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Optical properties of the 127° cylindrical energy analyzer used in LEIS experiments
Nuclear Instruments and Methods in Physics Research B 198 (2002) 208–219 www.elsevier.com/locate/nimb
Optical properties of the 127 cylindrical energy analyzer used in LEIS experiments N. Bundaleski *, Z. Rakocevic, I. Terzic Laboratory of Atomic Physics, Vin ca, Institute of Nuclear Sciences, P.O. Box 522, 11001 Belgrade, Yugoslavia Received 15 May 2002; received in revised form 21 August 2002
Abstract The optical properties of the 127 cylindrical energy analyzer used in low energy ion scattering (LEIS) experiments are studied by means of SIMION 3D version 6.0 program. The dependence of the acceptance solid angle X on the target plane coordinates ðx; yÞ and the relative particle energy e completely describes the optical properties of an analyzer. The Xðx; y; eÞ function is calculated from the computed trajectories of ions emitted from different points of the target plane. The influence of spherical aberrations to the error in the energy measurement is determined experimentally. The experimental results agree very well with the results obtained using the numerical simulations as well as, with the results obtained by means of the second order analytical approach. The optical properties are analyzed for different electrode potential configurations i.e. for different deflection voltage modes defined according to the potentials of the inner and the outer electrode. The applied deflection voltage mode does not change Xðx; y; eÞ significantly. However, there is an important influence of the deflection voltage mode to the analyzer constant due to the acceleration of ions traversing along the optical axis of the analyzer. The knowledge of Xðx; y; eÞ can be used to determine the dependence of the energy spectra on the optical properties of the analyzer as well as, on the primary beam profile. This is of particular interest in the analysis of LEIS spectra, because deviation of spectra caused by the optics of the analyzer can be a source of significant errors in quantitative surface composition analysis. 2002 Elsevier Science B.V. All rights reserved. Keywords: Ion optics; Electrostatic energy analyzer; Low energy ion scattering
1. Introduction Low energy ion scattering (LEIS) is a well-established technique for analyzing the composition and structure of a solid-state surface. When noble gas ions are used as projectiles, information depth * Corresponding author. Tel.: +38-11-455451; fax: +38-113410100. E-mail address: [email protected] (N. Bundaleski).
is restricted practically to the first atomic monolayer. This makes LEIS a powerful tool for studying the surface composition in different processes such as heterogeneous catalysis, adhesive and segregation processes [1]. The results of the surface composition measurements using LEIS cannot be directly compared with those obtained by Auger electron spectrometry or X-ray photoelectron spectrometry due to different information depths in these experiments. In many cases, surface segregation can induce a difference between the
0168-583X/02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 2 ) 0 1 4 7 0 - 2
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compositions of the outermost atomic monolayer and the inner monolayers [2]. The surface composition analysis of the first monolayer can hardly be performed without LEIS. Therefore, the quantitative composition analysis by this technique is of special interest in surface science. The energy spectra of low energy noble gas ions scattered from the solid-state surface consist of peaks, which approximately correspond to the single elastic-scattering event. Target atom masses can be calculated from the peak positions, according to the classical elastic binary collision model [1]. Peak intensities give the information about the concentration of the appropriate chemical element in the first monolayer. However, a standardless composition analysis using LEIS is not possible mainly due to the problems concerning the neutralization effects. The calibration can be performed using pure elemental targets as standards [3], bearing in mind that Ômatrix effectsÕ are found in only a few cases [4,5]. In this manner, relative sensitivity factors can be attributed to different elements. Unfortunately, there is a strong deviation in some cases between the relative sensitivity factors obtained in different experimental groups [6]. In order to understand the origin of these deviations, a close insight into different aspects of LEIS experiments and especially into the energy analysis of scattered ions should be performed. Electrostatic energy analyzers (ESA) are irreplaceable as high-resolution monochromators and energy spectrometers of ions and electrons in various atomic collision experiments, as well as in almost every experimental set-up for surface characterization [7]. The schematic of a cylindrical ESA analyzer is given in Fig. 1. For a defined deflection voltage, a charged particle traversing along the optical axis and passing through the centers of the entrance and exit slits of an analyzer possesses the tuning energy Ea . This trajectory is usually termed main path [8]. The main path coincides with the optical axis in the frame of the analytical approach. The tuning energy is directly proportional to the deflection voltage US defined as the voltage applied between the electrodes: Ea ¼ kUS , where k is the analyzer constant. As a first approximation, it is assumed that any particle
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Fig. 1. The definition of the acceptance solid angle Xðx; y; EÞ; U1 and U2 are potentials of the inner and the outer electrode, respectively; US is the deflection voltage. Dashed line represents the optical axis of the system.
detected for a specific deflection voltage has the same energy, equal to corresponding tuning energy Ea . However, this is not the case. For the fixed tuning energy Ea , particles having energies in a finite energy range will get to the detector. In LEIS experiments, ions entering the analyzer are coming from finite surface (Fig 1). For the fixed point on the target plane ðx; yÞ from which the ion is scattered, the ion energy E, and the tuning energy Ea , the acceptance solid angle Xðx; y; EÞ can be defined as a solid angle in which a particle should enter in order to pass through the analyzer and get to the detector placed behind the exit slit. It is usually assumed that there is an area in the target plane with a maximum and approximately constant magnitude of the acceptance solid angle. This area is referred to as the acceptance region [9]. The emitting surface represents an area from which ions are scattered. It is defined by the primary ion beam spot on the target plane. The maximum peak intensity will be obtained if the emitting surface lies inside the acceptance region. A systematic error in quantitative LEIS analysis will generally arise if the position of the emitting surface relative to the acceptance region is not the same in every measurement. This effect is identified as a major problem in quantitative surface analysis using LEIS [6].
