Space-charge effect in electron time-of-flight analyzer for high-energy photoemission spectroscopy

Highlights

• Two methods for the simulation of space-charge effect in time-resolved PES.

• Reliability and advantages in the use of the SIMION^® software.

• Simulation of the space-charge effect in an electron TOF analyzer.

• Feasibility of a TOF analyzer in time-resolved high-energy PES experiments at FEL.

Abstract

The space-charge effect, due to the instantaneous emission of many electrons after the absorption of a single photons pulse, causes distortion in the photoelectron energy spectrum. Two calculation methods have been applied to simulate the expansion during a free flight of clouds of mono- and bi-energetic electrons generated by a high energy pulse of light and their results have been compared. The accuracy of a widely used tool, such as SIMION^®, in predicting the energy distortion caused by the space-charge has been tested and the reliability of its results is verified. Finally we used SIMION^® to take into account the space-charge effects in the simulation of simple photoemission experiments with a time-of-flight analyzer.

Introduction

New frontiers in solid state physics can be explored with the advent of the next generation synchrotron radiation and laser sources. For example, the new free electron laser (FEL) facilities provide high-energy photon beams with an unprecedented high power within ultra-short pulses, allowing to investigate electron dynamics with a time resolution up to femtosecond. Observing electron exchange during a redox process [1], melting of charge density waves [2] or tuning the electron spin arrangement in a ferromagnet [3] are just some examples of the applications of these sources.

One of the most effective tools to investigate the electronic properties of the materials is the photoelectron spectroscopy (PES), but the effectiveness of this technique in connection with high peak power sources is still questionable. High peak power implies having a very high number of the photoelectrons in very short time intervals and this may limit the use of this spectroscopic technique with the new pulsed sources. As a matter of fact, in a standard PES experiment at synchrotron radiation sources, the typical flux is of the order of ∼10⁴ photons per pulse, while the femtosecond FEL pulses are several orders of magnitude more intense. This causes the emission of a large number of photoelectrons interacting with each other through the Coulomb repulsion and gives rise to the so called space-charge effect [4]. This effect may change the measured energy spectrum by introducing broadening and shift in the spectral feature to an extent that might obscure its physical meaning. For this reason it is of crucial importance to have an effective method capable of simulating this effect in order to find the optimal conditions to minimize it.

J.P. Long, B.S. Itchkawitz and M.N. Kabler (LIK) proposed in 1996 a very simple model to describe the broadening of photoemission peaks consequent to space-charge effect [5]. The LIK model predicts an energy broadening ΔE by the formula:

$$\mathrm{\Delta }E\approx \frac{e^2}{2\pi {ϵ}_0}\frac{N}{a}\approx 2.9·{10}^{-3}\mathrm{eV}\mu \mathrm{m}\frac{N}{a}$$

where N is the number of photoelectrons with charge −e emitted per pulse, a is the radius of the radiation spot on the sample surface (expressed in μm) and ΔE value is given in eV. The energy broadening does not depend on the initial kinetic energy of the electrons. Despite its strong approximations, and in particular the unrealistic hypothesis that all the photoelectrons have initially the same energy, the LIK model succeeds in predicting the order of magnitude of the energy broadening of the PES structures for low excitation energy, as pointed out by Hellmann et al. [6] and Verna et al. [7]. On the other hand, no simple model exists for quantifying the energy shift of the spectral features. Energy shifts have been reported to be of the same order of magnitude of the corresponding energy broadening [8].

Hellmann et al. [6] for the first time used the Treecode software [9], [10], a program for self-consistent N-body simulation, for calculating the effective coulomb repulsion among single electrons in a cloud during free flight. They obtained that an energy spread and a shift of the spectral features would result as a consequence of the space-charge effect.

This method based on deterministic calculation is effective in predicting values for energy broadening and shift, but the number of calculated trajectories must be equal to the number of electrons in the cloud. This could be a disadvantage when the number of electrons became very large (for examples ≥20,000 electrons). We would like to underline that Hellmann et al. treated only free-flying electrons while we want to understand how the space-charge affects the resolution of an electron analyzer. For evaluating the space-charge effect in an electron analyzer it can be useful to find a method where the number of calculated trajectories are limited and modifiable. Moreover, it is also important to have the possibility to calculate trajectories in the presence of the electrostatic field generated by the electrodes of the spectrometric apparatus.

A different approach to the investigation of the space-charge effect can be performed via the determination of electron trajectories in an electron-optics simulator, such as the SIMION^® software. SIMION^® uses a ray-tracing scheme based on the finite difference method that solves the Laplaces equation numerically [11], [12], [13]. Moreover, the SIMION software is one of the most used tools to simulate photoelectron analyzers. In the latest versions of this software package (7.0, 8.0 and 8.1) there is the possibility to introduce the space-charge effect in the calculation of the trajectories, but the utilization of this tool is subject to various caveats. According to the warning of the SIMION manual [11], the computation methods can give only a qualitative description of the effect of charge repulsion whereas a quantitative analysis of heavily space-charged environments is not guaranteed. One of the limitations is that only interactions between charged particles are taken into account, while the effect of the charge density on the local electric field is not computed [14]. Thus it is important to verify the reliability of SIMION in predicting the shift and broadening of spectral feature in PES experiments. As we will see, while we cannot define the limit of this approach in terms of space charge amount, the results of our work suggest that in the regime here explored SIMION is suited to correctly predict charged particles effects. It is important to note that Saito et al. [15] have written a deterministic algorithm to simulate the space-charge in SIMION, but this algorithm is not integrated into the standard version of the software. The SIMBUCA package [16] is an alternative software for charged particle optics used in particular to simulate ions in Penning traps and it also allows to calculate the space-charge effects on the particle trajectories [17].

In the present work we will compare the energy broadening obtained by two different deterministic calculations considering that at high-energy the stochastic term of the space-charge effect is negligible because the electrons are so fast that the probability to have scattering among the electrons is low [18]. The two deterministic calculation methods are represented by the SIMION and Treecode softwares and they have been applied to the expansion in free flight of a cloud of mono- and bi-energetic electrons generated by the same high-energy pulse of light. Once confirmed the accuracy of the calculation methods of SIMION in the free flight case, we will present the results of the simulation of a linear time-of-flight (TOF) analyzer optimized for high-energy time-resolved photoemission spectroscopy that takes into account space-charge effects. The paper is organized as it follows: in Section 2 the numerical methods at the base of the two programs are presented. In Section 3.1 the results of the two methods for clouds of mono-energetic and bi-energetic electrons in free flight are compared and the simulation for a realistic photoemission from the Cu 2p_3/2 core level is also presented. Section 3.2 is devoted to the space-charge effect in a linear TOF analyzer and the conclusions are reported in Section 4.