- Email: [email protected]
Improving the performance of the cylindrical mirror analyzer II: Reducing the dependence of energy resolution on sample position
Journal of Electron Spectroscopy and Related Phenomena 212 (2016) 62–73
Contents lists available at ScienceDirect
Journal of Electron Spectroscopy and Related Phenomena journal homepage: www.elsevier.com/locate/elspec
Improving the performance of the cylindrical mirror analyzer II: Reducing the dependence of energy resolution on sample position David Edwards Jr. IJL Research Center, Newark, VT 05871, United States
a r t i c l e
i n f o
Article history: Received 4 May 2016 Received in revised form 6 September 2016 Accepted 7 September 2016 Available online 9 September 2016 Keywords: Cylindrical mirror analyzer CMA Electron spectrometer High resolution
a b s t r a c t Electrode segmentation of the cylindrical mirror analyzer has been shown to significantly improve the resolution of the standard instrument. The high precision solution which had been found for the 42.3 device is shown in this report to exist in fact for an extended set of input angles or equivalently for a range of sample positions. This has allowed a significant limitation of the CMA, namely the critical dependence of instrument performance on sample position, to be largely overcome. In addition by placing the sample at the minimum in the curve of resolution vs sample position, a base width resolution of ∼.00005 has been obtained for a ±6◦ bundle an improvement of a factor of ∼40 over the standard spectrometer. Thus electrode segmentation has been found not only to significantly improve the device performance, but in fact to enable its high precision to be realized. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Electrostatic electron spectrometers may be divided into two classes: those with a real entrance aperture as in the hemispherical deflector or 127 ◦ analyzer and those without one as in the CMA [1], SEA type devices [2], the 90 ◦ sector analyzer, etc. (An aperture should be thought of in the above context as a real hole in a metal plate limiting the starting positions of rays which subsequently travel through the instrument.) In the HDA [3] there is such an aperture the location of which is of course known with the dimensional precision of the other mechanical parts of the spectrometer. This is not the situation with the latter cases (CMA, SEA, CDA . . .) in which there is no aperture, the starting positions of the trajectories being typically formed by the intersection of a particle or x-ray beam with a sample which may be only indirectly connected to the spectrometer, and thusly may not be at its designed position. This location problem for the CMA has been reported by both Khursheed [1] and Prutton & Giomati [2] as resulting in an unacceptable shift in the energy scale of the spectrometer and hence necessitates when possible a very careful external positioning of the sample to its design position. It was reported in [3] that segmentation significantly improved the resolution of the standard 42.3◦ CMA. It thusly appeared worthwhile to attempt to find high precision solutions for angles differing from 42.3◦ . It was somewhat surprisingly discovered that the
E-mail address: [email protected] http://dx.doi.org/10.1016/j.elspec.2016.09.002 0368-2048/© 2016 Elsevier B.V. All rights reserved.
enhanced resolution of the 42.3◦ device could be obtained for a range of design angles which of course implied that the enhanced resolution would be obtainable over a range of sample positions. It was thus apparent that if a table were constructed indexed by the sample position, each entry in the table, consisting of the segment voltages and the pass energy which resulted in a particular (high) resolution at each indexed point, one could use such a table to let the instrument itself determine the location of the sample In this manner the resolution associated with that location would be achieved without sample motion. The description and justification of these ideas are the subject of this paper. 2. Notation and terminology Sample position: the position of the point on the z axis from which particles emanate. It will be referred to as z. The index table: it is a table each entry consists of the following information: {|z0 |, v1. . .vn, Epass} where z0 is the sample position, v1 . . . vn the segment voltages, and Epass the kinetic energy of the input bundle which when the segment voltages are applied to the electrodes will produce the smallest spot size, resolution. Relaxation process in FDM: This refers to a process in which the potentials within an electrostatic geometry are determined. It starts with covering the geometry with a grid of points, considered a mesh, an initialization to some value of all points within the mesh and continues by stepping thru the mesh at each meshpoint replacing the current value with one determined using an algorithm itself using a selection
D. Edwards Jr. / Journal of Electron Spectroscopy and Related Phenomena 212 (2016) 62–73
63
Fig. 1. The various segment names are defined in this figure as well the assumed position of the sample at −z0 , the starting position of the standard CMA (z = −113), the median rays of the sample at z0 and −113. Also shown is a position sensitive detector (PSD) at location z = 124. (It is noted that the criterion for the exit plane was that the starting points of interest would intersect the exit plane above the axis. the plane at z = 124 was such a plane and no further adjustment of this parameter was made.).
of potentials at surrounding mesh points. When no further change is found upon subsequent iterations the mesh is said to be “relaxed” Mesh density: This term will be used here as representing the total number of meshpoints covering the geometry. Thus “higher density” will refer to a covering containing more mesh points than its comparison mesh. The units of energy will be considered to be in electron volts while that of potential will be in volts. The length unit of the geometry will be in dimensionless enabling the user to select any unit of his choice (i.e. mm, cm, inch . . .).
3. The geometry The basic geometry will be closely represented by the single segment solution of [3] and is illustrated Fig. 1. The sections of the inner pre and post segments are reasonably fine meshes allowing the beam to pass and the bundle definition itself is defined by real apertures in the inner segments determined by the bundle itself. Further clarification can be found in the “notes of caution” section. The inner and outer cylinder radii will be taken to be Rin = 37, Rout = 80 except as noted in a later section in which the geometry will be scaled by a factor of 2. As the sample position will be considered a non-constant parameter, it is necessary to define an angle 0 as the angle of the median ray traversing the path (-z0 , s1) where s1 is defined as the intercept with the inner electrode of the 42.3◦
ray having a starting position of −113. It should be clear from Fig. 1 that 0 is easily determined to be: 0 = arc tan (R in /(|z0 |-|s1 |)) which is a function of z0 alone for fixed Rin , s1 . The rays are assumed to emanate from a point and make angles of 0 +-n degrees with respect to the z axis. This set of rays will be denoted as a bundle and n = 6◦ will be the one considered here (denoted as a 6 ◦ bundle). The dimensions of the geometry of Fig. 1 are defined in Table 1. Although the geometry is quite similar to the single segment solution reported in [3] it is not identical, as the position of in1 start has been changed from −54.47[3] to −50. This change was necessitated in order to have the 6 ◦ bundle for 0 ∼33 ◦ sufficiently far from the singularity present at the contact point of inner presegment and in1 so that the ray would not be influenced by the lack of voltage precision quite close to the singularity. Recognizing the fact that the optimal solutions for separate z’s would exit the device at a position dependent on z itself, a posiTable 1 The mechanical dimensions of the 1 segment CMA are given. Rin
37
Rout
80
left end plate in1 start out1 start s1
−187 −50 −70 −72.333
right end plate in1 end out1 end
187 54.47 70
64
D. Edwards Jr. / Journal of Electron Spectroscopy and Related Phenomena 212 (2016) 62–73
Fig. 2. Optimized solutions for 6◦ bundles are plotted emanating from 4 distinct sample positions which will be later seen to encompass the range of sample positions which yield high resolutions. Seen also are the intercept locations, each point representing an annular ring about the z axis. The numbers to the left of the focal points in the PSD refer to the starting positions of the trajectories.