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Besides the peak intensity, the primary beam profile also influences the peak position and width due to the oblique incidence of ions into the analyzer. This deteriorates the properties of LEIS setup for the qualitative composition analysis. All these problems could be lowered if the primary ion beam profile is controlled and if the acceptance region is defined. In order to estimate the transparency and the position of the acceptance region, it is necessary to compute the acceptance solid angle Xðx; y; EÞ. However, the function Xðx; y; eÞ, where e ¼ E=Ea , is of particular interest. It will be shown in this article that this function does not only represent the transparency of an analyzer; it also includes all of its optical properties. The knowledge of Xðx; y; eÞ gives us an opportunity to compute the influence of the analyzer and the primary beam profile to the energy spectra and to obtain more realistic energy distribution from the experimental results. However, to the best of our knowledge, determining of Xðx; y; EÞ or Xðx; y; eÞ is rare in experimental practice (a similar kind of calculation is performed in [11]). In most of the cases, only the magnitude of the acceptance solid angle in the acceptance region is given. One of the most popular ESA analyzers is the 127 cylindrical energy analyzer. Although it has lower transparency than, for instance, 180 spherical ESA analyzer with the same resolution, it is still widely used due to its simplicity and significantly lower price. The optical properties of cylindrical analyzers were calculated in detail by means of the second order analytical approach [10–14]. Discrepancies between the experimental results and the theoretical predictions were accounted mainly to errors in manufacturing or mounting the system. However, the first computer calculations showed that particle trajectories in the fringing field region are not calculated properly using analytical approach [15]. Besides the greater precision of the numerical simulations as compared to the analytical methods, the former approach has two more advantages. Analytical methods are developed for specific analyzer geometries and it is supposed that the optical axis is on the earth potential – the potentials of the inner electrode U1 and the outer electrode U2 are US =2 and US =2, respectively (see Fig. 1). This kind of the
electrode potential configuration is usually referred to as the antisymmetrical deflection voltage mode. On the other hand, numerical simulations are not limited concerning the analyzer geometry and the electrode potential configuration. This allows studying the optical properties of analyzers with non-standard geometries as well as, of those that work in deflection voltage modes that are more convenient in some cases. Deflection ESA analyzers used in LEIS experiments generally work in the antisymmetrical deflection voltage mode. In practice, it is simpler to control the potential of only one of the electrodes, while the other is on the earth potential. There are two possibilities: U1 ¼ 0 V and U2 ¼ US or U1 ¼ US and U2 ¼ 0 V. These operating regimes are termed the positive and the negative deflection voltage modes, respectively. If antisymmetrical deflection voltage mode is not applied, the potential of the optical axis will not be 0 V, i.e. the particle traversing along the main path is going to be accelerated and decelerated inside the analyzer. The most important consequence of this effect is that the analyzer constant will be different as compared to the case of the antisymmetrical deflection voltage mode. Thus, we are introducing here three different analyzer constants corresponding to the antisymmetrical, the negative and the positive deflection voltage modes, respectively: k 0 , k and k þ . Other optical properties of the analyzer should also be influenced by the type of the applied operating regime. It is important to stress that these effects can be efficiently analyzed only by the use of numerical computations. In the present article, we are studying in detail the optical properties of the real 127 cylindrical energy analyzer. This energy analyzer is used in LEIS and direct recoil spectrometry (DRS) experiments [16]. Our analysis has been performed by means of numerical simulations. These simulations are realized using SIMION 3D program, version 6.0 [17]. From the numerically obtained trajectories, acceptance solid angle Xðx; y; eÞ is computed. Xðx; y; eÞ is determined for different types of the deflection voltage mode. The oblique incidence into the analyzer influences the analyzer constant and consequently contributes to an error in the energy measurement. This error is
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experimentally determined. Experimental results are compared to those obtained using the numerical simulations, and the second order analytical approach [8]. The influence of the applied deflection voltage mode on the analyzer constant is determined and discussed. The analyzer in our experimental set-up works in the negative deflection voltage mode. Thus, the difference between the optical properties of the analyzer working in the antisymmetrical and negative deflection voltage modes is particularly analyzed. Finally, the obtained results and corresponding consequences on the shape, the intensity and the position of the peaks in LEIS spectra are discussed.
2. Numerical simulations The 127 ESA analyzer used in LEIS and DRS experiments has the following geometric parameters (Fig. 2). The radii of the inside and the outside electrodes are r1 ¼ 112:5 mm and r2 ¼ 127:5 mm,
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respectively. The electrodes are 75 mm high. The real sector angle of the analyzer is 127. The distances between the apertures and the electrodes are both, d ¼ 4 mm. The widths of the entrance and the exit apertures are 2.8 and 4.3 mm, respectively. The positions of the slits define the main path, which coincides with the equipotential line U ðrr Þ in the framework of the analytical approach. Radius rr ¼ 120 mm, is slightly greater than the radius of the optical axis inside the analyzer r0 ¼ 119:76 mm [8]. Thus, U ðrr Þ is not exactly equal to the earth potential in case of the antisymmetrical deflection voltage mode. The distance between the slits and the electrodes is a1 ¼ 6:5 mm (entrance slit), i.e. a2 ¼ 8:6 mm (exit slit). The widths of the entrance and the exit slits are 0.7 and 1 mm, respectively; the height of the slits is h ¼ 10 mm. In order to compute the 3D trajectories of ions traversing the analyzer, SIMION 6.0 program was used for modeling the real 127 analyzer [17]. 2D field distribution was computed, because the ratio between the heights of the slits and the analyzer
Fig. 2. The schematic cross section of the non-ideal 127 cylindrical energy analyzer: (1) inner electrode; (2) outer electrode; (3) entrance aperture; (4) exit aperture; (5) entrance slit; and (6) exit slit. The radii of the inner and the outer electrode are denoted by r1 and r2 , respectively; d is the distance between the electrodes and the entrance or the exit aperture, a1 and a2 are distances between the electrodes and the entrance and the exit slit, respectively; the distance between the target and the entrance into the analyzer is denoted by l. The main path and the trajectory of the particle entering the analyzer with the incident angle ae are also presented.
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electrodes is low. The deflection voltage was kept constant. It was assumed that the particles are emitted from the surface positioned in front of the entrance slit of the analyzer. The distance between the entrance slit and the target plane (cf. Fig. 2) is l ¼ 75 mm in our LEIS experiment [16]. The trajectories were calculated for different ion energies E and target plane coordinates ðx; yÞ, as parameters. An incident angle in the deflection plane ae can be attributed to the x coordinate. It is defined as the angle between the particle trajectory in the deflection plane, and the main path on the entrance into the analyzer: ae ¼ arctgðx=lÞ (Fig. 2). Detailed calculations were performed for both, the antisymmetrical and the negative deflection voltage modes. Some computations were done for the positive deflection voltage mode, too. The grid density for computing the electrical field distribution was 0.2 mm. In order to estimate the error of the computations, some simulations were performed with resolution of 0.1 mm. Relative deviation between the corresponding results obtained using different resolution is less than 0.15%.
Fig. 3. The experimental set-up for determining the influence of oblique incidence to the error of the energy measurement.
and 2500 eV, in the negative deflection voltage mode – U1 ¼ US , U2 ¼ 0 V (Fig. 2).
3. Experimental
4. Results
The experimental set-up for measuring the influence of the oblique incidence to the analyzer constant is presented in Fig. 3. Nþ ions were accelerated, mass analyzed, and focused to the entrance slit of the 127 cylindrical analyzer. The angular width of the ion beam is about 0.3. The width of the ion beam profile is 3 mm [16]. The analyzer can be rotated round the axis that is perpendicular to the drawing plane. Different parts of the ion beam pass through the entrance slit, by rotating the analyzer. The ion energy distributions were determined for different incident angles, in the range ae 2 ½3; 3 . It was assumed that the ion energy distribution does not change across the ion beam profile. Thus, the discrepancies between the ion energies for the different incident angles can only be attributed to the energy measurement error due to the oblique incidence of ions into the energy analyzer. The experiments were performed with four different ion energies: 1000, 1500, 2000
The influence of the x coordinate and the particle energy E on the acceptance solid angle Xðx; y; EÞ was calculated from the computed trajectories. The tuning energy Ea was fixed. Acceptance solid angle was calculated for the x coordinate in the [)5, 5] mm range. The computations were performed for the antisymmetrical and the negative deflection voltage modes. For a fixed point ðx; yÞ, transfer function can be defined as the acceptance solid angle versus particle energy. Typical transfer function of this analyzer, shown in Fig. 4, has a triangular shape. The maximum of this function Em ðx; yÞ as well as its full width half-maximum DEðx; yÞ can be calculated. For the point ð0; 0Þ, Em Ea was obtained. From this result, the analyzer constant for different operating modes was calculated. The results are given in Table 1, together with appropriate experimental result for negative deflection voltage mode and analytical result according to the second
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Fig. 4. The instrument function of the energy analyzer for ae ¼ 0:74 in the case of the negative deflection voltage; Ea ¼ 352:20 eV; Em ¼ 351:77. The magnitude of entrance angle corresponds to the point (x ¼ 1 mm, y ¼ 0 mm) in the target plane.
order approximation for the antisymmetrical voltage mode. It can be seen from Fig. 4 that Em 6¼ Ea . This is a consequence of the spherical aberrations. The influence of the oblique incidence to the error of the energy measurement computed using the numerical simulations is expressed as ðEm ðae Þ Em ð0ÞÞ= Em ð0Þ ¼ ðkðae Þ kð0ÞÞ=kð0Þ (the deflection voltage US was constant in these calculations) and presented in Fig. 5. These results were obtained for the antisymmetrical deflection voltage mode as well as, for the negative deflection voltage mode. The deviation among the results is found to be negligible. The experimental results concerning the influence of the oblique incidence to the error of the energy measurement is also given in Table 2 as well as, in Fig. 5. The energy of the primary ion beam E was constant. However, the magnitude of US providing maximum signal, depends on the entrance angle ae . Thus, in this case, Dk ðae Þ=k ð0Þ ¼
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Fig. 5. The relative discrepancy of the analyzer constant versus entrance angle ae . Open marks correspond to the experimental results (the negative deflection voltage mode); solid squares are numerical simulation results obtained for the negative deflection voltage mode; the result obtained by the second order analytical approach (the antisymmetrical deflection voltage mode) is presented with the solid line.
