XPS spectrometer transmission function optimization by the differential evolution algorithm

Accepted Manuscript Title: XPS spectrometer transmission function optimization by the differential evolution algorithm Authors: J. Trigueiro, W. Lima, N. Bundaleski, O.M.N.D. Teodoro PII: DOI: Reference:

S0368-2048(17)30080-4 http://dx.doi.org/doi:10.1016/j.elspec.2017.07.004 ELSPEC 46691

To appear in:

Journal of Electron Spectroscopy and Related Phenomena

Received date: Revised date: Accepted date:

23-4-2017 7-7-2017 11-7-2017

Please cite this article as: J.Trigueiro, W.Lima, N.Bundaleski, O.M.N.D.Teodoro, XPS spectrometer transmission function optimization by the differential evolution algorithm, Journal of Electron Spectroscopy and Related Phenomenahttp://dx.doi.org/10.1016/j.elspec.2017.07.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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XPS spectrometer transmission function optimization by the differential evolution algorithm

J. Trigueiro, W. Lima, N. Bundaleski*, O.M.N.D. Teodoro CeFiTec, Departamento de Física, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, P-2829-516, Campus de Caparica, Caparica, Portugal *

Corresponding author

Email address: [email protected]

Highlights     

An XPS containing XSAM 800 setup, produced by KRATOS, was upgraded Acquisition and control systems, which were reconstructed, are now PC-based A sample-biasing method for transmission function measurement was introduced The novel method is superior to the well-established first principles method The transmission function was optimized by the differential evolution algorithm

An XPS containing XSAM 800 setup, produced by Kratos, was upgraded. This mainly included the reconstruction of the acquisition and control systems. The detection system non-linearity, and the transmission function of the upgraded system were measured. The dead time of the detection system was determined using two independent approaches, which provided practically the same result. The knowledge of the detector dead time allows us to anticipate the count rate loss. The transmission function was measured using the well-established first principles method. Since this approach did not provide satisfactory results for all pass energies, a novel procedure, here denoted as sample-biasing method, was proposed and successfully applied. The novel approach is much faster, and significantly reduces the measurement error as compared to the first principles method. The transmission function was then optimized by applying a differential evolution algorithm, which provided its relative increase in the range from 10 % to 110 % depending on the fixed analyzer transmission mode and the electron

2 kinetic energy. The optimization process is fully automatized, and can be readily applied for other similar problems such as tuning of charged particle beams.

Keywords: X-ray photoelectron spectroscopy, instrumentation, detector dead time, transmission function, optimization

1. Introduction X-ray Photoelectron Spectroscopy (XPS) is nowadays among the most important surface sensitive analytical techniques. Not only is it widely used in surface science, but has also become a standard tool for material characterization including the analysis of biological and other nonconductive samples [1]. Although mainly recognized as a tool providing information on the chemical bonds, it is at least equally important as the experimental technique for surface composition analysis having the smallest matrix effects [2, 3]. The instrumentation of an XPS setup is complex and therefore expensive. Modern XPS systems provide advanced options such as imaging [4], measurements of samples exposed to different gases under pressures typically up to 1 mbar (also known as environmental XPS) [5], or Angle Resolved Photoelectron Spectroscopy – ARPES performed on synchrotrons [6]. When compared with older systems, the most important progress concerns data acquisition and detection systems, the latter being frequently based on position sensitive detectors. However, the essential feature of any XPS system, highly precise measurement of the electron kinetic energy with a resolution of about 0.5 eV, was reached on standard instruments back in the seventies. Having all this in mind, it is not a surprise that many XPS instruments, twenty or more years old, are still under operation and are successfully employed for standard measurements. One very good example is the model Kratos XSAM 800 from the eighties, which is still being used ([7-10] for instance). Concerning the XPS data interpretation, the main focus is typically on the fitting of high resolution spectra, which is indeed a complex issue, while quantitative analysis is usually considered to be straight forward. The latter is based on the use of atomic sensitivity factors in the vast majority of

3 cases, which is very easy to implement. Quantitative analysis is, however, far from trivial due to several reasons related to the sample (sample uniformity is not always fulfilled, presence of matrix effects), data analysis (defining the region in which peak integration is performed, defining background, extra features such as shake-up satellites or multiplet splitting, etc.), but also due to the instrumentation. Today, XPS measurements are performed exclusively in fixed analyzer transmission (FAT) mode, in which the energy of detected electrons inside the analyzer (pass energy Epass) is kept constant. The energy scan is then performed by decelerating the electrons from the measured kinetic energy to the pass energy in a complex electron-optical system, situated between the sample and the energy analyzer. The measured intensity strongly depends on the collection efficiency of the spectrometer, i.e., on its transmission. In this context, the transmission function T(E) (transmission vs. the electron kinetic energy) encompasses all instrumental parameters of the XPS spectrometer such as acceptance area, acceptance solid angle and the detection efficiency. Sometimes, this magnitude is also named intensity/energy response function [11]. Being instrument dependent, the transmission function is the reason why specific atomic sensitivity factors have to be attributed to each XPS system. In their comprehensive work dedicated to the quantification in XPS, Seah et al. performed a systematic study of the transmission function of several XPS models [12]. It was shown that the transmission function of an XPS setup is changing with time. Moreover, it appeared that sometimes different instruments of the same model have large differences between their transmission functions, reaching 30 % in the particular case of the Kratos XSAM 800 model for pass energies of 80 eV, and even higher for 40 eV. This striking result emphasizes the necessity to periodically measure the transmission function of an XPS setup, in order to update its atomic sensitivity factors, and increase the reliability of the composition analysis. There are several methods reported in the literature for measuring the transmission function [13-15]. Since the energy of electrons inside the energy analyzer at the moment of their detection is kept constant in FAT mode, the transmission function actually reflects the energy dependence of the

4 collection efficiency of the retardation electron-optical system i.e. the acceptance area and particularly the acceptance solid angle. One commonly used approach for measuring the transmission function is known as the first principles method [13]. It is based on the measurement of the intensity ratio of different photoelectron lines of the same element, taken from the same previously cleaned uniform sample. The intensity S of any line is directly proportional to the atomic concentration of the element, the total photoionization cross-section σ, the attenuation length of photoelectrons λ, the transmission function T and the factor describing the cross-section anisotropy L(ψ) = 1 + (β/2)·(1.5·sin2ψ – 1). In this expression, β is an asymmetry parameter and ψ is the angle between the directions of impinging photons and analyzed photoelectrons. The intensity ratio of two photoelectron lines of the same element situated at kinetic energies Ej and Ei is then S j E j  Si Ei 



L j   j  j E j  T E j  .  Li   i i Ei  T Ei 

(1)