tion sensitive detector (PSD) was necessary and is essential to the following. Two orthogonal orientations of the device were considered, one being along the z axis, the other being, as shown in Fig. 1, in a plane normal to the z axis. The later was selected as not in itself limiting the source azimuthal angles allowed by the cylindrical symmetry of the geometry. For an on axis PSD the azimuthal angles would by construction be limited to 180◦ . The parallel − to − the − axis solution continues to be a potential option having the advantage that all azimuthal angles exit at a point rather than a ring as for the vertical PSD but with the disadvantage of not allowing the azimuthal variation of the source signal to be measured. The geometry together with 6◦ bundles from 4 distinct sample positions are shown in Fig. 2. This figure was constructed to give a perspective consisting of both the focal points on the exit plane and the range of sample positions from which high resolutions may be obtained. Thus one can see that the tolerance of positioning the sample for the high resolution instrument becomes ∼29% of Rin . It should be noted that in the past CMA’s have typically assumed the rays would leave at a definite sample position and focus on a ring of certain radius on the exit plane. For such a ring a real annular aperture or annular slit would be constructed in the exit plane with a detector placed behind. However seen in Fig. 2 the radius or position of the annular aperture is dependent on the sample position and so real apertures could not be considered. A solution was to consider using a PSD for the detector in which essentially virtual apertures could (in software) be constructed. Thus for a given
focal radius, only the current within the virtual aperture is to be measured. 4. The modelling 4.1. Potential and trajectory calculations The goal of the modelling is to accurately calculate the trajectories for rays traversing the geometry. A starting point will be to apply a set of voltages to the segment electrodes. Although there are 8 segments depicted in Fig. 1, the voltages on the endplates are fixed at 10 V while the inner pre and post segment electrodes are set at 0 V. Thus while the modelling involves 8 separate voltages, only 4 are variable and need be involved in the search for an optimal solution. As the trajectories depend on the electric field at all points along its path, the electric field is calculated by means of the potential within the geometry determined by means of FDM [4,5]. In this manner the potential on a uniform mesh placed over the geometry can be found, the details discussed in [3]. Here it is simply remarked that an order 10 algorithm [4,5] was used in all potential calculations. It is further noted that the effects of the singular points (the electrode interconnection points) were effectively neutralized by the construction of 10 telescoping regions convergent on the points themselves, each telescoping region having a height, width of 20 units. Again details may be found in [3] as well as [5]. Given the potential distribution within the geometry trajectories were found using a recently developed ray tracing process [6].
D. Edwards Jr. / Journal of Electron Spectroscopy and Related Phenomena 212 (2016) 62–73
65
Fig. 3. The trajectories are plotted of rays within 3 bundles having energies symmetric about E0. The resolution is clearly less than 0.05%.
One of the differences of this method from the more standard Runge Kutta is the order of the electric field determination, called here order s, is an input parameter independent of the order of the time determination (order t), another input parameter. While in all of the calculations order t was taken to be 10, the highest order currently available, for reasons of calculation time concerns a value of 4 was used for order s. It will be later shown that no significant degradation in the precision of the trajectory in general resulted from using order s of 4 rather than 10. 4.2. The resolution It is assumed that all rays within a bundle start their trajectory from a point z0 on the axis having an initial kinetic energy E0 and reach the exit plane (z = 124) at position ri . A plane constructed at any point on the median trajectory intercepts all of the rays of the bundle and the extent of these intercepts is considered the trace width of the bundle on this plane. When this plane coincides with the exit plane the trace width on this plane will be denoted by spot size or s. The resolution is found by creating three bundles having energies E0 , E0 (1 + x), E0 (1 − x). The xmin is defined as the minimal value of x such that the intercepts of 3 bundles on the exit plane are disjoint. The resolution is then: resolution = xmin /E0 . As an example consider the plots in Fig. 3 in which the segment voltages specific to z0 = −124 are applied to the geometry. Seen in this figure is that the sets of intercepts are clearly disjoint implying that the resolution is «0.05. The resolution determined by this method will be denoted by the base width resolution.
Another method will be used to determine the resolution based on the minimum trace width on the exit plane and is found from Estimated resolution = s (E0 )/(∂Gri (E0 )/∂GE0 ) (see also Hafner et al. [7]) : where ri (E0 ) is the r intercept on the exit plane of the median ray within a bundle having energy E0 . (This expression tacitly assumes that s (E0 ) is independent of E0. ) That the assumption of the above derivation (the independence of s on E0 ) is violated may be seen in Fig. 3 by observing s (E) is in fact the smallest for E = E0 being larger on either side of E0 . However the estimated resolution will be shown to be quite useful after a calibration constant is obtained relating the base width resolution to the estimated resolution. The calibration will be described in a later section. 4.3. The optimization process Given a general starting point (and corresponding angle), segment voltages must be found that minimize the resolution. A method for accomplishing this has been described in [3] for a situation in which a somewhat arbitrary selection of segment voltages is assumed as the starting condition. However in this work the optimized solution as a function of z0 starts from the known solution for 42.3◦ (z0 = −113) and then proceeds to z0 ± 1/2 etc. thus extending thru the useful range of z0 (-129 to −106). Thus the starting voltage set for any z0 is likely somewhat close to the solution at that z0 and a full search as presented in [3] is not necessary.
66
D. Edwards Jr. / Journal of Electron Spectroscopy and Related Phenomena 212 (2016) 62–73
Fig. 4. The base width resolution is plotted vs the estimated resolution.
Rather it was found that by slightly modifying large step − small step algorithm of [3] the search for the optimum solution would be considerably shortened. A brief summary of the idea behind the modification is as follows: given a starting point, two segment voltages from the 4 possible ones are selected (6 distinct ways) and increment/decremented by a small value (say 0.1 V). Thus each combination selected results in 4 distinct values for the pair (++ +−+ −). After this voltage selection has been made a scan through the segments is performed incrementing/decrementing by a much smaller value ∼.01 V. At each point in the second scan the minimum estimated resolution is found using E0 alone [3]. By means of this method a significant reduction in time was typically achieved for finding an optimized solution from a previous one. 5. The results 5.1. The resolution Starting from the known high resolution point of [3] (z0 = −113, theta = 42.3◦ ), z0 was extended to the surrounding points and at each point the optimized segment voltages and resolution were obtained. This process continued throughout the range −129–106 in steps of 0.5. Having obtained the optimized segment voltages for each z0 , the base width resolution was found using the method described above. A plot was then obtained of the base width resolution vs the estimated resolution and is shown in Fig. 4. In this figure it is seen that a linear approximation provides a reasonable fit to the data and allows a definition of the corrected resolution to be inferred from the estimated resolution by:
Table 2 The optimized voltage parameters for the highest resolution point of Fig. 5. inner pre segment
0V
outer pre segment
21.652V
inner v1 inner post segment z0 minimum spot size Epass
−0.083V 0V −124 0.004425 20.7399 eV
outer v1 outer post segment ri on plane z = 124 base width resolution dz(E)/dE
7.889V 13.608V 17.351 0.000048 132.82
corrected resolution ∼ = 1.351 * estimated resolution providing a correction to the resolution inferred from the estimated resolution. A comparison of the base width resolution and estimated resolution over the range of z0 , −129 to −106 is shown in Fig. 5. In this figure we see that the corrected resolution clearly provides a reasonable estimate to the base width resolution from the estimated resolution which is considerably easier to calculate. In all of the following resolution will be taken to be the corrected resolution inferred from the estimated resolution. The segment voltages corresponding to each −z0 in Fig. 5 not reported. However the data for the minimum result of Fig. 5 is given below in Table 2 enabling a comparison by others with the results of the present calculations. The complete table would be available upon request. It is noted that the data point z0 = −113 corresponding to a resolution of ∼.034% and a theta of 42.3 ◦ differs from that reported in [3] of 0.017% due to having a vertical detector plane rather than one on axis as in the previous study.
D. Edwards Jr. / Journal of Electron Spectroscopy and Related Phenomena 212 (2016) 62–73
67
Fig. 5. A comparison of the base width resolution and estimated resolution. The difference in the two curves likely results from the scatter seen in Fig. 4. The relation is not exact but only approximate.