DUS ðae Þ=US ð0Þ. The obtained data are also compared to the results of the second order analytical approach, for the case of the antisymmetrical deflection voltage mode [8]. Appropriate results are also presented in Fig. 5. As it was expected, there is no change of computed trajectories if the particle energy and the deflection voltage are multiplied by the same constant. This is the well-known characteristic of charged particles deflected in the electrostatic field. Two consequences of this fact are the linear dependences of the tuning energy Ea and the width of the transfer function DE on the deflection voltage. It would be useful in practice to describe optical properties of an analyzer using a function that does not depend on the deflection voltage i.e. on the tuning energy. This can be done by introducing the relative particle energy e: e ¼ E=Ea . Acceptance solid angle defined as a function of the target
Table 1 The analyzer constant determined by the numerical simulations, the second order analytical approach and the experiment Type of calculation
Numerical simulation
Analytical approach
Deflection voltage
Negative
Antisymmetrical
Positive
Antisymmetrical
Negative
k
3.522
4.022
4.521
4.011
3.409
The numerical simulation results are obtained for different types of the deflection voltage mode.
Experiment
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Table 2 Experimentally determined dependence of the analyzer constant k on the incident angle ae Dk ðae Þ=k ð0Þ (%) ae ()
E0 ¼ 1000 eV
E0 ¼ 1500 eV
E0 ¼ 2000 eV
E0 ¼ 2500 eV
)2.67 )1.83 )1.0 0.0 0.83 1.67 2.5
)0.2 )0.27 )0.03 0 0.2 0.47 0.74
)0.09 )0.22 )0.09 0 0.22 0.54 0.85
)0.47 )0.43 )0.20 0 0.23 0.44 0.64
)0.48 )0.51 )0.27 0 0.13 0.41 0.68
the energy widths is 5.7%. The maximum of Xðx; eÞy¼0 is about 5% greater in the case of the negative deflection voltage mode. Thus, applying the negative deflection voltage mode lowers the energy resolution and increases the transparency as compared to case of the antisymmetrical deflection voltage mode. The acceptance solid angle in case of for the negative deflection voltage mode versus the target plane coordinates for three magnitudes of e as a parameter is shown in Fig. 7.
5. Discussion plane coordinates and the relative particle energy Xðx; y; eÞ, entirely determines optical properties of an analyzer and does not depend on the tuning energy. The dependence of the acceptance solid angle on x and e for y ¼ 0 mm in the case of the negative deflection voltage mode is given in Fig. 6. It has a shape of a ridge, which is folded due to the spherical aberrations. Positions of the local maxima of the acceptance solid angle in the x–e plane correspond to the dependence of the relative error in the energy measurement on the entrance angle, given in Fig. 5. The shape of Xðx; eÞy¼0 does not depend qualitatively on the deflection voltage mode. It is generally wider with respect to the e coordinate, if the negative deflection voltage mode is applied. The average relative difference between
Fig. 6. The acceptance solid angle as a function of the target plane coordinate x and the relative particle energy e, in case of the negative deflecting voltage mode; y ¼ 0 mm.
The equipotential surfaces inside the analyzer, far from the ends of the electrodes, are cylinders. Particle trajectories obtained using the numerical simulations are close to the main path. This is valid even if the antisymmetrical deflection voltage mode is not applied. However, U ðrr Þ is changed with the change of the deflection voltage mode. This means that the particle energy is changed along the trajectory, if the deflection voltage mode is not antisymmetrical. It is clear that this effect has strong influence to the analyzer constant. Three different analyzer constants that are most interesting in practice are already introduced in Section 1: • Antisymmetrical deflection voltage mode U1 ¼ US =2, U2 ¼ US =2, k 0 ¼ E0 =US ; U 0 ðr0 Þ 0 V. • Positive deflection voltage mode U1 ¼ 0 V, U2 ¼ US , k þ ¼ Eþ =US ; U þ ðr0 Þ USþ =2. • Negative deflection voltage mode U1 ¼ US , U2 ¼ 0 V, k ¼ E =US ; U ðr0 Þ US =2. In case of the positive (negative) deflection voltage mode, particles are decelerated (accelerated) at the entrance into the analyzer. Their energy inside the analyzer is approximately E US =2 ðE þ US =2Þ. At the exit from the analyzer, particles are going to be accelerated (decelerated), and their final energy will be equal to the starting energy E. If the deflection voltage is equal for any of the applied deflection voltage modes (deflecting field is the same), the energy of the particle traversing along the main path will approximately be the same: E0 Eþ US =2 E þ US =2. After
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Fig. 7. The acceptance solid angle as a function of the target plane coordinates Xðx; yÞ in case of the negative deflection voltage mode; relative particle energy e is a parameter: (a) e ¼ 0:9920; (b) e ¼ 0:9993 and (c) e ¼ 1:0106.