Since all the parameters, apart from the transmission function, can be calculated, the ratio between the transmission function of the two energies can be readily obtained from the measured line intensities. The transmission function is normalized in such way that its value at some specific kinetic energy (usually 968 eV, corresponding to the energy of the C 1s line excited by the Mg Kα line) is fixed to 1. By measuring XPS line intensities of several samples and using different photon energies, it is possible to obtain the ratios of transmission functions, and from there reconstruct the overall T(E) dependences for any pass energy. Although rather straight forward, there is one major drawback of this approach: it is necessary to thoroughly prepare a considerable set of samples in order to determine the transmission function in a sufficiently wide energy range and with a satisfactory density of points. This approach is time consuming and introduces significant errors when some of the used lines have rather small intensity. Other commonly used methods also face different drawbacks, such as: a) direct comparison of spectra to an instrument of known transmission function [12] will require a reference sample prepared in exactly the same way;

5 b) when monitoring line intensity vs. pass energy, and its fit to S  T  Epass·(Epass/E)n in order to determine n [13], not necessarily correct T(Epass, E) dependence is presumed (as will be shown in this work); c) using the background signal (which is not only related to the actual photoelectron signal but also to the noise generated inside the energy analyzer) to acquire enough density of points [14]. Apart from the modification of the transmission function, quantitative XPS analysis can be affected by the detector nonlinearity, as pointed out in [16]. The nonlinearity may appear at high count rates, due to the saturation of the detection system, but also at very low rates, as observed in the case of micro-channelplate multipliers followed by the phosphor screen and a CCD camera [15, 16]. If the detector behaves non-linearly for count rates of XPS peaks, this problem will influence the transmission function measurement as well. In this work, we report the upgrade of the XSAM 800 XPS system manufactured by KratosCo, mainly consisting on the replacement of the obsolete acquisition system, based on a PDP-11 computer. The nonlinearity of the detection system was then measured, which allowed us to determine its dead time and eventually introduce an automatic correction of the detected count rate in the acquisition software. Having in mind the conclusions in [12], the transmission function of the XPS spectrometer was then measured, in order to improve the ability of the system concerning the quantitative XPS analysis. Although providing satisfactory results, T(E) determined from the first principles method is characterized by considerable measurement uncertainty, and relatively small density of points. For that purpose, a novel approach, here denoted as sample-biasing method, is introduced and validated by comparison with the first principles method. Its major advantages, apart from the reduced measurement error, are that a single sample can be used and that it provides an arbitrary density of points. Finally, once the transmission function was determined, we performed its optimization by applying one type of evolutionary algorithms, known as differential evolution [17]. It appears that the chosen algorithm is simple to implement and probably highly suitable for automatic optimization of versatile black-box-like systems, including tuning the intensity of electron and ion beams.

6

2. General Description of the Upgraded System The Kratos XSAM 800 is an UHV system designed for surface studies, with a base pressure in the low 10-10 mbar range. It contains an XPS setup, as well as other surface sensitive techniques. The system is equipped with a dual anode X-ray source (Mg/Al Kα lines) and a monochromatic X-ray source (Al Kα line). The spectrometer is composed of a deceleration electron-optical system (the column), a hemispherical energy analyzer with the main path radius of 127 mm, and the acquisition system. The measured value of the angle between the incident directions of the X-ray beams and the direction of emitted photoelectrons reaching the energy analyzer, is ψ = 69°. The acquisition system consists of three channel electron multipliers, each provided with its pre-amp channel, and the computer system with an acquisition card and software. Working with three detectors significantly increases the signal to noise ratio, and therefore reduces the data acquisition time. The position of each detector corresponds to a trajectory with a slightly different radius, and consequently different energy. Therefore, the acquired spectra have to be shifted with respect to each other for an energy amount proportional to the pass energy before performing the pile-up. The originally available operating pass energies were 5, 10, 20, 40, 80, 160 and 320 eV. Presently, only the pass energies in the range from 10 to 80 eV are being used. Their choice determines the analyzer transmission and resolution. There are two magnification modes, low and high, which determine the size of the acceptance area (the sample area ‘seen’ by the energy analyzer). Each mode has different voltages applied on the electrodes of the column. The energy analyzer is controlled by a single high voltage VHT, which defines the kinetic energy of detected electrons, E. The potentials of each electrode (entrance and exit slits, inner and outer hemispheres and fringing field correctors) are generated by another power supply, controlled by VHT. The electron-optical column consists of 6 electrodes, here denoted by numbers in ascending order along particle trajectories: the first electrode is closest to the entrance, and the sixth is in front of the analyzer entrance slit. The electrodes 3 and 4 are connected, and kept at voltage V3+4. The electrodes 1 and 2 are grounded in the low magnification mode, and at potentials representing fractions of VHT, thus

7 generated by adjustable voltage divider circuits, in the high magnification mode. The whole system is therefore controlled by four independent high voltages VHT, V3+4, V5 and V6, along with the relays that define the polarity, pass energy and gain. The original system was digitally controlled by a PDP-11 computer, which was regulating the high voltages with digital signals, switching the relays that determine the gain (ratio between output and control voltages), polarity and pass energy of the analyzer, and counting the pulses generated by the pre-amps. This system for control and data acquisition, obsolete and fully incompatible with the present information technologies, was therefore replaced with a newly developed system, schematically presented in Fig. 1.

2.1 Hardware upgrade The new system is based on a modern Windows PC, which is equipped with a National Instruments NI PCIe-6323 data acquisition board. The board contains four 16-bit analog output channels with a maximum sampling rate of 900 kS/s, 48 TTL digital I/O channels with a sample clock rate up to 1 MHz, and four 32-bit counters with a maximum internal clock of 100 MHz. The four analog outputs are used to control VHT and the voltage of electrodes 3 to 6 (electrodes 3 and 4 are connected). The digital outputs are used to control the relays of the power supplies that turn on/off the high voltage output, define the gain, polarity and pass energy. The signals coming from the three detectors are individually connected to the counter inputs of the board. The 4th counter is used as an internal clock for synchronization i.e. for the coordination between the output voltages and count numbering. The original high voltage power supplies were kept, but their circuits were modified in order to be compatible with the new control system. This consisted mainly in switching from the digital control of the power supplies, to their control by analog voltages: the Digital-to-Analog Converters inside the power supplies were bypassed, and the analog outputs of the NI PCIe-6323 board were directly connected to the control voltage inputs of the power supplies. At the same time, the digital channels of

8 the old PDP-11 system, used to control the relays, were replaced and connected to the digital outputs of the NI PCIe-6323 board. The gain of the high voltage supplies was also adjusted to assure a minimum step of 0.1 V on the output voltage.

2.2 Software New software used to perform the XPS analysis was developed using LabViewTM 2012 platform. It is designed to control the NI PCIe-6323 board, i.e. to define the scanning parameters (first energy, energy span, energy step, dwell time, pass energy, magnification mode) and perform the energy scan while measuring the signal intensity, among other functionalities. Perfect synchronization is essential to perform a reliable energy scan of the XPS signal, since the software has to define the electrodes’ voltages for each energy step, and count the pulses from the three electron multipliers during the dwell time. This is assured by using the 4th counter as a clock signal for triggering both the analog outputs and the counter input channels of the acquisition board. The data is displayed in real time as the scan progresses, and can be saved as an ASCII file according to ISO 14976:1998 (VAMAS) format [18]. The upgrade of the control and acquisition system to a modern PC significantly improves the versatility of this spectrometer, since most parameters of the system are now programmable and can be easily defined or adjusted. Besides, other functionalities can always be added to the software in the future.