5.2. Interpolation to points not within the index set The index set is considered to consist of optimized segment voltages, pass energies for all integer and half integer points between −129 and −106 and are indexed by |z0 |. In any physically realized situation, the sample position may be close to but not on one of the z0 values in the index set. The question is to find the optimized resolution for such points. The solution is straightforward, namely given a sample position z, scan through the index set and find the index point z’ that gives the highest resolution for z. Then one uses the segment values of z’ for those of z and further minimizes the resolution with respect to E0 . The result of this optimization gives the segment voltages (those of z’) and the pass energy (the value of E0 producing the minimum resolution) for z’. This procedure has been used in a scan of z between −129 and −106 with a step size of 0.1 and the results given in Fig. 6. In Fig. 6 the results of the interpolations are plotted along with the resolutions for points lying exactly on indices of the index set. Seen that there is measureable but reasonably small deterioration of the resolution for points in a neighborhood of but not on an index point. Also seen is that the deviation reaches a maximum for a points ∼½ way between points of the index set. While the interpolation deviation evidenced in Fig. 6 is in fact small, improvements could be made by a further refinement of the index set. A similar study was performed in which the sample position was placed at a random z in the interval −129,106. The procedure
sketched above was followed and the resultant resolution obtained at each random point. The results are given in Fig. 7. Seen again is that reasonable interpolations are made for the random locations of the sample. 5.3. Interpolation within a physical spectrometer In the previous section it was assumed that E0 (the kinetic energy of the particles leaving the sample) was both known and able to be varied. In contrast it will be assumed in this section that E0 is neither known nor variable; hence an equivalent process will be described that uses scaling to optimize the resolution. First recall that at any index zindex in the index table, an element at that index consists of a collection of segment voltages and the associated pass energy, Epass , the combination of the two yielding an optimized resolution for a bundle leaving the sample at −zindex having an energy E0 = Epass . This element is in fact a family of segment voltages, pass energies, each member of the family being related to the original by scaling each segment voltage and pass energy by a given parameter. Thus for example if a member of the index table is represented by {v1 , v2 . . . vn , Epass } a particle of kinetic energy Epass leaving the sample with appropriate voltages applied to the segments would be focused into the small optimized spot on the exit plane as would particles of kinetic energy Epass *f provided that the segment voltages were also scaled by f. In more succinct terminology the family of solutions for any index point zindex may be
68
D. Edwards Jr. / Journal of Electron Spectroscopy and Related Phenomena 212 (2016) 62–73
Fig. 6. The sample position is scanned in steps of 0.1 over the range −129, −106.
represented by {v1 *f, v2 *f . . . vn *f, Epass *f}.
5.4. Simulating the physical spectrometer To understand the operation of the simulated spectrometer, one must understand the separation that exists in the software used to model the spectrometer. In the software there are two separate modules: the first takes the desired segment voltages, applies them to the electrodes and finds the potential distribution within the geometry; the second uses this distribution and traces through the device rays emanating from a given sample position − z having initial kinetic energy at the sample of Epeak . In this simulation it is seen that −z is the actual location of the sample and Epeak is the kinetic energy of the rays emanating from the source. These two quantities are both fixed and unknown, and are to be determined along with the segment voltages that result in the smallest spot size (minimum resolution) for the bundle. The method is similar to the previous one: a scan through the index set is performed, and at each index point the segment voltages and associated pass energy (not E0 ) are scaled by a factor f, and the resolution (peak width) experimentally determined. The process is continued until an f results in the smallest resolution (peak width) possible. At the end of the above the index point with the smallest resolution has been located its pass energy determined which is the pass energy resulting from the scaling. It is also noted that the locus of peak intensity points in fact determines the annular aperture appropriate to the actual sample position. See a further discussion in the “notes of caution” section.
As an example consider the sample at position −119.72 and the kinetic energy of the rays is 11.3728. This point was chosen to be ∼1/2 way between index points −120 and 119.5. Results of a scan thru the index set are shown in Fig. 8. At each point in the index set a segment voltage scaling parameter f was found which minimized the resolution. It is noted that the pass energy of the index point is also scaled and recorded but not transmitted to the ray trace module. The scan in the above figure is seen to proceed from −125 to −111 in steps of 1/2 unit. A minimum can be seen in the curve at z0 = −120, the value of the minimum being within interpolation variations of Fig. 6 which is also plotted in Fig. 8. Thus for the minimal point of Fig. 8 the resolution, the determined segment voltages, the pass energy and ri are obtained. It is clear from this figure that were the density of index points increased a new minimum would be found likely between the two smallest points of Fig. 8. It thus appeared worthwhile to attempt to find this minimum from the coarse set by fitting the points near the minimum with a parabola and then finding the minimum of this curve. In this manner both estimates of the minimum resolution and actual sample position may be obtained. Recalling that the process used to obtain Fig. 8 also yields a plot of the resultant pass energy vs zindex (not shown) the pass energy at the calculated minimum can be found by interpolation using the newly found estimated sample position. To evaluate the possible utility of the above suggestion the error in the sample position and peak energy was determined for sample positions between −120.5 and −119.5 with a fixed peak energy of 11.3728. To obtain a higher precision than would be possible with the current index set (interval 0.5), the index set was expanded in
D. Edwards Jr. / Journal of Electron Spectroscopy and Related Phenomena 212 (2016) 62–73
69
Fig. 7. The optimized resolution obtained at random locations of the sample.
this restricted range using an interval spacing of 0.125. For each sample position taken (step 0.1) within this range the process described above was effected and yielded a plot of the errors in peak energy and sample position which is shown in Fig. 9. Plotted in this figure are the% errors in the peak energy and sample position. Seen is that the sample position error is close to but strictly less than the peak energy error. The average% peak energy error which has been found is ∼ 0.028 implying that a 1000 eV peak would have an error of ∼.3 eV. As all energies are referenced to the vacuum level of the sample this energy error is of the order of work function differences between the sample and spectrometer and hence no further subdivision of the index set to achieve higher precisions would appear propitious. 5.5. The effects on using order 2 on simulations However rereading the relevant section I see that improvements should be made. I have thus rewritten it as: As mentioned previously an order 10 algorithm was used for the determination of the potential distribution within the geometry and an order 4 algorithm for the electric field interpolation during the ray trace. (See above section “Potential and trajectory calculations”) In order to see the effects of degrading the algorithm precision two cases were considered: in the first the potential distribution was found using an order 2 (rather than 10) algorithm with the order s = 4 algorithm being used in the ray trace program. Using order s of 4 algorithm in ray trace was found equivalent to using an order s of 10). The second case was the converse, namely an order 10 algorithm was used for potential calculations while an
order s of 2 algorithm for the field interpolations during ray trace. It is noted that the time order (order t) in the ray trace calculations was fixed at 10. The effects of this study are given in Fig. 10. In this figure our high precision result (Fig. 5) is given together with the situations in which the order 2 algorithm was either used in the potential determination or in the field interpolations. Seen is that the effects are quite similar implying that even if quite accurate potentials were determined the resolution precision would be compromised by the use of a low order algorithm in the ray trace. It is also observed that the effects of the low order algorithm are quantitatively similar regardless if incorporated either in the potential or field determinations. Also displayed in Fig. 10 is the result of using an order 2 algorithm together with a higher density mesh (geometry scale factor 2, density enhancement 4). Seen is that this graph approaches and is reasonably close to the lower convergent curve (order 10 in mesh relaxation). The implication is that high precision algorithms are not necessarily required for the determination of the results presented in this report; low order algorithms would clearly suffice particularly if used in conjunction with higher density meshes.
5.6. Notes of caution Many concerns in relating the above model to a real spectrometer are the same as for the spectrometer described in [3] and as they have been covered in some detail in [3] are not repeated. Two additional comments are necessitated by the solution presented above. They are:
70
D. Edwards Jr. / Journal of Electron Spectroscopy and Related Phenomena 212 (2016) 62–73
Fig. 8. Fixing the sample at −119.72 and the kinetic energy of the rays at 11.3728, the resolution is obtained for each zindex in the index set. Also plotted are the data from Fig. 6 showing that the minimum found using segment voltage scaling is within the variations seen in Fig. 6.