dividing this expression by US the following relation is obtained: k 0 k þ 0:5 k þ 0:5. This rough estimation is remarkably good according to the results of the numerical simulation (cf. Table 1). The difference between the k magnitudes obtained experimentally and by use of the computer simulations are most probably due to the errors in manufacturing or mounting the system. In some cases, the optical properties of an analyzer can contribute to serious errors during the measurement of the energy spectra. The deviations of spectra can generally be manifested as errors in peak position, peak intensity, and peak shape. The errors are directly connected to the shape of the emitting surface i.e. to the primary beam profile on the target plane. Let us firstly discuss two extreme cases in order to clarify possible deviations of en-
ergy spectra of scattered ions due to different primary beam profiles. (a) The case of the narrow primary ion beam profile; if the maximum of the primary ion beam profile does not coincide with the center of the target (point ð0; 0Þ), the majority of scattered ions will enter the analyzer with ae 6¼ 0. The oblique incidence into the analyzer will contribute to the systematic error in the energy measurement due to the spherical aberrations. (b) The case of the wide primary ion beam profile; if the emitting surface is wider than the acceptance region, only a part of the primary ion current will contribute to the intensity of LEIS peak obtained in the spectrum. Thus, LEIS
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peak intensity may not be directly proportional to the total primary ion current, if the primary ion beam profile is not constant [6]. Wide primary beam profile i.e. large emitting area can also contribute to the significant peak broadening. A very good agreement between the experimental and the numerical results concerning the influence of the oblique incidence to the error in energy measurement (Fig. 5), indicates that performed numerical simulation approximates the real analyzer very well. It can also be seen that the results of the second order analytical approach give the same results as the numerical simulation. A peak shift due to very fine effects, such as inelastic energy losses in LEIS experiments, can be less than 1% in some cases [18]. The relative error of the energy measurement due to the oblique incidence of scattered ions can easily be of the same order (cf. Table 2) and even greater if the distance l is small. The ideal sector angle equals about 127.6 that is very close to the optimal magnitude needed for the first order focusing )127.3 [8]. Unfortunately, the relative error is increased to some extent because the positions of the entrance and the exit slits do not coincide with the ideal field boundaries. The dependence of the acceptance solid angle on the x coordinate and the relative particle energy e (Fig. 6) is qualitatively similar to the equivalent results given in [11] for the same type of the analyzer. The difference between the Xðx; eÞy¼0 for different deflection voltage modes is a consequence of the particle acceleration on the entrance into the analyzer in the case of the negative deflection voltage mode – the circular component of the particle velocity is increased, which contributes to folding the trajectory and decreasing the absolute magnitude of the angle between the trajectory and the optical axis. The particle acceleration increases the maximum of Xðx; eÞy¼0 function due to the trajectory folding in the non-dispersive plane, while the broadening of Xðx; eÞy¼0 along the energy axis is not significant. The trajectory folding in the dispersive plane contributes to the broadening of Xðx; eÞy¼0 along the energy axis. The maximum of Xðx; eÞy¼0 cannot be further increased by the tra-
jectory folding in the dispersive plane – it is limited by the entrance slit width. The dependence of the acceptance solid angle on the target plane coordinates and the relative particle energy Xðx; y; eÞ (Fig. 7) is the one that has major practical significance. As it was mentioned already, this function includes all the optical properties of the analyzer used for measuring the energy spectra of charged particles emitted from the target plane. There is a strong dependence of the acceptance solid angle on the relative particle energy. This complicates determining the acceptance region i.e. estimating the influence of the primary ion beam profile on the peak intensity. However, the knowledge of Xðx; y; eÞ allows us to compute the influence of the beam profile on the position, the intensity and the shape of the peaks obtained in the LEIS spectra. A possibility to precisely calculate the energy distribution of scattered ions from the experimentally obtained energy spectra, if primary ion beam profile is measured, should also be considered. All these effects are important in every single experiment in which an ESA analyzer is used. However, additional effects are present in LEIS experiments due to the dependence of the scattering angle on the target plane coordinates h ¼ hðx; yÞ. As the energy of scattered ions in LEIS experiment directly depends on the scattering angle h according to the elastical binary collision model [1], the deviation of h will contribute to the deviation of the energy of scattered ions i.e. to the LEIS peak position and/or broadening. According to the mentioned model, a relative deviation of the energy of scattered ions DE=E due to the deviation of the scattering angle Dh is defined by the expression DE 2 sin h Dh ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð1Þ E A2 sin2 h where A represents target mass to projectile mass ratio. The dependence of DE=E on h for several magnitudes of A as a parameter is given in Fig. 8. As in case of the oblique incidence into the analyzer, this effect can contribute to the peak position error and/or to the peak broadening. We find that the magnitude of Dh ¼ 1:5 is realistic for the
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Fig. 8. The relative deviation of the energy of scattered ions DE=E versus the scattering angle h. The error of the energy measurement is due to the deviation of the scattering angle Dh ¼ 1:5; target mass to projectile mass ratio A is a parameter.
typical LEIS systems, since it also includes the angular spread of the primary ion beam. It can be seen that in the case of small target mass to projectile mass ratio this effect contributes to greater deviations of the energy spectra than an analyzer with reasonable optical properties. Generally speaking, systematic errors concerning the peak positions and peak intensities rapidly increase with the decrease of the distance between the target and the entrance slit of the analyzer, l (cf. Fig. 2). The decrease of l contributes to the increase of the acceptance solid angle. Nevertheless, the area seen by the detector is decreased, the spherical aberrations are more pronounced, and the shift of the beam spot from the center of the target contributes to greater error in the energy measurement. The distance l is greater in our case than in the standard devices of similar type. This lowers the importance of the mentioned effects in our measurements as compared to other devices. Another general advantage of our system is that a small shift of the beam spot in the non-dispersive plane (along the y-axis) will not contribute to the significant error concerning the transparency and the energy measurement, because the magnitude of
the acceptance solid angle is almost constant in a wide range of y (cf. Fig. 7). This is in contrast to the analyzers having focusing action in both planes. Thus, in spite of using the analyzer with modest optical properties, our system is quite convenient for performing high quality experiments.
6. Conclusion In this work, we present a numerical and experimental study of the 127 energy analyzer used in the LEIS experiments. The main results are the following: (a) The dependence of the analyzer acceptance solid angle on the target plane coordinates and the relative particle energy X ¼ Xðx; y; eÞ is identified as a function that completely describes the realistic optical characteristics of an energy analyzer. This function is numerically computed in detail for the 127 cylindrical energy analyzer used in LEIS experiments by means of SIMION program. (b) The oblique incidence of ions into the energy analyzer contributes to the error of the energy
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measurement. The dependence of this error on the incident angle is experimentally measured. The same error is calculated from the numerically determined function Xðx; y; eÞ, as well as by means of the second order analytical approach. All these results agree very well. (c) The influence of the applied deflection voltage mode on the optical properties of the analyzer is computed, too. The major influence is on the analyzer constant, which can be explained very well by considering the particle acceleration along the optical axis. A simple relation between the analyzer constants for different deflection voltage modes is introduced: k 0 ¼ k þ 0:5 ¼ k þ 0:5. This relation is valid for the magnitudes of numerically obtained analyzer constants. Other characteristics of the analyzer are not significantly changed using different deflection voltage modes. The energy resolution is decreased and the transparency increased when the negative deflection voltage mode is applied instead of the antisymmetrical deflection voltage mode. (d) An additional peak broadening and eventual peak position shift is present in LEIS experiments due to the scattering angle dependence on the target plane coordinates. In case of low target mass to projectile mass ratio, this effect can have greater influence on the energy spectra than the above-mentioned effects. This phenomenon is specific for LEIS as well as, for other scattering experiments. The primary beam profile and the relative position of the target and the analyzer are very important in surface characterization. This problem is especially present in the LEIS technique: beam control and focusing is more difficult in the case of low energy ion beams as compared to electron beams and medium or high energy ion beams. A systematic error in the energy measurement will be made if the primary beam is not focused on the center of the target and/or if the optical axis of the analyzer does not intersect the center of the target. Thus, in case of fine measurements, such as determination of inelastic energy losses in LEIS experiments, extremely good determination of the experimental set-up geometry and of the primary beam profile is obligatory. The problem is also present in the case of the quantitative analysis:
primary ion beam profile will not contribute to the systematic errors as long as the beam spot is smaller than the acceptance region [6,9]. However, it is not simple to define this area – it can strongly depend on the relative particle energy (cf. Fig. 7). The most reliable way to quantitatively determine the influence of the analyzer on the energy spectra is to compute the spectra obtained by the analyzer for the defined primary beam profile and the energy distribution of particles emitted from the surface, using the knowledge of Xðx; y; eÞ. Unfortunately, modeling the energy distribution of emitted particles is generally a problem. The problem is increased in the case of LEIS, because distribution (i.e. scattering angle) depends on the target plane coordinates. Nevertheless, this type of computations can provide important information and further improve the analysis of the experimental results. The investigation of these problems is in progress.
Acknowledgements This work has been supported by the Project 2018 from the Ministry of Development, Science and Technology, Republic of Serbia.
References [1] H. Niehus, W. Heiland, E. Taglauer, Surf. Sci. Rep. 17 (1993) 213. [2] M. Prutton, Introduction to Surface Physics, Clarendon Press, Oxford, 1994. [3] S.N. Mikhailov, R.J.M. Elfrink, J.P. Jacobs, L.C.A. van den Oetelaar, P.J. Scanlon, H.H. Brongersma, Nucl. Instr. and Meth. B 93 (1994) 149. [4] S.N. Mikhailov, L.C.A. van den Oetelaar, H.H. Brongersma, Nucl. Instr. and Meth. B 93 (1994) 210. [5] M. Casagrande, S. Lacombe, L. Guillemot, V.A. Esaulov, Surf. Sci. Lett. 445 (2000) L36. [6] H.H. Brongersma et al., Nucl. Instr. and Meth. B 142 (1998) 377. [7] D.P. Woodruff, T.A. Delchar, Modern Techniques of Surface Science, Cambridge University Press, Cambridge, 1986. [8] H. Wollnik, in: A. Septier (Ed.), Focusing of Charged Particles, Vol. 2, Chapter 1, Academic press, New York, 1967.