3. Nonlinearity of the detection system By nonlinearity of a detection system we consider non-linear dependence of the rate of detected particles vs. the rate of particles impinging the detector. If the two events are too close in time, they will not be resolved. Linear properties of the detection system are determined by the characteristics of both, the detector and the data acquisition system. Qualitative tests of the control and data acquisition board were performed by connecting a pulse generator to its entrance and measuring the count rate as a function of the pulse frequency. In the

9 case of square pulses with 50 % duty cycle, this ratio appeared to be 1 up to the frequencies of about 10-30 MHz, when it was abruptly falling to zero. The pulse loss at higher count rates clearly suggests that the acquisition system should be described by a paralyzable model [19], and therefore characterized by a dead time τ. Further analysis of the acquisition process cleared out that the existence of dead time is caused by a) the finite width of the TTL pulse at the data acquisition system τw and b) a minimum (recovery) time interval τr between the end of a previous and the beginning of the next pulse so that the pulses can be resolved by the acquisition system. Consequently, τ = τw + τr. Existence of the detector dead time inevitably introduces detector nonlinearity. Oscilloscope measurements of the signal on the exit of the preamp revealed that τw ≈ 25 ns. Then, a square pulse generator was connected to the counter input of the NI PCIe-6323 board. By changing the duty cycle, we were able to detect the minimum time interval for which the peaks are still resolved. Further reduction of the duty cycle resulted in an abrupt count rate loss. Clearly, the minimum time interval, being about 40 ns, represents the recovery time τr. Since the dead time is expected to be a sum of the pulse width and the recovery time, we estimate from here the detection system dead time to about 65 ns. The detection system dead time was measured using the procedure proposed in [20]. If the measured count rate is R, the time fraction in which the detector was active equals 1 – R·τ. Consequently, the true count rate (i.e. the rate of particles impinging the detector) equals R0 = R/(1 – R·τ). This expression is correct as long as R << τ -1 i.e. until the probability of detecting a particle during a time interval [0, τ] becomes non-negligible. Although the true count rate cannot be directly measured, this magnitude can be controlled with the X-ray power Px, since R0  Px when the high voltage between the anode and the cathode of the X-ray gun is kept constant [16]. If the power Px is controlled by changing the emission current of the X-ray gun Ix, we may write R0 = a·Ix. The magnitude a is a proportionality constant. This linearity should be also manifested as the linear dependence between Ix and the photoelectron current Iph [15], the latter being a reliable measure of the intensity of X-rays irradiating the sample. The experimental Iph vs. Ix dependence was fitted to the linear function with the square of the correlation coefficient R2 = 0.99998, showing that measured Ix is

10 directly proportional to Px. Hence, we may write R/Ix = a – a·τ·R. A simple and reliable way to measure the detector nonlinearity is to measure the count rate as a function of Ix in a part of the energy spectrum in which the intensity is roughly constant [15, 16, 20]. By fitting R/Ix vs. R to a linear dependence, both detector dead time and the constant a can be readily obtained. The dead time of the detection system was determined by measuring the count rate of a single detector in the flat part of the secondary electron spectrum taken from a rutile TiO2 (110) sample as a function of Px, controlled by the electron emission current. The high voltage in the X-ray gun was kept constant at 10 kV. One typical experimental result is presented in Fig. 2, from which the detection system dead time was determined as 61±3 ns. Obviously, there is an excellent agreement between the preliminary tests of the acquisition system, performed with an oscilloscope and wave generator, and the dead time measured under realistic conditions.

4. Determination of the transmission function Our first choice in measuring the transmission function was to apply the well-established first principles method. Knowing that XPS analyses are typically performed in FAT 20 and 40 modes, we have performed T(E) determination for the pass energies of 20, 40 and 80 eV. The measurements were performed only for the low magnification mode, which is typically used in standard XPS analyses, and with a fully opened entrance slit. The intensity of lines was never exceeding 100000 counts per second, yielding relative error due to the detection system nonlinearity (R0 – R)/R0 < 0.6 %. Being considerably below other uncertainties (influence of sample cleanliness, calculated values of different parameters, uncertainty of the background determination, etc.), the relative error due to the detection system nonlinearity has been ignored.

11 4.1 The first principles method The transmission function was determined by measuring photoelectron lines from previously sputter cleaned nickel, gold, and copper samples using non-monochromatic Mg and Al Kα X-ray lines as irradiation sources. Since the area irradiated by non-monochromatic X-ray sources is typically larger than the acceptance area of the spectrometer, the measured area equals the latter. All samples were of the square shape with the surface area of 10×10 mm2, which secured that the measured area was encompassed by that of the sample surface. When measuring smaller samples with Epass = 80 eV, traces of copper from the sample holder were also observed. Knowing that even small amount of impurities affect the line intensities, particularly those of lower kinetic energies (due to the reduced attenuation length), perfect sample cleanliness is essential for these measurements. The photoionization cross sections and the asymmetry parameters were taken from [21] and [22], respectively. The value for quantitative attenuation length, obtained from the NIST database [23], was used for λ in order to properly take into account the elastic scattering of photoelectrons. The inelastic mean free path, used for the calculation of λ, was determined from the optical data available in the database. All parameters employed in calculations of the transmission function from the line intensities are summarized in Table 1. For each thoroughly cleaned sample, the characteristic lines stated in Table 1 were measured. Their intensity was determined as the area below the line after subtracting Shirley background. The ratio between the transmission functions at the corresponding energies was then obtained using eq. (1). The next step was a normalization of the data obtained for different samples, as already discussed in the introduction, so that T(968 eV) = 1. To the best of our knowledge, this procedure is not specified in the literature. We adopted the following algorithm: 

For each sample ‘i’, the measured transmission function was fitted to the exponential dependence Ti(E) = AiE-ni (being typical for T(E) [13]);



A normalization factor NFi, determined as NFi = Ti(968 eV), was attributed to each sample;

12 

The measured values of the transmission function in all samples were divided by the corresponding normalization factors NFi;



All (previously normalized) measured values of the transmission function were then fitted to a unique analytical expression T(E) = AE-n.

