How to analyse ion beam devices

By Nigel Atkinson

If we consider a general ion device, in terms of simulation, what components might we encounter? Very often we have a source of particles – occasionally the particles that are eventually required, but more often, they are electrons or ions in a lower ionization state than needed.  These enter a beam transport system, where the beam might be accelerated, focused and steered.

If the beam does not contain the required particle species, some interaction is required – either with a surface or volume. In this region exchange of charge and momentum takes place to give the required species. This often requires some sort of confinement system to increase the residence time of the particles in this region.

Once the required ions have been produced, they need to be extracted from the interaction region, usually electrostatically; unwanted particles that might also be extracted then need to be removed from the beam – using electric and/or magnetic fields – before the beam is transported for use. After transport, the beam is incident on its target, be it a further interaction volume or surface, and spent beams are incident on dumps, where the residual beam energy is absorbed.

So, what do we need a simulation tool to be able to do so that we can design and analyse an ion beam device?

We need to be able to generate primary particles; we need to be able to interact these particles with both electric and magnetic fields to accelerate, focus and steer the beam; and we need particle-particle, particle-surface and particle-volume interactions. In all of these we need to be able to simulate general scattering processes – where charge and momentum exchange can occur.

Having seen what we might need to simulate, I’d like to turn to how the features of Opera that help us to meet these needs. I’ll leave particle generation aside for the moment, and look at a couple of options for determining the interaction between charged particles, electric and magnetic fields.

We’ll start by considering the motion of charged particles in electric and magnetic fields. Since the particles are typically relativistic, it is convenient to cast Newton’s law of motion in terms of momentum, the rate of change of which is then given by the forces acting on the particle – the Lorentz force. Particle tracking using this simple formulation can be performed in the Opera Post-Processor. However, we have ignored the effect of the presence and motion of the charged particles on the fields. For a self consistent solution we need to include these, so we need some way to account for the interdependence.

If we limit the problem to static fields, than we can use an efficient approach that can significantly reduce potential run times. Instead of tracking superparticles we calculate the trajectories of current tracks.
And then find the self consistent trajectories in the presence of electric fields (both applied and from space charge), magnetic fields (both applied and from the beam self field). This is the principle behind Opera’s space charge solver. The process, in outline is that we initially solve Poisson’s equation without space charge (using Finite Elements). We then launch particle tracks from primary emitter regions-more on this later – and track them in the electrostatic and any applied magnetostatic field. Space charge is assigned to the FE mesh depending on the current in the tracks and the time spent in the proximity of the mesh element. The magnetic field from the tracks can also be calculated. We then re-solve for the updated fields, launch tracks again including any that are generated by secondary emission, and calculate the new trajectories. This continues until self-consistent trajectories are found. During the process, we can also introduce additional features – such as charging of dielectric surfaces – and the consequent current flow and effect on the electrostatic field.

In summary, the space charge solver in Opera incorporates an efficient technique that can track particles self-consistently in combined fields. The technique allows us to introduce primary particles, and enables interactions to produce secondary particles. The particle tracks carry current and have known momentum and power – so we can calculate power deposition. And we can do this fast enough to be able to simulate real-world devices.

I’ve mentioned tracking in combined fields several times, and the use of an applied magnetic field is common in ion beam devices. Opera provides several ways to add the applied magnetic field to the particle simulation. The simplest is the application of a uniform external field, specified by its components, Hx, Hy, Hz, in the Modeller.

A feature often used in Opera is to represent coils of various shapes as Biot-Savart conductors. These are extremely efficient, and may be included in a Opera’s space charge solver model; the resulting magnetic fields will be calculated, and used, during the space charge simulation. Alternatively you can import field data from measurements. The need to include the space charge from the particles is the norm, since its presence is often a performance determining factor. Conversely, very often, the magnetic field from the beam current is very small compared with the applied field, and is often ignored.