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[14] A.D. Johnstone, Rev. Sci. Instr. 43 (7) (1972) 1030. [15] P. Bruce, R.L. Dalglish, J.C. Kelly, Can. J. Phys. 51 (1973) 574. [16] I. Terzic, N. Bundaleski, Z. Rakocvic, N. Oklobdzija, J. Elazar, Rev. Sci. Instr. 7 (11) (2000) 4195. [17] D.A. Dahl, Simion 3D, version 6.0, Princeton Electronik Systems, 1995. [18] F. Shoji, T. Hanawa, Surf. Sci. Lett. 129 (1983) L261.
Optical properties of the 127 cylindrical energy analyzer used in LEIS experiments N. Bundaleski *, Z. Rakocevic, I. Terzic Laboratory of Atomic Physics, Vin ca, Institute of Nuclear Sciences, P.O. Box 522, 11001 Belgrade, Yugoslavia Received 15 May 2002; received in revised form 21 August 2002
Abstract The optical properties of the 127 cylindrical energy analyzer used in low energy ion scattering (LEIS) experiments are studied by means of SIMION 3D version 6.0 program. The dependence of the acceptance solid angle X on the target plane coordinates ðx; yÞ and the relative particle energy e completely describes the optical properties of an analyzer. The Xðx; y; eÞ function is calculated from the computed trajectories of ions emitted from different points of the target plane. The influence of spherical aberrations to the error in the energy measurement is determined experimentally. The experimental results agree very well with the results obtained using the numerical simulations as well as, with the results obtained by means of the second order analytical approach. The optical properties are analyzed for different electrode potential configurations i.e. for different deflection voltage modes defined according to the potentials of the inner and the outer electrode. The applied deflection voltage mode does not change Xðx; y; eÞ significantly. However, there is an important influence of the deflection voltage mode to the analyzer constant due to the acceleration of ions traversing along the optical axis of the analyzer. The knowledge of Xðx; y; eÞ can be used to determine the dependence of the energy spectra on the optical properties of the analyzer as well as, on the primary beam profile. This is of particular interest in the analysis of LEIS spectra, because deviation of spectra caused by the optics of the analyzer can be a source of significant errors in quantitative surface composition analysis. 2002 Elsevier Science B.V. All rights reserved. Keywords: Ion optics; Electrostatic energy analyzer; Low energy ion scattering
1. Introduction Low energy ion scattering (LEIS) is a well-established technique for analyzing the composition and structure of a solid-state surface. When noble gas ions are used as projectiles, information depth * Corresponding author. Tel.: +38-11-455451; fax: +38-113410100. E-mail address: [email protected] (N. Bundaleski).
is restricted practically to the first atomic monolayer. This makes LEIS a powerful tool for studying the surface composition in different processes such as heterogeneous catalysis, adhesive and segregation processes [1]. The results of the surface composition measurements using LEIS cannot be directly compared with those obtained by Auger electron spectrometry or X-ray photoelectron spectrometry due to different information depths in these experiments. In many cases, surface segregation can induce a difference between the
0168-583X/02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 2 ) 0 1 4 7 0 - 2
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compositions of the outermost atomic monolayer and the inner monolayers [2]. The surface composition analysis of the first monolayer can hardly be performed without LEIS. Therefore, the quantitative composition analysis by this technique is of special interest in surface science. The energy spectra of low energy noble gas ions scattered from the solid-state surface consist of peaks, which approximately correspond to the single elastic-scattering event. Target atom masses can be calculated from the peak positions, according to the classical elastic binary collision model [1]. Peak intensities give the information about the concentration of the appropriate chemical element in the first monolayer. However, a standardless composition analysis using LEIS is not possible mainly due to the problems concerning the neutralization effects. The calibration can be performed using pure elemental targets as standards [3], bearing in mind that Ômatrix effectsÕ are found in only a few cases [4,5]. In this manner, relative sensitivity factors can be attributed to different elements. Unfortunately, there is a strong deviation in some cases between the relative sensitivity factors obtained in different experimental groups [6]. In order to understand the origin of these deviations, a close insight into different aspects of LEIS experiments and especially into the energy analysis of scattered ions should be performed. Electrostatic energy analyzers (ESA) are irreplaceable as high-resolution monochromators and energy spectrometers of ions and electrons in various atomic collision experiments, as well as in almost every experimental set-up for surface characterization [7]. The schematic of a cylindrical ESA analyzer is given in Fig. 1. For a defined deflection voltage, a charged particle traversing along the optical axis and passing through the centers of the entrance and exit slits of an analyzer possesses the tuning energy Ea . This trajectory is usually termed main path [8]. The main path coincides with the optical axis in the frame of the analytical approach. The tuning energy is directly proportional to the deflection voltage US defined as the voltage applied between the electrodes: Ea ¼ kUS , where k is the analyzer constant. As a first approximation, it is assumed that any particle
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Fig. 1. The definition of the acceptance solid angle Xðx; y; EÞ; U1 and U2 are potentials of the inner and the outer electrode, respectively; US is the deflection voltage. Dashed line represents the optical axis of the system.
detected for a specific deflection voltage has the same energy, equal to corresponding tuning energy Ea . However, this is not the case. For the fixed tuning energy Ea , particles having energies in a finite energy range will get to the detector. In LEIS experiments, ions entering the analyzer are coming from finite surface (Fig 1). For the fixed point on the target plane ðx; yÞ from which the ion is scattered, the ion energy E, and the tuning energy Ea , the acceptance solid angle Xðx; y; EÞ can be defined as a solid angle in which a particle should enter in order to pass through the analyzer and get to the detector placed behind the exit slit. It is usually assumed that there is an area in the target plane with a maximum and approximately constant magnitude of the acceptance solid angle. This area is referred to as the acceptance region [9]. The emitting surface represents an area from which ions are scattered. It is defined by the primary ion beam spot on the target plane. The maximum peak intensity will be obtained if the emitting surface lies inside the acceptance region. A systematic error in quantitative LEIS analysis will generally arise if the position of the emitting surface relative to the acceptance region is not the same in every measurement. This effect is identified as a major problem in quantitative surface analysis using LEIS [6].