The measured values of the normalized transmission function for the pass energies of 20, 40 and 80 eV are presented in Fig. 3. The points corresponding to the 20 and 40 eV pass energies were fitted to the power law, yielding in very similar values for the coefficient n. The two curves, also shown in Fig. 3, pass through the point (968 eV, 1) as expected. In the case of the 80 eV pass energy, the situation is quite different: the points are scattered and the energy dependence does not appear to follow the power law. For this reason, the value of T(968 eV), which is necessary for normalizing the data, was not obtained from the fit, but from a linear interpolation of the neighbouring measured values. The unexpected shape of T(E) for Epass = 80 eV can hardly be due to the measurement error, since the signal to noise ratio of these measurements is actually the highest. It is more likely that the voltages in the electron-optical column do not provide optimal collection of the emitted electrons. Another point that should be stressed is a rather high relative error for some of the experimental points. The experimental error was estimated from the statistical noise, determined assuming the count rate obeys the Poisson distribution. Therefore, the points with particularly high error are those having small intensity and/or rather low ratios between the line intensity and its background. While the former problem can be overcome by increasing the acquisition time, the latter cannot be avoided. It should be also stressed that the measured areas of the small intensity peaks considerably depend on the chosen background and the range in which the integration has been performed. The choice of these parameters is somewhat arbitrary, and therefore strongly depends on the operator’s knowledge and experience. All this, together with the quite high experimental errors of some points, huge sensitivity on the sample cleanliness and necessity of working with several samples, disfavour employing the first principles method for transmission function measurement. This was our motivation to search for an alternative.

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4.2 The sample-biasing method 4.2.1 Method description As already discussed in the Introduction, we express the intensity of a photoelectron line measured from a monoatomic sample as Si = Jx·c·σi·Li·λi·T(Ei), where Jx is the X-ray intensity and c is atomic concentration. Biasing the sample to a positive voltage U should not affect any of these quantities, apart from the transmission function. If we assume that the latter is only shifted in energy to T(Ei – e·U), the transmission function can be determined by following the line intensity as a function of the biasing voltage, as pointed out by Ebel and coworkers [14]. An obvious problem with this approach is that the biasing will also modify electron trajectories in the vicinity of the sample surface due to the additional electrical field between the sample and the entrance of the electron-optical column. Consequently, the signal intensity will not be affected only by the change of the transmission function. In order to minimize this effect on the measurement of T(Ei – e·U), Ebel and coworkers were keeping the biasing very small with respect to the electron kinetic energy. However, this approach is highly questionable even for small biasing voltages, since the change of the transmission function from T(Ei) to T(Ei – e·U) should also be very small. Besides, neglecting the modification of trajectories due to the biasing is surely incorrect when applying higher biasing voltages, which is our aim. The influence of the biasing voltage on the transmission function measurement can actually be estimated. Moreover, this effect can be determined experimentally. The electron-optical column of the XPS spectrometer and the trajectories of photoelectrons that reach the detector are schematically presented in Fig. 4. These electrons will enter the energy analyzer passing through the entrance slit of width 2rs. The angle between their trajectories and the optical axis of the analyzer will not exceed the analyzer opening angle 2αs in the entrance slit plane. Here we assume that the analyzer opening angles are the same in the dispersive (αs) and non-dispersive (βs) planes, αs = βs, which is not generally the case. Electrons reaching the detector are emitted from the acceptance area A in the sample surface plane, into the acceptance solid angle Ω. Apart from the electron deceleration, the main role of the electron-optical column is to transfer the electrons emitted

14 from the sample surface into the energy analyzer, i.e., to focus the beam in the entrance slit plane. Therefore, the design of the electron-optical column and the voltages applied on its electrodes strongly influence the sizes of both A and Ω. Finally, the signal intensity is directly proportional to the acceptance solid angle Ω, and to the acceptance area A [11]. Assuming a cylindrical symmetry, the acceptance area is a circle with the radius r0, whilst Ω = 2π·(1 – cosα0), is described by its opening angle α0. Since for α0 << 1 we have Ω µ α02, S(E) will be proportional to (r0·α0)2. At each cross-section of the electron beam along its trajectory from the sample to the analyzer, we may define a range of electron distances from the optical axis r, and a range of angles between their velocities and the optical axis α, which contain all possible trajectories of electrons reaching the detector. At the sample and entrance slit planes, the beam is then characterized by the pair of magnitudes (r0, α0) and (rs, αs), respectively. We shall also be considering the plane at the electronoptical column entrance, located a few cm above the sample, described by the equivalent pair of magnitudes (rc, αc) (see Fig. 4). The entrance into the electron-optical column is on ground potential. Let us now define Ԑ, as the energy of electrons at any point along the optical axis. Photoelectrons with energy Ԑ = Ea at the moment of emission, will have Ԑ = Epass at the entrance slit plane and inside the analyzer. According to the Liouville’s theorem applied to the propagation of charged particles [24], the quantity r·α·Ԑ 1/2 is an invariant of the beam. Since rs and αs are defined by the slit width, the analyzer geometry and the pass energy when the spectrometer is operating in the FAT regime, we may consider them constant for a specified Epass. Consequently, the kinetic energy dependence of the transmission function is related exclusively to the electron-optical column, which transforms the shape of the electron beam cross-section (rc, αc) into (rs, αs). The kinetic energy actually affects the electron beam cross-section at the entrance into the electron-optical column. Moreover, we may write T(Ea)  (rc·αc)2. In standard XPS measurements, the sample is grounded, so that the region between the sample and the electron-optical column is field-free. This case is illustrated in Fig. 4a. By applying the Liouville’s theorem we have that r0·α0·Ea1/2 = rc·αc·Ea1/2 = rs·αs·Epass1/2, so that T(Ea)  (rc·αc)2 = (rs·αs)2·Epass/Ea. The latter implies the theoretically expected dependence T(Ea)  1/Ea (n = 1) [25].

15 Discrepancy from this dependence is common [11-15], mainly due to the beam restriction by the electron-optical system. Apart from the analyzer opening angles αs and βs, we have to consider an angular aperture of the electron-optical system θ, which define the solid angle in which electrons have to be emitted in order to reach the entrance slit. For high kinetic energies the acceptance angle is defined by the analyzer opening angles. α0 is becoming larger as the kinetic energy decreases, but it cannot exceed its maximum value – θ. Therefore, for lower kinetic energies transmission function may become proportional to E-1/2 or even energy independent. When including aberration effects of the spectrometer, this complex behaviour can be very well described by T  E-n shape [26]. Another consequence of the Liouville’s theorem is that S(Ea)  (r0·α0)2 = (rc·αc)2, indicating T(Ea) is directly proportional to A and Ω (as already discussed in the Introduction), i.e. to the line intensity I(Ea). When the sample is biased to a positive voltage U, which is the case illustrated in Fig. 4b, the electrons are decelerated in two steps. Their energy is firstly reduced to Ea – Eb (with Eb = eU) between the sample plane and the entrance into the column. Afterwards, the electron-optical column decelerates electrons from Ea – Eb to Epass at the entrance slit plane. The corresponding transmission function is now T(Ea – Eb), so that rc and αc are changed accordingly. However, the first deceleration will bend the trajectories and prevent some electrons to fall into the (rc, αc) area of the corresponding phase space. Hence, the magnitudes of r0 and α0 will be reduced to the respective values r0’ and α0’, due to the first deceleration step. When the sample is biased, we may write r0’·α0’·Ea1/2 = rc·αc·(Ea – Eb)1/2 = rs·αs·Epass1/2. Hence, the intensity of detected electrons Sa(Ea – Eb) is proportional to (rc·αc)2·(Ea – Eb)/Ea. Since T(Ea Eb)  (rc·αc)2, the ratio of the line intensities measured from the biased and unbiased sample is: Sa Ea  T Ea  Ea .   Sa Ea  Eb  T Ea  Eb  Ea  Eb