1. In our modelling the 6◦ bundle size was limited by a restriction on the rays emanating from the point source and not, as must be accomplished in an actual CMA, by a restriction of the entrance aperture. It has been determined that if an entrance aperture (in the inner cylinder)of the CMA is established by the limits of the 6◦ bundle for z0 = −124 (our maximum resolution point) that considering only z0 ’s having a resolution less than 0.0002 (see Fig. 5) i.e. −127 < z0 < −116 that the bundle size would vary between 5.5 ◦ and 7.4◦ . It is noted that the index table needs to take into account the actual bundle size permitted by this physical aperture when being formed. It is not considered that this would have a significant impact on the resolutions of the selected z0 ’s other than resulting in a slightly reduced transmission for z0 <-124 and a somewhat increased transmission for z0 >-124. 2. A position sensitive detector (PSD) has been used as the output rather than the more traditional aperture of the standard CMA. The process described above for solving the sample position problem would not permit using a real aperture in the exit plane. Hence this choice is not an option. However, considering the dimensions of Table 1 to be in mm, it is seen from Table 2 that the pixel size of such a PSD must be of the order the spot size ∼5 m This is either at or somewhat beyond the limit of current technology (see for example a recent article by Boronat et al. [8] describing progress in the 5 m range and below). It is noted that if one incorporates a PSD with resolution of 20 m, the spectrometer resolution at its highest resolution point (z0 = −124) would be ∼.0002 remaining a significant result and one which would
be improved to its design value by continued advances in this rapidly changing technology. 3. The intensity of the emitted beam leaving the sample with energy E0 is considered to be made by summing only the sub currents in the annular ring of the PSD centered on the radius of the focal point of this beam on the exit plane and of width given by its predicted spot size. This can be done in software coupled with the properties of the PSD itself. Further that the energy spectrum is measured by stepping the pass energy in discrete steps measuring the intensity at each step. 4. In the discussion determining the radius ri of the annular slot (assumed circular) in the detector plane (see Section below Fig. 8) it had been noted ri may be found by interpolation in the same manner as the energy E0 of the peak itself. However in the process involved in experimentally determining the sample position, E0, the actual annular ring (including eccentricities) has been in fact measured by the 2 dimensional plot of the locus of the peak intensity points on the PSD itself and hence an interpolation for ri is neither necessary nor desired. In fact it is clear that the above experimental determination will give actual annular slot including eccentricity corrections due to any slight cylindrical misalignments. Not having this ability (i.e. assuming a circular annular ring) would likely severely limit the performance of any experimentally constructed devices. Thus the use of a PSD has both allowed virtual apertures (not necessarily circular) to be constructed and enabled an essential modification of the annular aperture to an eccentric aperture, the eccentricity
D. Edwards Jr. / Journal of Electron Spectroscopy and Related Phenomena 212 (2016) 62–73
71
Fig. 9. The errors in determining the peak energy and peak position are plotted vs sample position. Also included is the average peak energy error.
depending upon the degree with which the cylinders themselves are aligned. 6. Verification In verifying the results the resolution vs z0 will be determined in several ways. The first is to simply increase the order s of the ray trace procedure from 4 to 8 using the standard mesh (Rin = 37, Rout = 80), while the second is to increase the density of the mesh by a factor of 4 by scaling which will produce a mesh of Rin = 74, Rout = 160. The results of these studies are given in Fig. 11. Seen is that the order s = 4 and 8 are essentially indistinguishable from the results from the scaled geometry. The most significant difference occurring in the lower portion of the curve near z0 ∼ −128.5. It is suspected that this discrepancy is caused by the rays nearing the singular point at the juncture of the inner presegment and the inner segment. The agreement suggests that the results are valid over the range of sample positions indicated. It is noted in all of these studies the potential distribution was determined using an order 10 algorithm. 7. Conclusion Several new developments of the cylindrical mirror analyzer have been found and presented. It has been shown that the high resolution afforded to the standard CMA (42.3◦ ) by segmentation can be extended to a continuous range of input angles and sample positions. This has allowed one of the significant problems of the CMA
to be improved if not overcome, namely the dependence of performance on sample position and has resulted in a high resolution to be effected over a range of sample positions. In addition provided that the sample can be precisely positioned (in the Rin = 37, Rout = 80 geometry) at z0 = −124 a resolution of ∼.0048% or a resolving power (1/resolution) of >10000 can be obtained. This represents a factor of ∼40 improvement over the standard geometry. Obtaining high resolution over a range of sample positions requires that in the instrument setup prior to operation segment voltages appropriate to the current sample position must be found (using a procedure described above) and when these voltages are applied a high resolution should be realized. From Fig. 7 for example a resolution of 0.025% or a resolving power of ∼4000 can be obtained for sample placements between −127 and −116 or within a span of 11 units (29% of Rin) the instrument could achieve its design performance. The procedure for determining the appropriate segment voltages for a given sample position results also in a side benefit, namely establishing of the energy of the peak used in the above calibration. It was found in the example studied that the absolute energy of the peak can be found with a precision of ∼.03%. Thus the benefits of spectrometer electrode segmentation clearly surpass those suggested in [3] in allowing high resolutions to be achieved not only for a given sample position but over a considerable range of positions. This process should be applicable to spectrometers not having a real aperture at its entrance as discussed in the introduction and likely represents a new paradigm for the design of high performance instruments.
72 D. Edwards Jr. / Journal of Electron Spectroscopy and Related Phenomena 212 (2016) 62–73
Fig. 10. The resolution is plotted vs z0 for various orders and 2 mesh densities. The labels “mesh Rin ,Rout ” reports the inner and outer cylinder radius of the CMA for that curve. (See text for further discussion.).
D. Edwards Jr. / Journal of Electron Spectroscopy and Related Phenomena 212 (2016) 62–73
73
Fig. 11. The scan of resolution vs sample position for several operational parameters.
Acknowledgments It is a pleasure to acknowledge discussions with Anjam Khursheed for making the author aware of this problem. Without this input this work would not exist. In addition appreciation is given to Omer Sise and Anjam Khursheed for their critical reading of the manuscript. Newark, Vermont. April 24, 2016.
[6] David Edwards Jr, The segmented cylindrical mirror analyzer (CMA), J. Electron Spectrosc. Relat. Phenom. 209 (May) (2016) 46–52. [7] David Edwards Jr, FDM for curved geometries in electrostatics II: the minimal algorithm, IMECS 2014 March 12–14, 2014, Hong Kong, in: Proceedings of the International Multi Conference of Engineers and Computer Scientists, Vol I, 2014. [8] David Edwards Jr, Finite difference method for boundary value problems. application: high precision electrostatics, IMECS 2014, March 18–20, 2015, Hong Kong, in: Proceedings of the International Multi Conference of Engineers and Computer Scientists, Vol I, 2014.
Further reading References [1] P.W. Palmberg, G.K. Bohn, J.C. Tracy, Appl. Phys. Lett. 15 (1969) 254 (number 8). [2] D. Cubric, N. Kholine, I. Konishi, Electron optics of spheroid charged particle energy analyzers, in: Proceedings of the Eighth International Conference on Charged Particle Optics, Singapore, July 12–16, 2010, p. 234. [3] M. Purcell Edward, The focusing of charged particles by a spherical condenser, Phys. Rev. 54 (1938) 818. [4] Anjam Khursheed, Scanning Electron Microscope Optics and Spectrometers, World Scientific Publishing Co Ptc. Ltd., 2001, pp. 99 (ISBN −13 978–981-283-667-0). [5] Martin Prutton, M. Mohamed El Gomati (Eds.), Scanning Auger Electron Microscopy, Wiley, 2006, pp. 0–978, ISBN: 978-0-470-86677-1. (See section on CMA).
[9] David Edwards Jr, High precision ray tracing in cylindrically symmetric electrostatics, J. Electron Spectrosc. Relat. Phenom. 205 (2015) 111–121. [10] H. Hafner, J. Arol Simson, C.E. Kuyatt, Rev. Sci. Instr. 39 (1968) 33. [11] M. Boronat, C. Marinas, A. Frey, I. Garcia, B. Schwenker, M. Vos, F. Wilk, Physical Limitations to the Spatial Resolution of Solidstate Detectors, Cite as ArXiv:1404.4535 [physics.ins-det], (2014).
