Modelling the line shape of very low energy peaks of positron beam induced secondary electrons measured using a time of flight spectrometer

In this paper, we present results of numerical modelling of the University of Texas at Arlington’s time of flight positron annihilation induced Auger electron spectrometer (UTA TOF-PAES) using SIMION® 8.1 Ion and Electron Optics Simulator. The time of flight (TOF) spectrometer measures the energy of electrons emitted from the surface of a sample as a result of the interaction of low energy positrons with the sample surface. We have used SIMION® 8.1 to calculate the times of flight spectra of electrons leaving the sample surface with energies and angles dispersed according to distribution functions chosen to model the positron induced electron emission process and have thus obtained an estimate of the true electron energy distribution. The simulated TOF distribution was convolved with a Gaussian timing resolution function and compared to the experimental distribution. The broadening observed in the simulated TOF spectra was found to be consistent with that observed in the experimental secondary electron spectra of Cu generated as a result of positrons incident with energy 1.5 eV to 901 eV, when a timing resolution of 2.3 ns was assumed.


Introduction
The UTA TOF-PAES is capable of generating and transporting a very low, ~1 eV, positron beam and detecting the subsequent electrons emitted from the sample with energy as low as ~0 eV. The spectrometer consists of: (1) a permanent neodymium magnet behind the sample used to parallelize the ejected electron momentum, (2) a field free TOF tube, (3) a set of E×B plates that deflects electrons onto a micro-channel plate (MCP), and (4) detector electronics used in determining the TOF. The energy of the incident positron beam can be adjusted by applying a voltage bias to the sample, resulting in incident positron beam energies of ~1.5 eV to ~901 eV [1].
The work in this paper was motivated by the desire to better understand the nature of the spectrometer's instrument response function in order to obtain a quantitative estimate of the true energy spectrum of low energy electrons emitted from a polycrystalline Cu sample as a result of Auger mediated positron sticking (AMPS) and related positron induced electron emission processes [2].

TOF-PAES Spectrometer Model
The measured TOF distribution of electrons emitted from the sample surface is broadened by the instrument response function of the TOF-PAES spectrometer. The parameters that determine this broadening include: (1) the magnetic field gradient produced by the permanent magnet behind the sample, (2) the electric fields along the flight path of the electrons, and (3) the timing resolution of the detection electronics [3]. SIMION® 8.1 simulations, using the experimental parameters listed in table 1, were used to model the TOF broadening due to the combined effect of the magnetic gradient and the perpendicular electric and magnetic fields between the E×B plates. A more detailed description of the experimental system modelled can be found in reference [1].
Single energy electrons generated randomly on a 2.6 mm diameter circle and emitted with isotropic angular distribution, were flown in the SIMION® 8.1. The TOF of the electrons reaching the MCP was calculated using reverse timing to be consistent with the timing scheme adopted in the experiment and was counted to generate a histogram. That way low time of flights correspond to lower energies and high time of flights correspond to higher energies as is the case in the experiment. The modelled TOF spectrum was then convolved with a Gaussian function, meant to represent the timing resolution of the TOF system, and was converted to the energy domain using an experimentally derived conversion formula. The output of the simulated spectrometer, to the input delta function energy distributions, was used to obtain the broadening of trial functions, representing the true secondary electron distribution, using the principle of superposition. The timing resolution function used was a ~2.35 ns (FWHM) Gaussian which was found by convolving the simulated secondary electron TOF data, for electrons generated as a result of positrons incident with ~901 eV, with a series of Gaussians of different FWHMs until a match with experiment was found. Table 1. Experimental parameters used in the SIMION® 8.1 model of the TOF-PAES spectrometer. The axial magnetic field is the magnetic field along the beam axis (south to north), the sample magnetic field is the magnetic field measured at the sample surface, and the east and west E×B plates refer to the plates east and west of the beam axis. See reference [1]

Results and Discussion
The choice of trial energy distribution functions used as inputs in our simulations was guided by the relevant physics of the positron induced electron processes measured. Following previous studies, we started with a parameterized trial function which has been used to describe the spectra of true secondary electrons produced by an electron beam [2,4].
The terms 0 , , and m are fitting parameters and were optimized to match the experimental data. Two regimes of parameters were found to match experiment best: (1)  If a positron transitions from a positive energy scattering state to the bulk ground state, then the maximum kinetic energy of the emitted secondary electron is given as: where is the maximum kinetic energy of the positron incident on the sample taking into account the maximum kinetic energy of the positron inside the TOF tube, the contact potential between the sample and sample chamber, and the sample bias, is the positron work function, and is the electron work function. However, if the incident positron becomes trapped in a surface state directly from the scattering state then the maximum kinetic energy of the emitted electron is given as: where is the surface state binding energy. This AMPS mechanism is dominant at very low energies but does not make a significant contribution to the electron yield for incident positron energies above ~10 eV [2,5]. The function in equation 1 cuts off at an energy determined by equation 2 or 3 depending on the maximum incident positron energy. Using equation 2, which represents the most probable process of electron emission at incident positron energies > 10 eV, we find the maximum kinetic energy for secondary electrons due to 901.25 eV positrons incident on a polycrystalline Cu to be 896.6 eV (taking φ cu -= 4.65 eV and φ cu + = 0 eV) [6]. Using the equation relevant at low positron energies, equation 3, we find the maximum kinetic energy for secondary electrons due to 3.75 eV positrons incident on a polycrystalline Cu sample to be 1.74 eV (taking = 2.64 eV) [1].      Figure 4 shows the simulated output compared with experimental Cu data. 50,500 particles were flown with the input electron energy distributions (figures 1 and 3) taking into account the distribution in the energy of positrons incident on the sample through a convolution of equation 1 with a Gaussian centred at ~0.5 eV and having a FWHM of 0.26 eV. The measured energy spectra can be seen to be significantly broadened in energy as compared to the model input spectra shown in figures 1 and 3. This broadening is well accounted for by the instrumental broadening found from our numerical modelling over a wide range of electron energies (~0.5 eV to ~900 eV) when a timing resolution of 2.35 ns was assumed.

Conclusion
In this paper, we have presented a model simulation of the University of Texas at Arlington's time of flight positron annihilation induced Auger electron spectrometer using SIMION® 8.1. Using this model we have determined the timing resolution of our system to be 2.3 ns. A parameterized function, equation 1, was used to model the initial kinetic energy of secondary electrons leaving the Cu sample as a result of positron annihilation inside the material. After comparison with experimental data, two regimes of parameters were found to match experiment best, one for incident positron energies abovẽ 10 eV and one for incident positron energies below ~10 eV. We believe regime 1 to represent positrons which mostly transition from a scattering state to a bulk state before thermalization and annihilation in a surface state while regime 2 represents positrons which mostly transition from a scattering state directly to a surface state. However, more work needs to be done in order to verify this.