(2)

We denote the second factor on the right side of eq. (2) a correction function, ξ:

 Eb , Ea  Eb   1 

Eb . Ea  Eb

(3)

16 Here demonstrated analysis is performed for specified pass energy. Clearly, rs, αs, rc, αc, r0, α0, r0’, α0’ and consequently the transmission function, will depend on Epass. Once the correction function is known, T(E) can be determined from measurements according to the following expression: T Ea  Eb  

Sa Ea  Eb   T Ea    Eb , Ea  Eb  . Sa Ea 

(4)

It should be stressed that the derived expression for the correction function is valid only when the entrance into the electron-optical system is on the ground potential. Although derived for the perfect cylindrical symmetry, the expression (3) should be also valid for more complicated shapes of the entrance slit and the acceptance area, as well as for αs ≠ βs, as long as the electron-optical system is cylindrically symmetric and does not restrict the beam. Cylindrical symmetry of the lens system secures decoupling of the 2D phase spaces attributed to different azimuths due to the absence of the circular field component. Under these circumstances, for any azimuth angle Liouville’s theorem actually states that the area containing all beam particles in (r, α·Ԑ1/2) phase space is constant along the beam. Another simplification introduced is invariance of r·α·Ԑ1/2 along the beam, which implies that the (r, α·Ԑ1/2) phase space area is rectangular. In the case of the sample plane, this assumption means that the acceptance solid angle Ω is considered to be constant in each point of the acceptance area A. However, this is only an approximation [27]. Since the assumptions used to derive eq. (3) are not generally fulfilled, it is necessary to experimentally determine the correction function. The correction function ξ(Eb, Ea – Eb) can be measured in a way similar to the first principles method for the transmission function measurement. Let us consider two photoelectron lines of the same element at energies Ea and Ek, taken from a clean monoatomic surface. The intensity ratio can be described by eq. (1), since the elemental concentrations will cancel. When the sample is biased to a voltage providing a shift of line ‘a’ to the original position of line ‘k’ (Ek = Ea – Eb), the intensity of line ‘a’ will be Sa(Ek) = Jx·c·σa·λa·La·T(Ek)·ξ(Eb, Ek). In this case, the intensity ratio of lines ‘a’ and ‘k’ can be simply written as

17 Sa Ek  σ a  La  λa   ξ Eb ,Ek  , Sk Ek  σ k  Lk  λk

(5)

since the transmission function term T(Ek), originally present in both the numerator and denominator, is cancelled. By measuring intensities of different lines taken from the same monoatomic sample and the known σ, L, and λ magnitudes of each line, ξ(Eb, Ek) can be determined for several biasing voltages, corresponding to the energy difference between the photoelectron lines of the element. Of course, the correction function is attributed only to the line ‘a’. Plenty of advantages with respect to the first principles method appear right away. First of all, the measurement can be theoretically performed with a single sample, which drastically shortens the time needed for the complete measurement, and reduces the dispersion of the results. Moreover, a single line is used for the T(E) evaluation, thus avoiding potential problems with the intense background (and therefore noise), but also with the correct choice of the integration range and the background type – since being kept constant, they should not influence the result. It should also be stressed that the surface purity is not essential for the T(E) measurement as long as the sample is laterally uniform, since we always measure the same photoelectron line. Finally, the density of points can be arbitrarily high. Although all the problems already stated in the case of the first principles method will appear when determining the correction function, these measurements should be typically performed for one sample only. Besides, we expect that small fluctuations of T(E) should not affect significantly the correction function. Therefore, ξ should be determined rarely (ideally only once), assuming the transmission measurements are performed always for the same distance between the sample surface and the entrance into the electron-optical column if the position of other objects inside the chamber is not changed. At the same time, the transmission function can be periodically measured, using the same correction function, on a sample which does not have to be perfectly clean.

4.2.2 Measurement of the transmission function

18 Transmission function measurement with the sample-biasing method was performed on a clean gold sample. In order to minimize the uncertainty of the measurement, its most intense line Au 4f7/2, was used, having the kinetic energy of about 1170 eV when excited by MgKα line. The correction function was determined from eq. (5), by sample biasing with voltages that shifted this line to the original positions of Au 4d, Au 4p3/2, Au 4p1/2 and Au 4s lines. This provided the data for the biasing voltages up to about 680 V. In order to increase the range further, a clean copper sample was used. The sample was biased to about 850 V, providing a shift of the Cu 3p line to the original position of the Cu 2p3/2 line. Since the position of the Cu 3p line is shifted for only 5 eV with respect to Au 4f7/2 line, we consider that the measurement made with copper fits very well to the ξ(1170 eV, 1170 eV – eU) for the main gold line. The used magnitudes of the photoionization cross sections, the asymmetry parameters and the quantitative attenuation lengths are shown in Table 1. As in the case of the first principles method, thorough sputter cleaning of samples was essential to provide reliable results. The measured values of the correction function for 20, 40 and 80 eV pass energies in low magnification mode are presented in Fig. 5, as a function of the ratio between the bias voltage and the kinetic energy. The results can be readily fitted to a linear dependence for lower pass energies, as suggested by eq. (3), while a somewhat worse linear fit was achieved for Epass = 80 eV. The slope of ξ(Eb, Ek) changes with Epass, implying that the conditions under which the expression (3) was derived are not fulfilled. This is actually expected, since none of the measured transmission functions are proportional to 1/E. The latter implies that the beam trajectories inside the optical system are most probably restricted by its optical aperture, as already discussed in the section 4.2.1. The restriction will be more pronounced when the angular magnification of the optical system is smaller i.e. for the larger is Epass/E ratio. Indeed, the discrepancy from the derived expression (3) increases with the pass energy, as can be seen from Fig. 5. Once ξ(Eb, Ek) is determined, the sample-biasing method based on eq. (4) was employed to measure the transmission function in the kinetic energy range from 313 to 1163 eV. The obtained results, shown in Fig. 6, are normalized so that T(968 eV) = 1. Transmission functions for 20 and 40 eV pass energies can be readily fitted to the power low, which is not the case with Epass = 80 eV. The

19 discrepancy of the transmission function from the 1/E behaviour increases with the pass energy, as in the case of the correction function. This further supports assumption that the discrepancy originates from the restriction of the beam trajectories in the optical system. The obtained results are very close to the ones obtained using the first principles method. However, as we have already anticipated, the measurement uncertainty is now much less pronounced, and the higher density of points further strengthens the reliability of the obtained data. By comparing the results given in Figs. 3 and 5, it can be concluded that the biasing method not only produces a reliable measurement of the transmission function, but is in fact superior to the commonly used first principles method.