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Besides the peak intensity, the primary beam profile also influences the peak position and width due to the oblique incidence of ions into the analyzer. This deteriorates the properties of LEIS setup for the qualitative composition analysis. All these problems could be lowered if the primary ion beam profile is controlled and if the acceptance region is defined. In order to estimate the transparency and the position of the acceptance region, it is necessary to compute the acceptance solid angle Xðx; y; EÞ. However, the function Xðx; y; eÞ, where e ¼ E=Ea , is of particular interest. It will be shown in this article that this function does not only represent the transparency of an analyzer; it also includes all of its optical properties. The knowledge of Xðx; y; eÞ gives us an opportunity to compute the influence of the analyzer and the primary beam profile to the energy spectra and to obtain more realistic energy distribution from the experimental results. However, to the best of our knowledge, determining of Xðx; y; EÞ or Xðx; y; eÞ is rare in experimental practice (a similar kind of calculation is performed in [11]). In most of the cases, only the magnitude of the acceptance solid angle in the acceptance region is given. One of the most popular ESA analyzers is the 127 cylindrical energy analyzer. Although it has lower transparency than, for instance, 180 spherical ESA analyzer with the same resolution, it is still widely used due to its simplicity and significantly lower price. The optical properties of cylindrical analyzers were calculated in detail by means of the second order analytical approach [10–14]. Discrepancies between the experimental results and the theoretical predictions were accounted mainly to errors in manufacturing or mounting the system. However, the first computer calculations showed that particle trajectories in the fringing field region are not calculated properly using analytical approach [15]. Besides the greater precision of the numerical simulations as compared to the analytical methods, the former approach has two more advantages. Analytical methods are developed for specific analyzer geometries and it is supposed that the optical axis is on the earth potential – the potentials of the inner electrode U1 and the outer electrode U2 are US =2 and US =2, respectively (see Fig. 1). This kind of the
electrode potential configuration is usually referred to as the antisymmetrical deflection voltage mode. On the other hand, numerical simulations are not limited concerning the analyzer geometry and the electrode potential configuration. This allows studying the optical properties of analyzers with non-standard geometries as well as, of those that work in deflection voltage modes that are more convenient in some cases. Deflection ESA analyzers used in LEIS experiments generally work in the antisymmetrical deflection voltage mode. In practice, it is simpler to control the potential of only one of the electrodes, while the other is on the earth potential. There are two possibilities: U1 ¼ 0 V and U2 ¼ US or U1 ¼ US and U2 ¼ 0 V. These operating regimes are termed the positive and the negative deflection voltage modes, respectively. If antisymmetrical deflection voltage mode is not applied, the potential of the optical axis will not be 0 V, i.e. the particle traversing along the main path is going to be accelerated and decelerated inside the analyzer. The most important consequence of this effect is that the analyzer constant will be different as compared to the case of the antisymmetrical deflection voltage mode. Thus, we are introducing here three different analyzer constants corresponding to the antisymmetrical, the negative and the positive deflection voltage modes, respectively: k 0 , k and k þ . Other optical properties of the analyzer should also be influenced by the type of the applied operating regime. It is important to stress that these effects can be efficiently analyzed only by the use of numerical computations. In the present article, we are studying in detail the optical properties of the real 127 cylindrical energy analyzer. This energy analyzer is used in LEIS and direct recoil spectrometry (DRS) experiments [16]. Our analysis has been performed by means of numerical simulations. These simulations are realized using SIMION 3D program, version 6.0 [17]. From the numerically obtained trajectories, acceptance solid angle Xðx; y; eÞ is computed. Xðx; y; eÞ is determined for different types of the deflection voltage mode. The oblique incidence into the analyzer influences the analyzer constant and consequently contributes to an error in the energy measurement. This error is
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experimentally determined. Experimental results are compared to those obtained using the numerical simulations, and the second order analytical approach [8]. The influence of the applied deflection voltage mode on the analyzer constant is determined and discussed. The analyzer in our experimental set-up works in the negative deflection voltage mode. Thus, the difference between the optical properties of the analyzer working in the antisymmetrical and negative deflection voltage modes is particularly analyzed. Finally, the obtained results and corresponding consequences on the shape, the intensity and the position of the peaks in LEIS spectra are discussed.
2. Numerical simulations The 127 ESA analyzer used in LEIS and DRS experiments has the following geometric parameters (Fig. 2). The radii of the inside and the outside electrodes are r1 ¼ 112:5 mm and r2 ¼ 127:5 mm,
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respectively. The electrodes are 75 mm high. The real sector angle of the analyzer is 127. The distances between the apertures and the electrodes are both, d ¼ 4 mm. The widths of the entrance and the exit apertures are 2.8 and 4.3 mm, respectively. The positions of the slits define the main path, which coincides with the equipotential line U ðrr Þ in the framework of the analytical approach. Radius rr ¼ 120 mm, is slightly greater than the radius of the optical axis inside the analyzer r0 ¼ 119:76 mm [8]. Thus, U ðrr Þ is not exactly equal to the earth potential in case of the antisymmetrical deflection voltage mode. The distance between the slits and the electrodes is a1 ¼ 6:5 mm (entrance slit), i.e. a2 ¼ 8:6 mm (exit slit). The widths of the entrance and the exit slits are 0.7 and 1 mm, respectively; the height of the slits is h ¼ 10 mm. In order to compute the 3D trajectories of ions traversing the analyzer, SIMION 6.0 program was used for modeling the real 127 analyzer [17]. 2D field distribution was computed, because the ratio between the heights of the slits and the analyzer
Fig. 2. The schematic cross section of the non-ideal 127 cylindrical energy analyzer: (1) inner electrode; (2) outer electrode; (3) entrance aperture; (4) exit aperture; (5) entrance slit; and (6) exit slit. The radii of the inner and the outer electrode are denoted by r1 and r2 , respectively; d is the distance between the electrodes and the entrance or the exit aperture, a1 and a2 are distances between the electrodes and the entrance and the exit slit, respectively; the distance between the target and the entrance into the analyzer is denoted by l. The main path and the trajectory of the particle entering the analyzer with the incident angle ae are also presented.
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electrodes is low. The deflection voltage was kept constant. It was assumed that the particles are emitted from the surface positioned in front of the entrance slit of the analyzer. The distance between the entrance slit and the target plane (cf. Fig. 2) is l ¼ 75 mm in our LEIS experiment [16]. The trajectories were calculated for different ion energies E and target plane coordinates ðx; yÞ, as parameters. An incident angle in the deflection plane ae can be attributed to the x coordinate. It is defined as the angle between the particle trajectory in the deflection plane, and the main path on the entrance into the analyzer: ae ¼ arctgðx=lÞ (Fig. 2). Detailed calculations were performed for both, the antisymmetrical and the negative deflection voltage modes. Some computations were done for the positive deflection voltage mode, too. The grid density for computing the electrical field distribution was 0.2 mm. In order to estimate the error of the computations, some simulations were performed with resolution of 0.1 mm. Relative deviation between the corresponding results obtained using different resolution is less than 0.15%.
Fig. 3. The experimental set-up for determining the influence of oblique incidence to the error of the energy measurement.
and 2500 eV, in the negative deflection voltage mode – U1 ¼ US , U2 ¼ 0 V (Fig. 2).
3. Experimental
4. Results
The experimental set-up for measuring the influence of the oblique incidence to the analyzer constant is presented in Fig. 3. Nþ ions were accelerated, mass analyzed, and focused to the entrance slit of the 127 cylindrical analyzer. The angular width of the ion beam is about 0.3. The width of the ion beam profile is 3 mm [16]. The analyzer can be rotated round the axis that is perpendicular to the drawing plane. Different parts of the ion beam pass through the entrance slit, by rotating the analyzer. The ion energy distributions were determined for different incident angles, in the range ae 2 ½3; 3 . It was assumed that the ion energy distribution does not change across the ion beam profile. Thus, the discrepancies between the ion energies for the different incident angles can only be attributed to the energy measurement error due to the oblique incidence of ions into the energy analyzer. The experiments were performed with four different ion energies: 1000, 1500, 2000
The influence of the x coordinate and the particle energy E on the acceptance solid angle Xðx; y; EÞ was calculated from the computed trajectories. The tuning energy Ea was fixed. Acceptance solid angle was calculated for the x coordinate in the [)5, 5] mm range. The computations were performed for the antisymmetrical and the negative deflection voltage modes. For a fixed point ðx; yÞ, transfer function can be defined as the acceptance solid angle versus particle energy. Typical transfer function of this analyzer, shown in Fig. 4, has a triangular shape. The maximum of this function Em ðx; yÞ as well as its full width half-maximum DEðx; yÞ can be calculated. For the point ð0; 0Þ, Em Ea was obtained. From this result, the analyzer constant for different operating modes was calculated. The results are given in Table 1, together with appropriate experimental result for negative deflection voltage mode and analytical result according to the second
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Fig. 4. The instrument function of the energy analyzer for ae ¼ 0:74 in the case of the negative deflection voltage; Ea ¼ 352:20 eV; Em ¼ 351:77. The magnitude of entrance angle corresponds to the point (x ¼ 1 mm, y ¼ 0 mm) in the target plane.
