar, click

.

2

button in the toolbar.

3

In the window for , locate the section.

4

From the list, choose .

5

On the object , select Boundary 3 only.

6

.

Work Plane 1 (wp1)>Plane Geometry

button in the toolbar.

Work Plane 1 (wp1)>Circle 1 (c1)

1

In the toolbar, click

.

2

In the window for , locate the section.

3

In the text field, type 2.

4

.

Last, create a circular edge to be used in the feature as a reference edge.

Work Plane 2 (wp2)

1

In the window, right-click and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

On the object , select Boundary 13 only.

5

.

Work Plane 2 (wp2)>Circle 1 (c1)

1

In the toolbar, click

.

2

In the window for , locate the section.

3

From the list, choose .

4

Locate the section. In the text field, type 8.

5

In the toolbar, click

.