5. Optimization of the Transmission Function The transmission function of the spectrometer can be divided in two contributions, corresponding to the transmissions of the energy analyzer and the electron-optical column. While the former cannot be improved, since it is defined by the pass energy and the slit widths, the latter is a function of the geometry of electrodes and the voltages applied, and can generally be adjusted. For a known geometry, a theoretical approach for the optimization of the electrodes’ voltages can be performed using electron optics software (such as SIMION) and automatic optimization procedures of the electron trajectories [28]. It is not possible to apply this method to our system, as the detailed geometry of the optical column is not available from the manufacturer. Therefore, another approach is necessary, based on experimental measurements. By individually varying the voltages of each electrode and measuring the magnitude of the output signal, we can search for a set of voltages that maximizes the signal. However, since there are 5 independent voltages in total and the voltage span is large, this approach generates so many combinations that an algorithm has to be employed in order to find an optimal voltage set in a timely manner.

20 Evolutionary algorithms are a type of heuristic optimization procedures, inspired by biological evolution processes, initially introduced by Rochenberg to optimize experimental parameters of scientific instruments [29]. We propose the use of Differential Evolution, developed by Storn and Price [17], a type of evolutionary algorithm widely used in scientific and engineering applications [30, 31]. It can be easily implemented due to the small number of control parameters and a very simple algorithm, which exhibits a fast conversion capable of handling most types of functions (nondifferential and non-linear), regardless of the initial conditions.

5.1 Description of the optimization algorithm Consider an arbitrary real function of m variables f: Rm → R (R represents the set of real numbers), also called the fitness function, which is to be maximized. In our case f is the count rate, being a function of m voltages. Any set of m voltages which determine f can be represented as an mdimensional vector. We can randomly create a set of NP m-dimensional vectors, where NP is a predetermined number of vectors that form a population. Each vector in the frame of the population is a candidate for the global maximum of the function f. Apart from NP, two additional parameters has to be defined: differential weight F  [0, 2], and crossover probability CR  [0, 1]. The optimization procedure, applied to each population member z = (z1, z2, … zm), then runs as follows. 1. Mutation phase. Three distinct vectors a, b and c, which are members of the population and different from z, are randomly chosen and used to form another vector x = a + F×(b – c). 2. Crossover phase. A random integer N  [1, m] is generated, as well as a set of m random real numbers ri  [0, 1]. Then, another vector z’ = (z’1, z’2, … z’m) is formed from z and x in the following manner: if ri > CR and i ≠ N, z’i = xi; otherwise, z’i = zi. 3. Selection phase. Function f is evaluated for vectors z and z’. If f(z’) > f(z), vector z is replaced by z’; otherwise, z is kept. By applying the steps 1-3 through all population members, we create a new generation (new set of NP vectors) from the previous population. This cycle is repeated until a fitness criterion, closely related to the convergence of the obtained maximum value of the function f, is satisfied by one of the

21 population members. Alternatively, if a pre-determined number of cycles (iterations) is reached, the best population member is chosen as the optimal. A proper choice of values for the control parameters (NP, F and CR) can have a large impact on the method’s performance. In our case these parameters were chosen according to the corresponding meta-optimization analysis, performed by Pedersen [32]. We stress here that the lack of any knowledge concerning the shape of the fitness function f(z), which is the case with the transmission function of the electron-optical column, does not affect the efficiency of the algorithm. This fact, together with the great simplicity of the algorithm, makes the method highly versatile and particularly suitable for automatized optimization.

5.2 Optimization of the transmission function The optimization of the electron-optical column transmission function consisted of the count rate maximization in the whole range of kinetic energies relevant for XPS analysis. However, two additional conditions had to be fulfilled: a) increase of the count rate should not be due to the background increase; b) the resolution had to be preserved. In order to fulfil both conditions, the fitness function was defined as the difference between the count rate of a photoelectron peak’s maximum, and its base (≤ 5% of the peak’s intensity). Having this in mind, and the fact that three channeltrons are used to detect electrons with different energies, the optimization has been performed using a single detector, as evaluating the fitness function for two more energies would greatly increase the total time necessary for the optimization procedure. The schematic of the electron-optical column is presented in figure 7. The optimization has been performed for the low magnification mode, in which electrodes 1 and 2 are grounded. Therefore, the transmission of the electron-optical column is defined by the voltages of electrodes 3-6 (electrodes 3 and 4 are on the same potential): V3+4, V5 and V6. Consequently, each population member is a 3dimensional vector. The values of the parameters NP, F and CR were chosen according to the recommendation from [32] as 13, 0.9096 and 0.745, respectively. The adopted convergence criterion

22 was that the fluctuation of the fitness function between subsequent iterations is in the frame of the noise level, the latter being evaluated under the assumption the count rate follows the Poisson distribution. The above algorithm for maximizing the transmission function was implemented in LabView, allowing control of the voltages and fitness function evaluation. The time required for optimization is governed by the acquisition time for each measurement of the fitness function and the time necessary for changing the electrode voltages. The latter is significantly longer than during the normal spectra acquisition, when the relative changes of the electrode voltages are very small, as opposed to jumping between the (random) values of each population member. For a given photoelectron line, the transmission function optimization was performed automatically for the pass energies of 20, 40 and 80 eV. The optimization took place on a set of photoelectron lines spread along the whole energy range of interest. A sufficiently high density of points is achieved by performing the optimization on several samples. The used samples, with a typical size of 10×10 mm2, and the corresponding photoelectron lines are listed in Table 2. All lines were measured using the Mg Kα line with the exception of Au 4f7/2 line, which was also measured using the Al Kα line. This choice of lines allowed us to perform the optimization in the kinetic energy range of 200-1400 eV. The time necessary for the optimization of one energy depends on the signal intensity and, consequently, on the FAT mode. Since we needed 10-50 min for the optimization of single energy, full optimization in one FAT mode lasted around 10 h in average. Energy dependencies of the original and optimized electrode voltages for Epass = 40 eV are presented in Fig. 8. Very similar results were obtained for the other two pass energies. Unlike the original voltages, all optimized ones are characterized by a linear dependence with the electron kinetic energy. V6 remains practically unchanged by the optimization, and the slope of V3+4 was increased to a value closer to V6. V5 changed more dramatically, to a high energy independent positive value. The ‘focusing strategy’ for the two sets of voltages appears to be the same, i.e., the electrons are initially decelerated (V3+4 < 0), then accelerated by V5, and finally decelerated towards the pass energy. The