Contents lists available at ScienceDirect
Journal of Electron Spectroscopy and Related Phenomena journal homepage: www.elsevier.com/locate/elspec
Improving the performance of the cylindrical mirror analyzer II: Reducing the dependence of energy resolution on sample position David Edwards Jr. IJL Research Center, Newark, VT 05871, United States
a r t i c l e
i n f o
Article history: Received 4 May 2016 Received in revised form 6 September 2016 Accepted 7 September 2016 Available online 9 September 2016 Keywords: Cylindrical mirror analyzer CMA Electron spectrometer High resolution
a b s t r a c t Electrode segmentation of the cylindrical mirror analyzer has been shown to significantly improve the resolution of the standard instrument. The high precision solution which had been found for the 42.3 device is shown in this report to exist in fact for an extended set of input angles or equivalently for a range of sample positions. This has allowed a significant limitation of the CMA, namely the critical dependence of instrument performance on sample position, to be largely overcome. In addition by placing the sample at the minimum in the curve of resolution vs sample position, a base width resolution of ∼.00005 has been obtained for a ±6◦ bundle an improvement of a factor of ∼40 over the standard spectrometer. Thus electrode segmentation has been found not only to significantly improve the device performance, but in fact to enable its high precision to be realized. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Electrostatic electron spectrometers may be divided into two classes: those with a real entrance aperture as in the hemispherical deflector or 127 ◦ analyzer and those without one as in the CMA [1], SEA type devices [2], the 90 ◦ sector analyzer, etc. (An aperture should be thought of in the above context as a real hole in a metal plate limiting the starting positions of rays which subsequently travel through the instrument.) In the HDA [3] there is such an aperture the location of which is of course known with the dimensional precision of the other mechanical parts of the spectrometer. This is not the situation with the latter cases (CMA, SEA, CDA . . .) in which there is no aperture, the starting positions of the trajectories being typically formed by the intersection of a particle or x-ray beam with a sample which may be only indirectly connected to the spectrometer, and thusly may not be at its designed position. This location problem for the CMA has been reported by both Khursheed [1] and Prutton & Giomati [2] as resulting in an unacceptable shift in the energy scale of the spectrometer and hence necessitates when possible a very careful external positioning of the sample to its design position. It was reported in [3] that segmentation significantly improved the resolution of the standard 42.3◦ CMA. It thusly appeared worthwhile to attempt to find high precision solutions for angles differing from 42.3◦ . It was somewhat surprisingly discovered that the
E-mail address: [email protected] http://dx.doi.org/10.1016/j.elspec.2016.09.002 0368-2048/© 2016 Elsevier B.V. All rights reserved.
enhanced resolution of the 42.3◦ device could be obtained for a range of design angles which of course implied that the enhanced resolution would be obtainable over a range of sample positions. It was thus apparent that if a table were constructed indexed by the sample position, each entry in the table, consisting of the segment voltages and the pass energy which resulted in a particular (high) resolution at each indexed point, one could use such a table to let the instrument itself determine the location of the sample In this manner the resolution associated with that location would be achieved without sample motion. The description and justification of these ideas are the subject of this paper. 2. Notation and terminology Sample position: the position of the point on the z axis from which particles emanate. It will be referred to as z. The index table: it is a table each entry consists of the following information: {|z0 |, v1. . .vn, Epass} where z0 is the sample position, v1 . . . vn the segment voltages, and Epass the kinetic energy of the input bundle which when the segment voltages are applied to the electrodes will produce the smallest spot size, resolution. Relaxation process in FDM: This refers to a process in which the potentials within an electrostatic geometry are determined. It starts with covering the geometry with a grid of points, considered a mesh, an initialization to some value of all points within the mesh and continues by stepping thru the mesh at each meshpoint replacing the current value with one determined using an algorithm itself using a selection
D. Edwards Jr. / Journal of Electron Spectroscopy and Related Phenomena 212 (2016) 62–73
63
Fig. 1. The various segment names are defined in this figure as well the assumed position of the sample at −z0 , the starting position of the standard CMA (z = −113), the median rays of the sample at z0 and −113. Also shown is a position sensitive detector (PSD) at location z = 124. (It is noted that the criterion for the exit plane was that the starting points of interest would intersect the exit plane above the axis. the plane at z = 124 was such a plane and no further adjustment of this parameter was made.).
of potentials at surrounding mesh points. When no further change is found upon subsequent iterations the mesh is said to be “relaxed” Mesh density: This term will be used here as representing the total number of meshpoints covering the geometry. Thus “higher density” will refer to a covering containing more mesh points than its comparison mesh. The units of energy will be considered to be in electron volts while that of potential will be in volts. The length unit of the geometry will be in dimensionless enabling the user to select any unit of his choice (i.e. mm, cm, inch . . .).
3. The geometry The basic geometry will be closely represented by the single segment solution of [3] and is illustrated Fig. 1. The sections of the inner pre and post segments are reasonably fine meshes allowing the beam to pass and the bundle definition itself is defined by real apertures in the inner segments determined by the bundle itself. Further clarification can be found in the “notes of caution” section. The inner and outer cylinder radii will be taken to be Rin = 37, Rout = 80 except as noted in a later section in which the geometry will be scaled by a factor of 2. As the sample position will be considered a non-constant parameter, it is necessary to define an angle 0 as the angle of the median ray traversing the path (-z0 , s1) where s1 is defined as the intercept with the inner electrode of the 42.3◦
ray having a starting position of −113. It should be clear from Fig. 1 that 0 is easily determined to be: 0 = arc tan (R in /(|z0 |-|s1 |)) which is a function of z0 alone for fixed Rin , s1 . The rays are assumed to emanate from a point and make angles of 0 +-n degrees with respect to the z axis. This set of rays will be denoted as a bundle and n = 6◦ will be the one considered here (denoted as a 6 ◦ bundle). The dimensions of the geometry of Fig. 1 are defined in Table 1. Although the geometry is quite similar to the single segment solution reported in [3] it is not identical, as the position of in1 start has been changed from −54.47[3] to −50. This change was necessitated in order to have the 6 ◦ bundle for 0 ∼33 ◦ sufficiently far from the singularity present at the contact point of inner presegment and in1 so that the ray would not be influenced by the lack of voltage precision quite close to the singularity. Recognizing the fact that the optimal solutions for separate z’s would exit the device at a position dependent on z itself, a posiTable 1 The mechanical dimensions of the 1 segment CMA are given. Rin
37
Rout
80
left end plate in1 start out1 start s1
−187 −50 −70 −72.333
right end plate in1 end out1 end
187 54.47 70
64
D. Edwards Jr. / Journal of Electron Spectroscopy and Related Phenomena 212 (2016) 62–73
Fig. 2. Optimized solutions for 6◦ bundles are plotted emanating from 4 distinct sample positions which will be later seen to encompass the range of sample positions which yield high resolutions. Seen also are the intercept locations, each point representing an annular ring about the z axis. The numbers to the left of the focal points in the PSD refer to the starting positions of the trajectories.