order approximation for the antisymmetrical voltage mode. It can be seen from Fig. 4 that Em 6¼ Ea . This is a consequence of the spherical aberrations. The influence of the oblique incidence to the error of the energy measurement computed using the numerical simulations is expressed as ðEm ðae Þ Em ð0ÞÞ= Em ð0Þ ¼ ðkðae Þ kð0ÞÞ=kð0Þ (the deflection voltage US was constant in these calculations) and presented in Fig. 5. These results were obtained for the antisymmetrical deflection voltage mode as well as, for the negative deflection voltage mode. The deviation among the results is found to be negligible. The experimental results concerning the influence of the oblique incidence to the error of the energy measurement is also given in Table 2 as well as, in Fig. 5. The energy of the primary ion beam E was constant. However, the magnitude of US providing maximum signal, depends on the entrance angle ae . Thus, in this case, Dk ðae Þ=k ð0Þ ¼
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Fig. 5. The relative discrepancy of the analyzer constant versus entrance angle ae . Open marks correspond to the experimental results (the negative deflection voltage mode); solid squares are numerical simulation results obtained for the negative deflection voltage mode; the result obtained by the second order analytical approach (the antisymmetrical deflection voltage mode) is presented with the solid line.
DUS ðae Þ=US ð0Þ. The obtained data are also compared to the results of the second order analytical approach, for the case of the antisymmetrical deflection voltage mode [8]. Appropriate results are also presented in Fig. 5. As it was expected, there is no change of computed trajectories if the particle energy and the deflection voltage are multiplied by the same constant. This is the well-known characteristic of charged particles deflected in the electrostatic field. Two consequences of this fact are the linear dependences of the tuning energy Ea and the width of the transfer function DE on the deflection voltage. It would be useful in practice to describe optical properties of an analyzer using a function that does not depend on the deflection voltage i.e. on the tuning energy. This can be done by introducing the relative particle energy e: e ¼ E=Ea . Acceptance solid angle defined as a function of the target
Table 1 The analyzer constant determined by the numerical simulations, the second order analytical approach and the experiment Type of calculation
Numerical simulation
Analytical approach
Deflection voltage
Negative
Antisymmetrical
Positive
Antisymmetrical
Negative
k
3.522
4.022
4.521
4.011
3.409
The numerical simulation results are obtained for different types of the deflection voltage mode.
Experiment
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Table 2 Experimentally determined dependence of the analyzer constant k on the incident angle ae Dk ðae Þ=k ð0Þ (%) ae ()
E0 ¼ 1000 eV
E0 ¼ 1500 eV
E0 ¼ 2000 eV
E0 ¼ 2500 eV
)2.67 )1.83 )1.0 0.0 0.83 1.67 2.5
)0.2 )0.27 )0.03 0 0.2 0.47 0.74
)0.09 )0.22 )0.09 0 0.22 0.54 0.85
)0.47 )0.43 )0.20 0 0.23 0.44 0.64
)0.48 )0.51 )0.27 0 0.13 0.41 0.68
the energy widths is 5.7%. The maximum of Xðx; eÞy¼0 is about 5% greater in the case of the negative deflection voltage mode. Thus, applying the negative deflection voltage mode lowers the energy resolution and increases the transparency as compared to case of the antisymmetrical deflection voltage mode. The acceptance solid angle in case of for the negative deflection voltage mode versus the target plane coordinates for three magnitudes of e as a parameter is shown in Fig. 7.
5. Discussion plane coordinates and the relative particle energy Xðx; y; eÞ, entirely determines optical properties of an analyzer and does not depend on the tuning energy. The dependence of the acceptance solid angle on x and e for y ¼ 0 mm in the case of the negative deflection voltage mode is given in Fig. 6. It has a shape of a ridge, which is folded due to the spherical aberrations. Positions of the local maxima of the acceptance solid angle in the x–e plane correspond to the dependence of the relative error in the energy measurement on the entrance angle, given in Fig. 5. The shape of Xðx; eÞy¼0 does not depend qualitatively on the deflection voltage mode. It is generally wider with respect to the e coordinate, if the negative deflection voltage mode is applied. The average relative difference between
Fig. 6. The acceptance solid angle as a function of the target plane coordinate x and the relative particle energy e, in case of the negative deflecting voltage mode; y ¼ 0 mm.
The equipotential surfaces inside the analyzer, far from the ends of the electrodes, are cylinders. Particle trajectories obtained using the numerical simulations are close to the main path. This is valid even if the antisymmetrical deflection voltage mode is not applied. However, U ðrr Þ is changed with the change of the deflection voltage mode. This means that the particle energy is changed along the trajectory, if the deflection voltage mode is not antisymmetrical. It is clear that this effect has strong influence to the analyzer constant. Three different analyzer constants that are most interesting in practice are already introduced in Section 1: • Antisymmetrical deflection voltage mode U1 ¼ US =2, U2 ¼ US =2, k 0 ¼ E0 =US ; U 0 ðr0 Þ 0 V. • Positive deflection voltage mode U1 ¼ 0 V, U2 ¼ US , k þ ¼ Eþ =US ; U þ ðr0 Þ USþ =2. • Negative deflection voltage mode U1 ¼ US , U2 ¼ 0 V, k ¼ E =US ; U ðr0 Þ US =2. In case of the positive (negative) deflection voltage mode, particles are decelerated (accelerated) at the entrance into the analyzer. Their energy inside the analyzer is approximately E US =2 ðE þ US =2Þ. At the exit from the analyzer, particles are going to be accelerated (decelerated), and their final energy will be equal to the starting energy E. If the deflection voltage is equal for any of the applied deflection voltage modes (deflecting field is the same), the energy of the particle traversing along the main path will approximately be the same: E0 Eþ US =2 E þ US =2. After
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Fig. 7. The acceptance solid angle as a function of the target plane coordinates Xðx; yÞ in case of the negative deflection voltage mode; relative particle energy e is a parameter: (a) e ¼ 0:9920; (b) e ¼ 0:9993 and (c) e ¼ 1:0106.