23 main difference between the two is in the middle stage, which now provides a much faster acceleration, due to the high value of V5. The optimization result is summarized in Fig. 9. We define the optimization factor as the ratio between the line intensities after and before the optimization, Tnew/Told. The optimization factor, calculated before the normalization of the transmission function, is given in Fig 9a. For pass energies of 20 and 40 eV, the signal was improved in the range of plus 10–50 %, and the optimization factor vs. the kinetic energy is practically of the same shape, monotonically decreasing with the kinetic energy. The situation with the 80 eV pass energy is quite different: the optimization provided much higher signal increase, yielding in the optimization factor in the range of plus 125–210 %. The improvement is particularly high at both ends of the kinetic energy range. While the general trend of T(E) is expected to be preserved for lower pass energies, a significant change should appear for Epass = 80 eV. This was clearly confirmed by T(E) measurement of the optimized system using the modified biasing method, shown in Fig. 9b. All three transmission functions have the same shape after the optimization, following the power law. For lower pass energies the transmission functions practically overlap with the power constant n ≈ 0.985, whilst a somewhat smaller power constant is obtained for 80 eV pass energy (n = 0.78). The correction functions, presented in Fig. 9c, now have a linear dependence for all three pass energies, with the slope varying from ~1.0 (for 20 and 40 eV pass energies) to 0.88 (for Epass = 80 eV). It should be emphasized that T(E) is practically proportional to 1/E in FAT 20 and 40 modes, whilst the corresponding measured correction functions equal the derived expression (3) in the frame of the experimental error. At the same time, both T(E) and ξ(Eb, Ek) in FAT 80 mode approach the dependencies predicted by the Liouville’s theorem after the optimization. All this suggests that the optimization yielded in reduction or even elimination (for lower pass energies) of the beam restriction inside the electron-optical system. As already mentioned, the optimization can be considered useful only if the quality of XPS spectra is preserved. Therefore, true validation of the optimization can only be established by comparing the photoelectron spectra obtained with the two sets of voltages in the electron-optical column. In Fig. 10 we show XPS spectra of Au 4f and Cu 2p3/2 photoelectron lines taken in FAT 20,

24 40 and 80 modes from gold and copper samples, respectively. Each line was taken using original and optimized voltages, while keeping all other parameters constant (X-ray intensity, dwell time, sample position, etc.). The lines look very similar apart from the increased intensities when the optimized voltage set is used. The line widths w (FWHM) and areas of the Cu 2p3/2 and Au 4f7/2 lines are summarized in Table 3. The resolution is preserved while the intensity is increased, clearly confirming the superiority of the optimized voltage set. The reported results show that the optimization procedure yielded in improved transmission function for all pass energies. Furthermore, the optimized transmission functions for all pass energies have now the same shape, described by the power law, which is frequently encountered in the literature [11-15]. It is an open question whether the transmission increase is due to an increase of the acceptance solid angle and/or the acceptance area. In the latter case, a higher increase of the transmission function is expected when performing XPS with non-monochromatic X-ray beam (as it was done here) than in the case of the monochromatic XPS and/or samples of smaller surface area. Nevertheless, since the optimization was performed in the low magnification mode, we were not interested in the lateral resolution of the system. The optimized voltage set is fully applicable for the XPS characterization of samples having large area, which was our main goal. These results can be considered as a confirmation that the proposed optimization method can be readily applied on any XPS system. The approach can be further extended for the transmission optimization in the high magnification mode. One possibility might be to apply the same optimization algorithm, with a set of small samples, having an area comparable to that of the desired acceptance area. Further improvement of the optimization procedure in high magnification mode can be reached by adding two programmable analog outputs to the computer, and connecting them to the high-voltage power supplies of electrodes 1 and 2, which are grounded in low magnification mode. There is however a possibility that introducing two additional variables will greatly increase the number of iterations in the optimization process, and hence significantly prolong the time needed to reach the convergence. Finally, with a good choice of the fitness function, any experimental setup can be optimized in a similar way, as long as the signal is stable enough.

25

6. Summary An upgrade of the XSAM 800 Kratos system was reported, as well as the characterization of its detection system nonlinearity and transmission function. The transmission function was optimized for the three most frequently used pass energies, 20, 40 and 80 eV, in the low magnification mode. Both transmission function determination and optimization have been performed by applying novel procedures: 

The sample-biasing method for determining the transmission function was introduced and successfully applied;



The optimization of the transmission function was performed by applying the differential evolution algorithm to the voltage set of the electron-optical column.

From the experimental investigation of the detection system non-linearity, its dead time was estimated to be about 61 ns. This allows us to make eventual corrections of the measured count rates in future measurements. The superiority of the sample-biasing method with respect to the commonly used first principles method was demonstrated in terms of the measurement error, simplicity and execution time. It was also shown that the experimentally determined correction function, which quantifies the effects of electron deceleration in front of the electron-optical column, perfectly fits the derived expression in the case of optimized voltages for 20 and 40 eV pass energies. The optimization procedure yielded a gain of 10-50 % in the transmission function in the case of 20 and 40 eV pass energies, and 40-110 % for the 80 eV pass energy. Additionally, in the latter case the shape of the T(E) was significantly modified, and all three transmission functions can now be well described by the power law T µ E-n, with n equal 0.99, 0.98 and 0.78 for the 20, 40 and 80 eV pass energies, respectively.

26

Acknowledgments This work was supported by Fundaҫão para a Ciência e Tecnologia do Ministério da Ciência, Tecnologia e Ensino Superior (FCT/MCTES), under Contract PTDC/FIS-NAN/1154/2014, and the Portuguese Research Grant Pest-UID/FIS/00068/2013 through FCT-MEC. João Trigueiro also acknowledges FCT/MEC for his PhD scholarship.

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27 [9] L. Chen, J. Li, Y. Lin, X. Liu, J. Li, X. Gong, D. Li, Preparation of γ-Fe2O3/ZnFe2O4 nanoparticles by enhancement of surface modification with NaOH, Chemistry Central Journal 8 (2014) 40. [10] S. Ptasinska, A. Stypczynska, T. Nixon, N. Mason, D.V. Klyachko, L. Sanche, X-ray induced damage in DNA monitored by X-ray photoelectron spectroscopy. J. Chem. Phys. 129(6) (2008) 129134. [11] M.P. Seah, A system for the intensity calibration of electron spectrometers. J. Elec.Spec. Rel. Phenom. 71 (1995) 191-204. [12] M.P. Seah, XPS Reference Procedure for the Accurate Intensity Calibration of Electron Spectrometers - Results of a BCR Intercomparison co-sponsored by the VAIMAS SCA TWA., 20 (October 1992), 243–266. (1993) [13] L.T. Weng, G. Vereecke, M.J. Genet, P. Bertrand W.E.E. Stone, Quantitative XPS. Part I: Experimental Determination of the Relative Analyser Transmission Function of Two Different Spectrometers – A Critical Assessment of Various Methods, Parameters Involved and Errors Introduced, Surf. Interf. Anal. 20 (1993) 179-192. [14] H. Ebel, G. Zuba, M.F. Ebel, A modified bias-method for the determination of spectrometer functions, J. Elec. Spec. Rel. Phenom. 31 (1983) 123–130. [15] R.S. Wicks, N.J.C. Ingle, Characterizing the detection system nonlinearity, internal inelastic background, and transmission function of an electron spectrometer for use in x-ray photoelectron spectroscopy, Rev.Sci.Instr. 80 (2009) 053108. [16] N. Mannella, S. Marchesini, A.W. Kay, A. Nambu, T. Gresch, S.-H. Yang, B.S. Mun, J.M. Bussat, A. Rosenhahn, C.S. Fadley, Correction of non-linearity effects in detectors for electron spectroscopy. J.Elec.Spec.Rel.Phenom. 141 (2004) 45-49. [17] R. Storn, K. Price, Differential Evolution – A Simple and Efficient Heuristic for global Optimization over Continuous Spaces, Journal of Global Optimization, 11(4) (1997) 341–359. [18] M. P. Seah, Summary of ISO Standard: I ISO 14976:1998 - Surface Chemical Analysis - Data Transfer Format, Surf. Interf. Anal. 27 (1999) 693-694.