tion sensitive detector (PSD) was necessary and is essential to the following. Two orthogonal orientations of the device were considered, one being along the z axis, the other being, as shown in Fig. 1, in a plane normal to the z axis. The later was selected as not in itself limiting the source azimuthal angles allowed by the cylindrical symmetry of the geometry. For an on axis PSD the azimuthal angles would by construction be limited to 180◦ . The parallel − to − the − axis solution continues to be a potential option having the advantage that all azimuthal angles exit at a point rather than a ring as for the vertical PSD but with the disadvantage of not allowing the azimuthal variation of the source signal to be measured. The geometry together with 6◦ bundles from 4 distinct sample positions are shown in Fig. 2. This figure was constructed to give a perspective consisting of both the focal points on the exit plane and the range of sample positions from which high resolutions may be obtained. Thus one can see that the tolerance of positioning the sample for the high resolution instrument becomes ∼29% of Rin . It should be noted that in the past CMA’s have typically assumed the rays would leave at a definite sample position and focus on a ring of certain radius on the exit plane. For such a ring a real annular aperture or annular slit would be constructed in the exit plane with a detector placed behind. However seen in Fig. 2 the radius or position of the annular aperture is dependent on the sample position and so real apertures could not be considered. A solution was to consider using a PSD for the detector in which essentially virtual apertures could (in software) be constructed. Thus for a given
focal radius, only the current within the virtual aperture is to be measured. 4. The modelling 4.1. Potential and trajectory calculations The goal of the modelling is to accurately calculate the trajectories for rays traversing the geometry. A starting point will be to apply a set of voltages to the segment electrodes. Although there are 8 segments depicted in Fig. 1, the voltages on the endplates are fixed at 10 V while the inner pre and post segment electrodes are set at 0 V. Thus while the modelling involves 8 separate voltages, only 4 are variable and need be involved in the search for an optimal solution. As the trajectories depend on the electric field at all points along its path, the electric field is calculated by means of the potential within the geometry determined by means of FDM [4,5]. In this manner the potential on a uniform mesh placed over the geometry can be found, the details discussed in [3]. Here it is simply remarked that an order 10 algorithm [4,5] was used in all potential calculations. It is further noted that the effects of the singular points (the electrode interconnection points) were effectively neutralized by the construction of 10 telescoping regions convergent on the points themselves, each telescoping region having a height, width of 20 units. Again details may be found in [3] as well as [5]. Given the potential distribution within the geometry trajectories were found using a recently developed ray tracing process [6].
D. Edwards Jr. / Journal of Electron Spectroscopy and Related Phenomena 212 (2016) 62–73
65
Fig. 3. The trajectories are plotted of rays within 3 bundles having energies symmetric about E0. The resolution is clearly less than 0.05%.
One of the differences of this method from the more standard Runge Kutta is the order of the electric field determination, called here order s, is an input parameter independent of the order of the time determination (order t), another input parameter. While in all of the calculations order t was taken to be 10, the highest order currently available, for reasons of calculation time concerns a value of 4 was used for order s. It will be later shown that no significant degradation in the precision of the trajectory in general resulted from using order s of 4 rather than 10. 4.2. The resolution It is assumed that all rays within a bundle start their trajectory from a point z0 on the axis having an initial kinetic energy E0 and reach the exit plane (z = 124) at position ri . A plane constructed at any point on the median trajectory intercepts all of the rays of the bundle and the extent of these intercepts is considered the trace width of the bundle on this plane. When this plane coincides with the exit plane the trace width on this plane will be denoted by spot size or s. The resolution is found by creating three bundles having energies E0 , E0 (1 + x), E0 (1 − x). The xmin is defined as the minimal value of x such that the intercepts of 3 bundles on the exit plane are disjoint. The resolution is then: resolution = xmin /E0 . As an example consider the plots in Fig. 3 in which the segment voltages specific to z0 = −124 are applied to the geometry. Seen in this figure is that the sets of intercepts are clearly disjoint implying that the resolution is «0.05. The resolution determined by this method will be denoted by the base width resolution.
Another method will be used to determine the resolution based on the minimum trace width on the exit plane and is found from Estimated resolution = s (E0 )/(∂Gri (E0 )/∂GE0 ) (see also Hafner et al. [7]) : where ri (E0 ) is the r intercept on the exit plane of the median ray within a bundle having energy E0 . (This expression tacitly assumes that s (E0 ) is independent of E0. ) That the assumption of the above derivation (the independence of s on E0 ) is violated may be seen in Fig. 3 by observing s (E) is in fact the smallest for E = E0 being larger on either side of E0 . However the estimated resolution will be shown to be quite useful after a calibration constant is obtained relating the base width resolution to the estimated resolution. The calibration will be described in a later section. 4.3. The optimization process Given a general starting point (and corresponding angle), segment voltages must be found that minimize the resolution. A method for accomplishing this has been described in [3] for a situation in which a somewhat arbitrary selection of segment voltages is assumed as the starting condition. However in this work the optimized solution as a function of z0 starts from the known solution for 42.3◦ (z0 = −113) and then proceeds to z0 ± 1/2 etc. thus extending thru the useful range of z0 (-129 to −106). Thus the starting voltage set for any z0 is likely somewhat close to the solution at that z0 and a full search as presented in [3] is not necessary.
66
D. Edwards Jr. / Journal of Electron Spectroscopy and Related Phenomena 212 (2016) 62–73
Fig. 4. The base width resolution is plotted vs the estimated resolution.
Rather it was found that by slightly modifying large step − small step algorithm of [3] the search for the optimum solution would be considerably shortened. A brief summary of the idea behind the modification is as follows: given a starting point, two segment voltages from the 4 possible ones are selected (6 distinct ways) and increment/decremented by a small value (say 0.1 V). Thus each combination selected results in 4 distinct values for the pair (++ +−+ −). After this voltage selection has been made a scan through the segments is performed incrementing/decrementing by a much smaller value ∼.01 V. At each point in the second scan the minimum estimated resolution is found using E0 alone [3]. By means of this method a significant reduction in time was typically achieved for finding an optimized solution from a previous one. 5. The results 5.1. The resolution Starting from the known high resolution point of [3] (z0 = −113, theta = 42.3◦ ), z0 was extended to the surrounding points and at each point the optimized segment voltages and resolution were obtained. This process continued throughout the range −129–106 in steps of 0.5. Having obtained the optimized segment voltages for each z0 , the base width resolution was found using the method described above. A plot was then obtained of the base width resolution vs the estimated resolution and is shown in Fig. 4. In this figure it is seen that a linear approximation provides a reasonable fit to the data and allows a definition of the corrected resolution to be inferred from the estimated resolution by:
Table 2 The optimized voltage parameters for the highest resolution point of Fig. 5. inner pre segment
0V
outer pre segment
21.652V
inner v1 inner post segment z0 minimum spot size Epass
−0.083V 0V −124 0.004425 20.7399 eV
outer v1 outer post segment ri on plane z = 124 base width resolution dz(E)/dE
7.889V 13.608V 17.351 0.000048 132.82
corrected resolution ∼ = 1.351 * estimated resolution providing a correction to the resolution inferred from the estimated resolution. A comparison of the base width resolution and estimated resolution over the range of z0 , −129 to −106 is shown in Fig. 5. In this figure we see that the corrected resolution clearly provides a reasonable estimate to the base width resolution from the estimated resolution which is considerably easier to calculate. In all of the following resolution will be taken to be the corrected resolution inferred from the estimated resolution. The segment voltages corresponding to each −z0 in Fig. 5 not reported. However the data for the minimum result of Fig. 5 is given below in Table 2 enabling a comparison by others with the results of the present calculations. The complete table would be available upon request. It is noted that the data point z0 = −113 corresponding to a resolution of ∼.034% and a theta of 42.3 ◦ differs from that reported in [3] of 0.017% due to having a vertical detector plane rather than one on axis as in the previous study.
D. Edwards Jr. / Journal of Electron Spectroscopy and Related Phenomena 212 (2016) 62–73
67
Fig. 5. A comparison of the base width resolution and estimated resolution. The difference in the two curves likely results from the scatter seen in Fig. 4. The relation is not exact but only approximate.
