Vector Potential Photoelectron Microscopy Instrument Design

Highlights

• Introduction to vector potential photoelectron microscopy.

• Design rules for a vector potential photoelectron microscope.

• Implementations of vector potential photoelectron microscopes.

• Lessons learned from a prototype microscope.

Abstract

This report covers the main aspects of designing a vector potential photoelectron microscope (VPPEM). While the VPPEM is straightforward in concept, there are several areas where the optimum configuration of the optics is not immediately obvious, and compromises must be made to make a practical instrument. This report summarizes our instrumental setups, and some basic design issues.

Introduction

Vector potential photoelectron microscopy (VPPEM) is a new technique for materials microanalysis at the meso scale [1], [2], [3]. VPPEM images the elemental and chemical makeup of the top few nanometers of a sample using near edge x-ray absorption fine structure (NEXAFS) as the main image contrast mechanism. VPPEM will be able to image many types of samples including: samples with rough or contoured surfaces, poor conductors, or even poorly consolidated samples such as powder residues. The source of x-rays needs to be a bright tunable x-ray source such as a synchrotron beam line. This report concentrates on main aspects of the design which are seen as necessary to produce a practical instrument.

The VPPEM optical system needs some introduction. The optical system is a new class of imaging system, it has no analog in any other imaging system either in light optics or electron optics. VPPEM uses the canonical momentum of the photoelectrons emitted into magnetic field to form an image. The canonical momentum of an electron in a magnetic field is [4]:

$$\mathit{p}=m\dot{\mathit{r}}+q\mathbf{A}\left(\mathbf{r}\right)$$

Where p is the momentum, $\dot{\mathit{r}}$ the velocity, A is the vector potential, and m and q are the mass and charge of the photoelectron. Where, from Maxwell's equations, we have the magnetic field is the curl of the vector potential, $\mathbf{B}=\nabla \mathrm{x}\mathbf{A}$.

For a solenoid consisting of a circular loop of current I of radius R:

$$\mathbf{A}\left(r\right)=\frac{{\mu }_0}{4R}r\mathit{I}$$

The direction of A is around the axis in the direction of the current. This is illustrated in Figure 1 where the arrows indicate the direction, and magnitude of the vector field. The vector potential field is radially symmetric, zero on the axis, and increases linearly with radius out to the loop radius. This distribution is a two-dimensional vector field each point defined by a unique direction and magnitude mapped onto a two dimensional space in angle θ around the axis, and radius r away from the axis,

Photoelectrons emitted into the vacuum from a sample surface inside a solenoid have an additional momentum due to their position in the vector potential field. It is this additional two-dimensional distribution of momentum that is projected through the electron optical system to form an image.

In producing an image, the VPPEM electron optical system is required to convert a wide energy range two-dimensional photoelectron momentum distribution into a monochromatic distribution projected onto a two-dimensional image plane. The VPPEM system has three main electron optical elements to achieve this: a magnetic enclosure, an electron energy analyzer, and focusing optics.

The magnetic enclosure comprises a solenoid magnet enclosed in a ferromagnetic shield with an aperture on the axis for the photoelectrons to exit the magnetic field. The energy analyzer is a concentric hemispherical analyzer (CHA). The focusing optics comprises two parts, one a condensing lens into the CHA, and the other a focusing lens out of the CHA to form a real mage on a detector. A schematic of the overall electron optical system is shown in Figure 2.

The novel part of the VPPEM optics is the action of the magnetic field in producing an angular image. The photoelectrons leave the sample at the center of the magnetic field along cyclotron orbits approximately parallel to the optical axis following the magnetic field lines. The maximum size of the cyclotron orbits depends on the energy of the photoelectrons. As the photoelectrons move towards a weaker field, the cyclotron orbits start to unwind, and become more collimated. The cyclotron orbits also start to move away from the axis as the trajectories follow down the expanding field lines. When the photoelectrons reach the aperture, there is a sudden change of the vector potential. This change of momentum leads to a force which deflects the electrons producing a two-dimensional angular image that is mapped to the angle and magnitude of the vector potential at the sample. (see Supplemental file 1 for further details of the deflection action, and Section 7 for a discussion of the electron optical consequences of having an angular image).

This angular image has a full spectrum of energies. The angular image is focused into the CHA using an electrostatic condenser lens. A CHA is double focusing in both angular directions, and a monochromatic angular image passes through the energy defining exit slit of the CHA. This monochromatic angular image is projected as a real image onto a microchannel plate electron image detector.

A cutaway illustration of a simple magnetic circuit with a numerical simulation of the action of the aperture on electron trajectories is shown in Figure 3, Figure 4.