28 [19] R.D. Evans, The Atomic Nucleus, McGraw-Hill, New York, 1955. [20] Seah, 1995, Effective Dead Time in Pulse Counting Systems, Surf. Interf. Anal. 23 (1995) 729732. [21]J.H. Scofield, Hartree-Slater subshell photoionization cross-sections at 1254 and 1487 eV, J. Elec. Spec. Rel. Phenom. 8 (1976) 129-137. [22] J.J. Yeh, I. Lindau, Atomic subshell photoionization cross sections and asymmetry parameters: 1 ≤ Z ≤ 103, Atomic Data and Nuclear Data Tables 32 (1985) 1-155. [23] C.J. Powel, A. Jablonski, NIST Electron Effective-Attenuation-Length Database, ver. 3.1, SRD 82, National Institute of Standards and Technology, Gaithersburg, MD (USA) (2011) [24] S.Jr. Humphries, Charged Particle Beams, first ed., John Wiley&Sons New York, 1990. [25] M.P. Seah, The quantitative analysis of surfaces by XPS: a review, Surface and Interface Analysis, 2 (1980) 222–239. [26] M.P. Seah, G.C. Smith, Quantitative AES and XPS: Determination of the Electron Spectrometer Transmission Function and the Detector Sensitivity Energy Dependencies for the Production of True Electron Emission Spectra in AES and XPS, Surf. Interf. Anal. 15 (1990) 751-766 [27] N. Bundaleski, Z. Rakočević, I. Terzić, Optical properties of the 127° cylindrical energy analyzer used in LEIS experiments, Nucl.Instr.Meth. B 198 (2002) 208-219. [28] O. Sise, G. Martínez, I. Madesis, A. Laoutaris, A. Dimitriou, M. Fernández-Martín, T.J.M. Zouros, The voltage optimization of a four-element lens used on a hemispherical spectrograph with virtual entry for highest energy resolution, J. Elec. Spec. Rel. Phenom. 211 (2016) 19–31. [29] M. Mitchel, The Introduction to Genetic Algorithms, first ed., A Bradford Book the MIT Press, Cambridge, Massachusetts 1998. [30] S. Das, P.N. Suganthan, Differential evolution: A survey of the state-of-the-art, IEEE Transactions on Evolutionary Computation 15 (2011) 4–31. [31] S. Das, S.S. Mullick, P.N. Suganthan, Recent advances in differential evolution - An updated survey, Swarm and Evolutionary Computation 27 (2016) 1–30.

29 [32] M.E.H. Pedersen, Good parameters for differential evolution, Hvass Laboratories Technical Report no. HL1002. (2010) https://pdfs.semanticscholar.org/48aa/36e1496c56904f9f6dfc15323e0c45e34a4c.pdf

Figure captions Figure 1 – Schematic diagram of the new control and acquisition system developed for the Kratos XSAM 800 spectrometer.

Figure 2 – The measured R/Ix vs. R dependence, fitted to the linear function.

30 Figure 3 – Normalized transmission function determined by the first principles method for pass energies of 20, 40 and 80 eV, fitted to the power law.

Figure 4 – Schematic of the electron beam inside the electron-optical column when the sample is a) grounded and b) biased to a positive voltage U.

31 Figure 5 – Correction function for Au 4f7/2 line and Epass of 20, 40 and 80 eV, fitted to a linear dependence.

Figure 6 – Normalized transmission function determined by the sample-biasing method for pass energies of 20, 40 and 80 eV

32 Figure 7. Scheme of the electron-optical column operating in the low magnification mode

Figure 8. Original and optimized electron-optical column voltages vs. kinetic energy, for Epass = 40 eV.

33 Figure 9. Results of the optimization: a) the ratio Tnew/Told vs. kinetic energy (the lines are a guide for the eye); b) normalized transmission function vs. kinetic energy; c) correction factor due to the sample biasing. The fitting results are shown in b) and c).

34 Figure 10. XPS spectra of Cu 2p3/2 and Au 4f lines taken from the copper and gold samples, respectively, after subtracting the Shirley background. The spectra were taken in FAT 20, 40 and 80 modes, using the original and optimized sets of voltages.

Table 1. Photoionization cross sections [21], asymmetry parameters [22] and quantitative attenuation lengths [23] for photoelectron lines of interest excited by Mg Kα photons. The data corresponding to Al Kα irradiation is also given for gold lines measured using both photon energies (Mg Kα / Al Kα).

Line Au 4f Au 4d Au 4p3/2 Au 4p1/2 Au 4s Ni 3p Ni 3s Ni 2p3/2 Ni 2p1/2 Cu 3p Cu 2p3/2

E (eV) 1168 / 1401 909 / 1142 707 / 940 611 492 / 725 1184 1140 398 381 1179 322

σ (arb.un.) 17.47 / 17.12 15.92 / 19.8 4.55 / 5.89 1.53 1.45 / 1.92 0.0451 0.0168 0.3128 0.1564 0.0501 0.3565

β 1.006 / 1.032 1.165 / 1.241 1.565 / 1.625 1.565 2/2 1.535 2 1.396 1.396 1.548 1.344

λ (nm) 1.378 / 1.585 1.134 / 1.343 0.933 / 1.144 0.844 0.608 / 0.937 1.498 1.437 0.662 0.643 1.678 0.648

Table 2. Photoelectron lines used for the optimization of the transmission function and their kinetic energies. All lines were excited by Mg Kα photons, apart from Au 4f7/2 which was also excited by Al Kα X-ray line.

Sample

Line

Gold

Au 4f7/2

E (eV) 1170 (Mg Kα) 1403 (Al Kα)

Sample Indium TiO2

Line In 3d3/2 Ti 2p3/2

E (eV) 589 795

35 Tantalum Molybdenum Graphite Silver Indium

Ta 4f Mo 3d5/2 C 1s Ag 3d5/2 Ag 3p3/2 Ag 3p1/2 In 3d5/2

1227 1026 969 886 681 650 810

Cobalt Nickel Copper Zinc

O 1s Co 2p3/2 Ni 2p3/2 Cu 2p3/2 Cu 2p1/2 Zn 2p3/2 Zn 2p1/2

724 476 402 322 303 232 209

Table 3. Intensities and FWHMs of the lines taken in different FAT modes using original and optimized voltage sets.

Line Au 4f7/2 Cu 2p3/2

Voltage set original optimized original optimized

Epass = 20 eV w (eV) area 0.99 3097 0.98 3558 1.27 6518 1.26 9381

Epass = 40 eV w (eV) area 1.06 12553 1.06 14713 1.27 25566 1.27 37490

Epass = 80 eV w (eV) area 1.44 35638 1.43 57536 1.56 67385 1.54 122450