5.2. Interpolation to points not within the index set The index set is considered to consist of optimized segment voltages, pass energies for all integer and half integer points between −129 and −106 and are indexed by |z0 |. In any physically realized situation, the sample position may be close to but not on one of the z0 values in the index set. The question is to find the optimized resolution for such points. The solution is straightforward, namely given a sample position z, scan through the index set and find the index point z’ that gives the highest resolution for z. Then one uses the segment values of z’ for those of z and further minimizes the resolution with respect to E0 . The result of this optimization gives the segment voltages (those of z’) and the pass energy (the value of E0 producing the minimum resolution) for z’. This procedure has been used in a scan of z between −129 and −106 with a step size of 0.1 and the results given in Fig. 6. In Fig. 6 the results of the interpolations are plotted along with the resolutions for points lying exactly on indices of the index set. Seen that there is measureable but reasonably small deterioration of the resolution for points in a neighborhood of but not on an index point. Also seen is that the deviation reaches a maximum for a points ∼½ way between points of the index set. While the interpolation deviation evidenced in Fig. 6 is in fact small, improvements could be made by a further refinement of the index set. A similar study was performed in which the sample position was placed at a random z in the interval −129,106. The procedure
sketched above was followed and the resultant resolution obtained at each random point. The results are given in Fig. 7. Seen again is that reasonable interpolations are made for the random locations of the sample. 5.3. Interpolation within a physical spectrometer In the previous section it was assumed that E0 (the kinetic energy of the particles leaving the sample) was both known and able to be varied. In contrast it will be assumed in this section that E0 is neither known nor variable; hence an equivalent process will be described that uses scaling to optimize the resolution. First recall that at any index zindex in the index table, an element at that index consists of a collection of segment voltages and the associated pass energy, Epass , the combination of the two yielding an optimized resolution for a bundle leaving the sample at −zindex having an energy E0 = Epass . This element is in fact a family of segment voltages, pass energies, each member of the family being related to the original by scaling each segment voltage and pass energy by a given parameter. Thus for example if a member of the index table is represented by {v1 , v2 . . . vn , Epass } a particle of kinetic energy Epass leaving the sample with appropriate voltages applied to the segments would be focused into the small optimized spot on the exit plane as would particles of kinetic energy Epass *f provided that the segment voltages were also scaled by f. In more succinct terminology the family of solutions for any index point zindex may be
68
D. Edwards Jr. / Journal of Electron Spectroscopy and Related Phenomena 212 (2016) 62–73
Fig. 6. The sample position is scanned in steps of 0.1 over the range −129, −106.
represented by {v1 *f, v2 *f . . . vn *f, Epass *f}.
5.4. Simulating the physical spectrometer To understand the operation of the simulated spectrometer, one must understand the separation that exists in the software used to model the spectrometer. In the software there are two separate modules: the first takes the desired segment voltages, applies them to the electrodes and finds the potential distribution within the geometry; the second uses this distribution and traces through the device rays emanating from a given sample position − z having initial kinetic energy at the sample of Epeak . In this simulation it is seen that −z is the actual location of the sample and Epeak is the kinetic energy of the rays emanating from the source. These two quantities are both fixed and unknown, and are to be determined along with the segment voltages that result in the smallest spot size (minimum resolution) for the bundle. The method is similar to the previous one: a scan through the index set is performed, and at each index point the segment voltages and associated pass energy (not E0 ) are scaled by a factor f, and the resolution (peak width) experimentally determined. The process is continued until an f results in the smallest resolution (peak width) possible. At the end of the above the index point with the smallest resolution has been located its pass energy determined which is the pass energy resulting from the scaling. It is also noted that the locus of peak intensity points in fact determines the annular aperture appropriate to the actual sample position. See a further discussion in the “notes of caution” section.
As an example consider the sample at position −119.72 and the kinetic energy of the rays is 11.3728. This point was chosen to be ∼1/2 way between index points −120 and 119.5. Results of a scan thru the index set are shown in Fig. 8. At each point in the index set a segment voltage scaling parameter f was found which minimized the resolution. It is noted that the pass energy of the index point is also scaled and recorded but not transmitted to the ray trace module. The scan in the above figure is seen to proceed from −125 to −111 in steps of 1/2 unit. A minimum can be seen in the curve at z0 = −120, the value of the minimum being within interpolation variations of Fig. 6 which is also plotted in Fig. 8. Thus for the minimal point of Fig. 8 the resolution, the determined segment voltages, the pass energy and ri are obtained. It is clear from this figure that were the density of index points increased a new minimum would be found likely between the two smallest points of Fig. 8. It thus appeared worthwhile to attempt to find this minimum from the coarse set by fitting the points near the minimum with a parabola and then finding the minimum of this curve. In this manner both estimates of the minimum resolution and actual sample position may be obtained. Recalling that the process used to obtain Fig. 8 also yields a plot of the resultant pass energy vs zindex (not shown) the pass energy at the calculated minimum can be found by interpolation using the newly found estimated sample position. To evaluate the possible utility of the above suggestion the error in the sample position and peak energy was determined for sample positions between −120.5 and −119.5 with a fixed peak energy of 11.3728. To obtain a higher precision than would be possible with the current index set (interval 0.5), the index set was expanded in
D. Edwards Jr. / Journal of Electron Spectroscopy and Related Phenomena 212 (2016) 62–73
69
Fig. 7. The optimized resolution obtained at random locations of the sample.
this restricted range using an interval spacing of 0.125. For each sample position taken (step 0.1) within this range the process described above was effected and yielded a plot of the errors in peak energy and sample position which is shown in Fig. 9. Plotted in this figure are the% errors in the peak energy and sample position. Seen is that the sample position error is close to but strictly less than the peak energy error. The average% peak energy error which has been found is ∼ 0.028 implying that a 1000 eV peak would have an error of ∼.3 eV. As all energies are referenced to the vacuum level of the sample this energy error is of the order of work function differences between the sample and spectrometer and hence no further subdivision of the index set to achieve higher precisions would appear propitious. 5.5. The effects on using order 2 on simulations However rereading the relevant section I see that improvements should be made. I have thus rewritten it as: As mentioned previously an order 10 algorithm was used for the determination of the potential distribution within the geometry and an order 4 algorithm for the electric field interpolation during the ray trace. (See above section “Potential and trajectory calculations”) In order to see the effects of degrading the algorithm precision two cases were considered: in the first the potential distribution was found using an order 2 (rather than 10) algorithm with the order s = 4 algorithm being used in the ray trace program. Using order s of 4 algorithm in ray trace was found equivalent to using an order s of 10). The second case was the converse, namely an order 10 algorithm was used for potential calculations while an
order s of 2 algorithm for the field interpolations during ray trace. It is noted that the time order (order t) in the ray trace calculations was fixed at 10. The effects of this study are given in Fig. 10. In this figure our high precision result (Fig. 5) is given together with the situations in which the order 2 algorithm was either used in the potential determination or in the field interpolations. Seen is that the effects are quite similar implying that even if quite accurate potentials were determined the resolution precision would be compromised by the use of a low order algorithm in the ray trace. It is also observed that the effects of the low order algorithm are quantitatively similar regardless if incorporated either in the potential or field determinations. Also displayed in Fig. 10 is the result of using an order 2 algorithm together with a higher density mesh (geometry scale factor 2, density enhancement 4). Seen is that this graph approaches and is reasonably close to the lower convergent curve (order 10 in mesh relaxation). The implication is that high precision algorithms are not necessarily required for the determination of the results presented in this report; low order algorithms would clearly suffice particularly if used in conjunction with higher density meshes.
5.6. Notes of caution Many concerns in relating the above model to a real spectrometer are the same as for the spectrometer described in [3] and as they have been covered in some detail in [3] are not repeated. Two additional comments are necessitated by the solution presented above. They are:
70
D. Edwards Jr. / Journal of Electron Spectroscopy and Related Phenomena 212 (2016) 62–73
Fig. 8. Fixing the sample at −119.72 and the kinetic energy of the rays at 11.3728, the resolution is obtained for each zindex in the index set. Also plotted are the data from Fig. 6 showing that the minimum found using segment voltage scaling is within the variations seen in Fig. 6.