In the simulation of Figure 4 the fan of trajectories leaving the magnetic field have started at different off-axis positions at the sample position: ± 10, 20, 30, 40, and 50 microns. The central field is 0.2 Tesla, the ferromagnet shield is a cylindrical box of 100 mm radius, the aperture is 4 mm diameter, and the trajectories are simulations for 50 eV electron energies. The trajectory plot of Figure 4 is deceptively simple, only the radial distance of the trajectories from the axis is plotted. In fact, the image is rotated around the axis, and it is not a simple angular image.

Figure 5 shows a simulated trajectory of a 500 eV electron from a 1.0 Tesla field emitted at 45^o to the magnetic axis, and at a small distance off-axis in the x direction. In Figure 5 we are looking down the magnetic axis. We see the cyclotron orbit move away from the y axis, and expand as the field decreases. As the trajectory exits the field, it undergoes a deflection into its final direction. Clearly, the final direction will depend on exactly where the orbit crosses the aperture as the vector potential varies across the aperture. Two things to note about the final direction of the trajectory. Firstly, it is aligned at an angle to the x,y axis, and secondly it does not cross the x, and y axis in the same place. This is what we expect as the vector potential is pointing around the axis, and the starting position of the cyclotron orbit is off axis. The orbit moves further off-axis as magnetic field weakens. If we plot the trajectories in the x, and y directions separately, Figure 6, we see the image is rotated, and that the image has its apparent origin on the optical axis in different positions on the x (the meridional) and the y (the transverse) directions.

The different apparent object positions is a complication when it comes to focusing the image out of the VPPEM aperture into the CHA. In fact, the input lens could be viewed as a condenser because a true focus is not obtainable.

In the VPPEM, electrons of all energies from a sample are deflected into an angular image by the change in vector potential at the aperture. This angular image has to be converted into a monochromatic real image focused at an image plane. Figure. 2 illustrates the electrostatic elements needed. Diverging electron trajectories exiting the VPPEM aperture are focused into a concentric hemispherical analyzer (CHA) by a multielement cylindrical electrostatic lens. After energy analysis by the CHA, which is double focusing, the angular monochromatic image is projected as a real image onto an image detector using a second cylindrical electrostatic lens.

As shown in Figure 2 the CHA is operated without an input slit, but with an energy defining output slit. This is the reverse of the normal operation for a CHA. The utility of this arrangement is that with no input slit blocking the line of sight, the photon beam can enter the instrument through the alignment port at the back of the CHA, pass down the axis of the instrument, and illuminate the sample at normal incidence.

If we run the simulation of Figure 5 with a range of off axis angles from 0^o to 90^o, and with a weighting for a cosine emission distribution, we would expect to get a cone of confusion equivalent to the disk of confusion from a magnetic projection microscope [5]. This is what we find. By ray tracing over the full range of angles, we can see the VPPEM angular cone of confusion is caused by the variation of the exit point in the vector potential field at the aperture.

The calculated VPPEM spatial resolution is dependent on the cyclotron radius of the photoelectrons at the sample. The cyclotron radius at the sample depends on the magnetic field, and the imaged electron energy. A good approximation for the 20-80% edge spatial resolution ρ in microns is:

$$\rho =3\frac{\sqrt{E}}{B}$$

Where E is the electron energy in electron volts, and B the magnetic field in Tesla [6]. A useful feature of the strong magnetic field containment of the emitted electrons is that they can be accelerated, or decelerated within the magnetic field without distorting the image. Very low energy electrons, with energies below 1eV, can be imaged by accelerating them off the sample to several tens of electron volts. It has been our practice to collect NEXAFS spectra with different low energy take-off energies to maximize spatial resolution. Interestingly, the spatial resolution at compositional interfaces appears to be greater than expected. One possible explanation is the surface fields from chemical inhomogeneities are influencing the electron trajectories[2].

The VPPEM electron optical path is unusual for a microscope in that the sample, and the image are not on conjugate planes. Although there is a one to one mapping of points from the sample to the image, the sample is not sitting at a particular plane in the electron optics, and the virtual image of the sample is at the field termination aperture. The sample itself is sitting in the center of a cylinder formed by the vector potential field distribution. If the vector potential field is constant over a distance near the center of the solenoid axis, then the sample will be imaged at the same magnification within this distance. It will not go out of ‘focus’, the effective depth of field can be relatively large.

This introduction gives an outline of the electron optics of VPPEM. There are clearly some complications in how the image is formed, and this impacts the actual design and implementation of a VPPEM system. The design of the optical system must be done as a whole because design of the different parts impact each of the others. However, it is useful to begin by discussing the different sections separately.