dividing this expression by US the following relation is obtained: k 0 k þ 0:5 k þ 0:5. This rough estimation is remarkably good according to the results of the numerical simulation (cf. Table 1). The difference between the k magnitudes obtained experimentally and by use of the computer simulations are most probably due to the errors in manufacturing or mounting the system. In some cases, the optical properties of an analyzer can contribute to serious errors during the measurement of the energy spectra. The deviations of spectra can generally be manifested as errors in peak position, peak intensity, and peak shape. The errors are directly connected to the shape of the emitting surface i.e. to the primary beam profile on the target plane. Let us firstly discuss two extreme cases in order to clarify possible deviations of en-
ergy spectra of scattered ions due to different primary beam profiles. (a) The case of the narrow primary ion beam profile; if the maximum of the primary ion beam profile does not coincide with the center of the target (point ð0; 0Þ), the majority of scattered ions will enter the analyzer with ae 6¼ 0. The oblique incidence into the analyzer will contribute to the systematic error in the energy measurement due to the spherical aberrations. (b) The case of the wide primary ion beam profile; if the emitting surface is wider than the acceptance region, only a part of the primary ion current will contribute to the intensity of LEIS peak obtained in the spectrum. Thus, LEIS
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peak intensity may not be directly proportional to the total primary ion current, if the primary ion beam profile is not constant [6]. Wide primary beam profile i.e. large emitting area can also contribute to the significant peak broadening. A very good agreement between the experimental and the numerical results concerning the influence of the oblique incidence to the error in energy measurement (Fig. 5), indicates that performed numerical simulation approximates the real analyzer very well. It can also be seen that the results of the second order analytical approach give the same results as the numerical simulation. A peak shift due to very fine effects, such as inelastic energy losses in LEIS experiments, can be less than 1% in some cases [18]. The relative error of the energy measurement due to the oblique incidence of scattered ions can easily be of the same order (cf. Table 2) and even greater if the distance l is small. The ideal sector angle equals about 127.6 that is very close to the optimal magnitude needed for the first order focusing )127.3 [8]. Unfortunately, the relative error is increased to some extent because the positions of the entrance and the exit slits do not coincide with the ideal field boundaries. The dependence of the acceptance solid angle on the x coordinate and the relative particle energy e (Fig. 6) is qualitatively similar to the equivalent results given in [11] for the same type of the analyzer. The difference between the Xðx; eÞy¼0 for different deflection voltage modes is a consequence of the particle acceleration on the entrance into the analyzer in the case of the negative deflection voltage mode – the circular component of the particle velocity is increased, which contributes to folding the trajectory and decreasing the absolute magnitude of the angle between the trajectory and the optical axis. The particle acceleration increases the maximum of Xðx; eÞy¼0 function due to the trajectory folding in the non-dispersive plane, while the broadening of Xðx; eÞy¼0 along the energy axis is not significant. The trajectory folding in the dispersive plane contributes to the broadening of Xðx; eÞy¼0 along the energy axis. The maximum of Xðx; eÞy¼0 cannot be further increased by the tra-
jectory folding in the dispersive plane – it is limited by the entrance slit width. The dependence of the acceptance solid angle on the target plane coordinates and the relative particle energy Xðx; y; eÞ (Fig. 7) is the one that has major practical significance. As it was mentioned already, this function includes all the optical properties of the analyzer used for measuring the energy spectra of charged particles emitted from the target plane. There is a strong dependence of the acceptance solid angle on the relative particle energy. This complicates determining the acceptance region i.e. estimating the influence of the primary ion beam profile on the peak intensity. However, the knowledge of Xðx; y; eÞ allows us to compute the influence of the beam profile on the position, the intensity and the shape of the peaks obtained in the LEIS spectra. A possibility to precisely calculate the energy distribution of scattered ions from the experimentally obtained energy spectra, if primary ion beam profile is measured, should also be considered. All these effects are important in every single experiment in which an ESA analyzer is used. However, additional effects are present in LEIS experiments due to the dependence of the scattering angle on the target plane coordinates h ¼ hðx; yÞ. As the energy of scattered ions in LEIS experiment directly depends on the scattering angle h according to the elastical binary collision model [1], the deviation of h will contribute to the deviation of the energy of scattered ions i.e. to the LEIS peak position and/or broadening. According to the mentioned model, a relative deviation of the energy of scattered ions DE=E due to the deviation of the scattering angle Dh is defined by the expression DE 2 sin h Dh ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð1Þ E A2 sin2 h where A represents target mass to projectile mass ratio. The dependence of DE=E on h for several magnitudes of A as a parameter is given in Fig. 8. As in case of the oblique incidence into the analyzer, this effect can contribute to the peak position error and/or to the peak broadening. We find that the magnitude of Dh ¼ 1:5 is realistic for the
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Fig. 8. The relative deviation of the energy of scattered ions DE=E versus the scattering angle h. The error of the energy measurement is due to the deviation of the scattering angle Dh ¼ 1:5; target mass to projectile mass ratio A is a parameter.
typical LEIS systems, since it also includes the angular spread of the primary ion beam. It can be seen that in the case of small target mass to projectile mass ratio this effect contributes to greater deviations of the energy spectra than an analyzer with reasonable optical properties. Generally speaking, systematic errors concerning the peak positions and peak intensities rapidly increase with the decrease of the distance between the target and the entrance slit of the analyzer, l (cf. Fig. 2). The decrease of l contributes to the increase of the acceptance solid angle. Nevertheless, the area seen by the detector is decreased, the spherical aberrations are more pronounced, and the shift of the beam spot from the center of the target contributes to greater error in the energy measurement. The distance l is greater in our case than in the standard devices of similar type. This lowers the importance of the mentioned effects in our measurements as compared to other devices. Another general advantage of our system is that a small shift of the beam spot in the non-dispersive plane (along the y-axis) will not contribute to the significant error concerning the transparency and the energy measurement, because the magnitude of
the acceptance solid angle is almost constant in a wide range of y (cf. Fig. 7). This is in contrast to the analyzers having focusing action in both planes. Thus, in spite of using the analyzer with modest optical properties, our system is quite convenient for performing high quality experiments.
6. Conclusion In this work, we present a numerical and experimental study of the 127 energy analyzer used in the LEIS experiments. The main results are the following: (a) The dependence of the analyzer acceptance solid angle on the target plane coordinates and the relative particle energy X ¼ Xðx; y; eÞ is identified as a function that completely describes the realistic optical characteristics of an energy analyzer. This function is numerically computed in detail for the 127 cylindrical energy analyzer used in LEIS experiments by means of SIMION program. (b) The oblique incidence of ions into the energy analyzer contributes to the error of the energy
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measurement. The dependence of this error on the incident angle is experimentally measured. The same error is calculated from the numerically determined function Xðx; y; eÞ, as well as by means of the second order analytical approach. All these results agree very well. (c) The influence of the applied deflection voltage mode on the optical properties of the analyzer is computed, too. The major influence is on the analyzer constant, which can be explained very well by considering the particle acceleration along the optical axis. A simple relation between the analyzer constants for different deflection voltage modes is introduced: k 0 ¼ k þ 0:5 ¼ k þ 0:5. This relation is valid for the magnitudes of numerically obtained analyzer constants. Other characteristics of the analyzer are not significantly changed using different deflection voltage modes. The energy resolution is decreased and the transparency increased when the negative deflection voltage mode is applied instead of the antisymmetrical deflection voltage mode. (d) An additional peak broadening and eventual peak position shift is present in LEIS experiments due to the scattering angle dependence on the target plane coordinates. In case of low target mass to projectile mass ratio, this effect can have greater influence on the energy spectra than the above-mentioned effects. This phenomenon is specific for LEIS as well as, for other scattering experiments. The primary beam profile and the relative position of the target and the analyzer are very important in surface characterization. This problem is especially present in the LEIS technique: beam control and focusing is more difficult in the case of low energy ion beams as compared to electron beams and medium or high energy ion beams. A systematic error in the energy measurement will be made if the primary beam is not focused on the center of the target and/or if the optical axis of the analyzer does not intersect the center of the target. Thus, in case of fine measurements, such as determination of inelastic energy losses in LEIS experiments, extremely good determination of the experimental set-up geometry and of the primary beam profile is obligatory. The problem is also present in the case of the quantitative analysis:
primary ion beam profile will not contribute to the systematic errors as long as the beam spot is smaller than the acceptance region [6,9]. However, it is not simple to define this area – it can strongly depend on the relative particle energy (cf. Fig. 7). The most reliable way to quantitatively determine the influence of the analyzer on the energy spectra is to compute the spectra obtained by the analyzer for the defined primary beam profile and the energy distribution of particles emitted from the surface, using the knowledge of Xðx; y; eÞ. Unfortunately, modeling the energy distribution of emitted particles is generally a problem. The problem is increased in the case of LEIS, because distribution (i.e. scattering angle) depends on the target plane coordinates. Nevertheless, this type of computations can provide important information and further improve the analysis of the experimental results. The investigation of these problems is in progress.
Acknowledgements This work has been supported by the Project 2018 from the Ministry of Development, Science and Technology, Republic of Serbia.
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