1. In our modelling the 6◦ bundle size was limited by a restriction on the rays emanating from the point source and not, as must be accomplished in an actual CMA, by a restriction of the entrance aperture. It has been determined that if an entrance aperture (in the inner cylinder)of the CMA is established by the limits of the 6◦ bundle for z0 = −124 (our maximum resolution point) that considering only z0 ’s having a resolution less than 0.0002 (see Fig. 5) i.e. −127 < z0 < −116 that the bundle size would vary between 5.5 ◦ and 7.4◦ . It is noted that the index table needs to take into account the actual bundle size permitted by this physical aperture when being formed. It is not considered that this would have a significant impact on the resolutions of the selected z0 ’s other than resulting in a slightly reduced transmission for z0 <-124 and a somewhat increased transmission for z0 >-124. 2. A position sensitive detector (PSD) has been used as the output rather than the more traditional aperture of the standard CMA. The process described above for solving the sample position problem would not permit using a real aperture in the exit plane. Hence this choice is not an option. However, considering the dimensions of Table 1 to be in mm, it is seen from Table 2 that the pixel size of such a PSD must be of the order the spot size ∼5 m This is either at or somewhat beyond the limit of current technology (see for example a recent article by Boronat et al. [8] describing progress in the 5 m range and below). It is noted that if one incorporates a PSD with resolution of 20 m, the spectrometer resolution at its highest resolution point (z0 = −124) would be ∼.0002 remaining a significant result and one which would
be improved to its design value by continued advances in this rapidly changing technology. 3. The intensity of the emitted beam leaving the sample with energy E0 is considered to be made by summing only the sub currents in the annular ring of the PSD centered on the radius of the focal point of this beam on the exit plane and of width given by its predicted spot size. This can be done in software coupled with the properties of the PSD itself. Further that the energy spectrum is measured by stepping the pass energy in discrete steps measuring the intensity at each step. 4. In the discussion determining the radius ri of the annular slot (assumed circular) in the detector plane (see Section below Fig. 8) it had been noted ri may be found by interpolation in the same manner as the energy E0 of the peak itself. However in the process involved in experimentally determining the sample position, E0, the actual annular ring (including eccentricities) has been in fact measured by the 2 dimensional plot of the locus of the peak intensity points on the PSD itself and hence an interpolation for ri is neither necessary nor desired. In fact it is clear that the above experimental determination will give actual annular slot including eccentricity corrections due to any slight cylindrical misalignments. Not having this ability (i.e. assuming a circular annular ring) would likely severely limit the performance of any experimentally constructed devices. Thus the use of a PSD has both allowed virtual apertures (not necessarily circular) to be constructed and enabled an essential modification of the annular aperture to an eccentric aperture, the eccentricity
D. Edwards Jr. / Journal of Electron Spectroscopy and Related Phenomena 212 (2016) 62–73
71
Fig. 9. The errors in determining the peak energy and peak position are plotted vs sample position. Also included is the average peak energy error.
depending upon the degree with which the cylinders themselves are aligned. 6. Verification In verifying the results the resolution vs z0 will be determined in several ways. The first is to simply increase the order s of the ray trace procedure from 4 to 8 using the standard mesh (Rin = 37, Rout = 80), while the second is to increase the density of the mesh by a factor of 4 by scaling which will produce a mesh of Rin = 74, Rout = 160. The results of these studies are given in Fig. 11. Seen is that the order s = 4 and 8 are essentially indistinguishable from the results from the scaled geometry. The most significant difference occurring in the lower portion of the curve near z0 ∼ −128.5. It is suspected that this discrepancy is caused by the rays nearing the singular point at the juncture of the inner presegment and the inner segment. The agreement suggests that the results are valid over the range of sample positions indicated. It is noted in all of these studies the potential distribution was determined using an order 10 algorithm. 7. Conclusion Several new developments of the cylindrical mirror analyzer have been found and presented. It has been shown that the high resolution afforded to the standard CMA (42.3◦ ) by segmentation can be extended to a continuous range of input angles and sample positions. This has allowed one of the significant problems of the CMA
to be improved if not overcome, namely the dependence of performance on sample position and has resulted in a high resolution to be effected over a range of sample positions. In addition provided that the sample can be precisely positioned (in the Rin = 37, Rout = 80 geometry) at z0 = −124 a resolution of ∼.0048% or a resolving power (1/resolution) of >10000 can be obtained. This represents a factor of ∼40 improvement over the standard geometry. Obtaining high resolution over a range of sample positions requires that in the instrument setup prior to operation segment voltages appropriate to the current sample position must be found (using a procedure described above) and when these voltages are applied a high resolution should be realized. From Fig. 7 for example a resolution of 0.025% or a resolving power of ∼4000 can be obtained for sample placements between −127 and −116 or within a span of 11 units (29% of Rin) the instrument could achieve its design performance. The procedure for determining the appropriate segment voltages for a given sample position results also in a side benefit, namely establishing of the energy of the peak used in the above calibration. It was found in the example studied that the absolute energy of the peak can be found with a precision of ∼.03%. Thus the benefits of spectrometer electrode segmentation clearly surpass those suggested in [3] in allowing high resolutions to be achieved not only for a given sample position but over a considerable range of positions. This process should be applicable to spectrometers not having a real aperture at its entrance as discussed in the introduction and likely represents a new paradigm for the design of high performance instruments.
72 D. Edwards Jr. / Journal of Electron Spectroscopy and Related Phenomena 212 (2016) 62–73
Fig. 10. The resolution is plotted vs z0 for various orders and 2 mesh densities. The labels “mesh Rin ,Rout ” reports the inner and outer cylinder radius of the CMA for that curve. (See text for further discussion.).
D. Edwards Jr. / Journal of Electron Spectroscopy and Related Phenomena 212 (2016) 62–73
73
Fig. 11. The scan of resolution vs sample position for several operational parameters.
Acknowledgments It is a pleasure to acknowledge discussions with Anjam Khursheed for making the author aware of this problem. Without this input this work would not exist. In addition appreciation is given to Omer Sise and Anjam Khursheed for their critical reading of the manuscript. Newark, Vermont. April 24, 2016.
[6] David Edwards Jr, The segmented cylindrical mirror analyzer (CMA), J. Electron Spectrosc. Relat. Phenom. 209 (May) (2016) 46–52. [7] David Edwards Jr, FDM for curved geometries in electrostatics II: the minimal algorithm, IMECS 2014 March 12–14, 2014, Hong Kong, in: Proceedings of the International Multi Conference of Engineers and Computer Scientists, Vol I, 2014. [8] David Edwards Jr, Finite difference method for boundary value problems. application: high precision electrostatics, IMECS 2014, March 18–20, 2015, Hong Kong, in: Proceedings of the International Multi Conference of Engineers and Computer Scientists, Vol I, 2014.
Further reading References [1] P.W. Palmberg, G.K. Bohn, J.C. Tracy, Appl. Phys. Lett. 15 (1969) 254 (number 8). [2] D. Cubric, N. Kholine, I. Konishi, Electron optics of spheroid charged particle energy analyzers, in: Proceedings of the Eighth International Conference on Charged Particle Optics, Singapore, July 12–16, 2010, p. 234. [3] M. Purcell Edward, The focusing of charged particles by a spherical condenser, Phys. Rev. 54 (1938) 818. [4] Anjam Khursheed, Scanning Electron Microscope Optics and Spectrometers, World Scientific Publishing Co Ptc. Ltd., 2001, pp. 99 (ISBN −13 978–981-283-667-0). [5] Martin Prutton, M. Mohamed El Gomati (Eds.), Scanning Auger Electron Microscopy, Wiley, 2006, pp. 0–978, ISBN: 978-0-470-86677-1. (See section on CMA).
[9] David Edwards Jr, High precision ray tracing in cylindrically symmetric electrostatics, J. Electron Spectrosc. Relat. Phenom. 205 (2015) 111–121. [10] H. Hafner, J. Arol Simson, C.E. Kuyatt, Rev. Sci. Instr. 39 (1968) 33. [11] M. Boronat, C. Marinas, A. Frey, I. Garcia, B. Schwenker, M. Vos, F. Wilk, Physical Limitations to the Spatial Resolution of Solidstate Detectors, Cite as ArXiv:1404.4535 [physics.ins-det], (2014).