Analytical formulae for trajectory displacement in electron beam and generalized slice method
Introduction
The Coulomb interactions between particles of a charged particle beam can be separated into three distinct effects. The space charge effect is the effect of averaged charge density. This manifests itself mostly by a defocus and higher order geometrical aberrations of the particle beam. The other two effects are caused by the stochastic nature of the particle beam. The Boerch effect describes the statistical longitudinal acceleration and deceleration of the particles and therefore broadening of the beam energy distribution. On the other hand, the trajectory displacement covers the transverse stochastic repulsion of the particles and describes the perturbation of a particle’s trajectory. We will focus on the trajectory displacement.
The stochastic part of the Coulomb interactions is fully described by the solution to the many-body problem of all the individual particles of the beam. However, an analytical solution for such a problem is not feasible.
One method of determining the trajectory displacement in a charged particle beam is using a Monte Carlo method to simulate the beam. This approach can give very reliable results, however, it does not provide much insight into the general behaviour of the trajectory displacement and it is very computationally demanding.
Another approach is to apply some simplifications and approximations to the many-body problem, which allow it to be solved. When the beam current is not very large, the particle interactions are rare so most particles are interacting with at most one other at any given time. This means the individual particle interactions are uncorrelated and thus can be treated separately. This is called the two–particle interaction model which forms the basis of the theory of statistical Coulomb interactions in particle beams developed by Jansen [1].
Using the two–particle interaction model, two methods to determine the trajectory displacement were established. First is the so-called slice method which expresses the trajectory displacement as an integral of differential contributions from thin slices of the beam along the trajectory. The second method is to go one step further and express the integral as a closed analytical function of the parameters of the system. This gives a much clearer picture of the dependence of the trajectory displacement on the various parameters.
Both the slice method and the analytical formulae have been used extensively to calculate trajectory displacement in a particle beam, but both of them have their limitations. The analytical formulae require a limitation to a particular regime of Coulomb interactions and become inaccurate for the intermediate regimes. Additionally, both methods require a large simplification of the system in the form of dividing the system into multiple segments separated by thin lenses. The contribution to the trajectory displacement is then calculated separately for each segment, but it is not clear how to combine the contributions to obtain the final result. Jansen suggests power-sum of the contributions [1], while Jiang uses a linear sum without further explanation of its correctness [2].
Here we review the two different approaches – slice method and analytical formulae, explain their connection in detail and address their limitations. We derive a completely new formula for trajectory displacement that unifies two of the regimes into a single expression. Furthermore, we generalize the slice method to more complex systems with varying beam energy and thick lenses.
Section snippets
Slice method
We are considering the trajectory displacement in electron beam systems with relatively low beam current, such as a scanning electron microscopes. In those systems, Coulomb interactions between the particles can be described as uncorrelated individual scattering events. The trajectory displacement then can be described by the so-called extended two-particle model. Using the model the Coulomb interaction effect of a certain field particle on a test particle can be expressed. By taking into
Integral formulae for Holtzmark and pencil-beam regime
The approach pioneered by Jansen in [1] is to integrate each regime contribution independently and then combine the results from the regime using the power sum rule. Integration of differential contributions (8) and (9) leads to trajectory displacements:The functions SHT and SPT are obtained by integrating over S ∈ [0, 1] in (8) and (9) respectively. Let us first examine the function SHT:
Analytical formula for the funnel regime
Although the power sum in Eq. (11) is not analytically integrable, we can note that the power sum rule for the differential contribution of both regimes uses an exponent of . This exponent is close to and therefore we can approximate the differential contribution as . This approximate differential contribution is analytically integrable. The result again depends on the position of the crossover with respect to the segment. Let us first investigate the case of Sc ≤ 0
General trajectory and variable energy
The slice method described with (8)–(12) as well as the integral formula (22) are both a convenient way to calculate the trajectory displacement in a system with thin lenses and constant energy across each segment. However, for example in the case of electrostatic lenses, the length of the lens can be significant with respect to the whole trajectory and the energy of the particle changes in the region of the lens. Moreover, in the case of probe-forming systems, it is in fact usually the region
Conclusion
The calculation of trajectory displacement in charged particle beam optical systems is a complicated subject. A substantial amount of approximations must be done along the way to a useable expression for the trajectory displacement. Most of the work was done by Jansen who developed analytical formulae for the trajectory displacement. However, these formulae have their limits. Some systems can not be simply divided into several segments with straight trajectories because their lenses are not
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The research was supported by the TA CR ( TE01020118 ), the MEYS CR (LO1212), its infrastructure by the MEYS CR and the EC (CZ.1.05/2.1.0 0/01.0 017) and by the CAS (RVO:68081731). Jan Stopka would like to thank Tomáš Radlička of Institute of Scientific Instruments of the CAS and Pieter Kruit of TU Delft for their helpful remarks. The author would also like to thank his wife Adéla Stopka for the enormous help with creating figures for this article.
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Cited by (1)
Trajectory displacement in a multi beam scanning electron microscope
2021, UltramicroscopyCitation Excerpt :This is an expression which converges to Holtzmark and pencil-beam expressions in the corresponding limiting cases, but interpolates well in the case a transition between the two regimes occurs. Since the formula is rather complicated, we omit it here and instead refer the reader to [15], where both the (generalized) slice method and the funnel regime formula are described in detail. Consequentially, there are parts of the multi-beam path where the trajectory displacement is in fact decreasing.