However, with an intense beam, the self-field can be significant, and must be calculated self-consistently in the simulation – Opera allows this as an option. Returning to what we need to calculate, we have described this so far – the interaction between particles and fields. We’ve mentioned the generation of particles, and their interactions, we’ll now look at this in a little more detail. In Opera, the generation of particles, and their interaction with other particles, surfaces and volumes, is enabled by a set of emission models. These models embody the physics of emission and interaction. Opera includes a wide range of such models that provide the ability to generate arbitrary species of particles, and to define their interactions. Among these are a number specifically for ion devices.

Ions may be generated in Opera by the interaction of particles from a primary emitter with a surface on which a secondary emitter has been defined. These are among the simplest types of devices to simulate.  Alternative, the primary particle beam can interact with volume secondary emitters – generating multiple species. This is typical of the interaction of charged particles with volumes of gases.
Plasma sources and plasma ion sources can be simulated by the bombardment of electrons and ions interacting with the gas to produce ion-electron pairs and the space-charge interaction of these particles trapped in electric and magnetic fields. Multiple emitter models within the volume implement the individual components of the process – for example secondary ions, secondary electrons, primary particle energy loss etc. While this kinetic approach is often performed, it is something possible to simplify the process further, and characterize the behaviour of the plasma by a few simple, measurable parameters.
This leads us to a simple method of simulating such systems – the use of the Bohm plasma sheath approximation for generating a plasma meniscus from which beams may be extracted. This method ignores the actual behaviour of the plasma inside the ionisation chamber and instead assumes a uniform and constant current density on the meniscus surface and a constant potential. The plasma characteristics are then encapsulated by its meniscus potential and beam current and electron temperature of the plasma. The shape of the plasma meniscus is then determined from the self-consistent potential solution in the presence of the space charge from the beam.

This is adequate for many applications and produces results quickly. This type of emitter is included in Opera, where it is known as the plasma free surface emitter. A quick mention for a post-processing feature -the two views in the centre show the ion beam trajectories coloured by their time of flight, and the potential distribution on device structure.

Opera also includes an emitter intended for the analysis of magnetron sputtering systems. It generates ions from gas interaction with high energy electrons that enter the system, partly from secondary emission from the sputter target. In this case, no primary particles are required – the production of ions, and target neutrals, is self-starting. Returning to what we need to calculate again, we have now described the features that we need to simulate.

But what about extraction of results? Having simulated these processes, we need to extract useful metrics from the result. Opera as standard includes a very capable Post-Processor, with very flexible tools for generating many useful types of result.

A standard feature of the Post-Processor is the ability to display field quantities on the surfaces of the model, and on patches and lines placed anywhere within the model space. All such quantities may then be save in simple text format as required. While many of the field quantities are common across a range of Opera solvers, some apply specifically to particular applications, including space charge modelling.  One of these is the voltage in the absence of space charge. We can see the significance of the space charge by comparing this with the actual voltage.

When it comes to evaluating the statistics of an ion beam, the starting point is again a patch placed across the beam at some user-defined location. Standard tools in the Post-Processor can then extract intersection data, such as the current and velocity components of each intersecting trajectory and the number and current densities.

From these, simple additional computations can generate beam metrics, including the moments, emittance and phase space. This may be performed using Opera’s scripting language – and if required example scripts can be made available as a starting point for user-specific processing. When performing magnetron sputtering simulations the Post-processor can also display number density of the depositing species in the beam, and of surface material particles produced by secondary emission for erosion. This you can produce graphical output representing deposition and erosion rates.

The power deposited on surfaces of the model by the particle tracks can be captured on all surfaces on which a secondary emitter is defined. The deposited beam power density is then derived as the difference between incident power and the power in secondary emission species, and can be exported simply to Opera’s thermal simulator. Thermal analysis may then be performed – either steady state, or, by time-limiting the excitation, a transient thermal analysis may be run to simulate the effect of pulsed beams.

The steady state thermal profile can then be used in Opera’s stress solver to give a mechanically deformed model that can be re-analysed in the space charge solver.

In summary the features of the Opera Space Charge solver provide a powerful and flexible tool for the design of ion beam devices. The techniques that it uses allow rapid and accurate analysis Opera provides all of the features required to perform a complete multiphysics analysis The Optimizer allows users to rapidly improve designs, even with competing requirements.