REVIEW OF SCIENTIFIC INSTRUMENTS
VOLUME 71, NUMBER 9
SEPTEMBER 2000
A technique for accurate measurements of ion beam current density using a Faraday cup C. E. Sosolik,a) A. C. Lavery, E. B. Dahl, and B. H. Cooperb) Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853-2501
共Received 18 April 2000; accepted for publication 22 May 2000兲 We have performed measurements of the spatial distribution of current in various alkali and reactive ion beams over the energy range 5–600 eV using a Faraday cup. Ion beam current densities have been extracted from these measurements using a simple deconvolution procedure. Our results reveal that the beams are Gaussian in shape with a constant width, , for energies greater than approximately 75 eV and for all ion species investigated. This width is consistent with that determined from the distribution of oxygen on a Cu共001兲 crystal after an O⫹ ion beam deposition, measured using Auger electron spectroscopy. Using the measurement technique outlined in this article, together with the linear relationship between current density and Faraday cup current, it is possible to determine the beam current density using a single current measurement. © 2000 American Institute of Physics. 关S0034-6748共00兲03309-8兴
I. INTRODUCTION
mental apparatus and the technique used for determining the spatial extent of the beam current. In Sec. III we discuss the results of our current density measurements. This includes a comparison of our deconvolution results for the twodimensional current density, or the beam shape, to a separate measurement that utilized the trapping of O⫹ ions incident on a Cu共001兲 surface. In Sec. IV we show that, for a beam with a well-defined and constant shape, there is a simple, linear relationship between the current density and the measured Faraday cup current, a result that is in agreement with our measurements. The deconvolution procedure used in this work is outlined in the Appendix.
The characterization of hyperthermal and low energy ion beams is important in a variety of surface analysis techniques. For instance, depth-profiling studies and scattering cross-section and sputter yield measurements require accurate, and repeatable, determinations of total beam current and current density. The techniques available for determining the current density of an ion beam vary widely, from timeresolved measurements with microchannel plates to methods which rely on beam-induced surface damage.1–4 In this article we focus on the use of a Faraday cup to accurately determine the current density of hyperthermal energy ion beams. Faraday cups are used routinely to measure beam current in experiments that employ ion beams. The main difficulty in extracting the ion beam current density from a Faraday cup current measurement is that, typically, the width of the ion beam is similar in size to the radius of the aperture in the Faraday cup. Consequently a Faraday cup current measurement is an integration of the beam current density over a finite, and significant, area. We show here that by using a Faraday cup of known dimensions and by making careful measurements of the spatial distribution of the beam current, one can easily extract the beam current density through a straightforward deconvolution procedure. Furthermore, we show that after assembling a relatively small number of these current density measurements, one can reproducibly extract information on beam current density from just a single current measurement. The current densities measured in this study have spatial widths which are independent of the ion species and constant for energies above approximately 75 eV. Below this energy, divergence due to spacecharge repulsion becomes progressively more important in determining the spatial width of the current density. In Sec. II we briefly describe the details of our experi-
II. EXPERIMENT
The beamline and ultrahigh vacuum 共UHV兲 chamber used to make these measurements have been described in detail elsewhere.5–7 Here we outline only the components necessary for this discussion. All ions were produced in a Colutron ion source.8 Noble gas and reactive ion beams were created by introducing gas into the source and using electrons from a resistively heated thoria-coated iridium filament for ionization. In particular, O⫹ and O⫹ 2 ions were produced using a gas mixture of 15% O2 and 85% Ne, and Ar⫹ and Ne⫹ were obtained using pure Ar and Ne gas, respectively. The source has also been modified to allow highly efficient alkali ion beam production from a solid state source.9 In this article, we will present results for the alkali beams Na⫹ and K⫹ only. The beams and energies used in this work are listed in Table I. Ions produced in the source are accelerated for transport through the beamline and pass first through a Wien filter for mass selection followed by a 90° sector spherical electrostatic analyzer or monochromator. The monochromator filters neutral particles from the beam and determines the energy resolution at ⌬E/E ⬇1%. Throughout the beamline, the ion beam is focused and steered by a series of electrostatic lenses and deflectors.6,7 To reduce the effects of space
a兲
Electronic mail:
[email protected] Recently deceased.
b兲
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TABLE I. The ion beam species and energies for which spatial current grids were obtained. Ion beam
Energy 共eV兲
No. of grids
O⫹ O⫹ 2 Ne⫹ Na⫹ K⫹
5–600 400 400 5–612 100
15 1 1 19 1
charge spreading, beams with a final energy below 400 eV are transported through the beamline at 400 eV and then decelerated prior to entering a Faraday cup mounted in the chamber. The three main components of the system, namely, the source, monochromator, and chamber, are separated by pumping impedances and differentially pumped to reduce the gas load in the scattering chamber. Typical pressures when the gas source is operating are 1⫻10⫺5 Torr in the source section, 1⫻10⫺8 Torr in the monochromator section, and less than 1⫻10⫺10 Torr in the scattering chamber. Typical pressures when the solid state source is in operation are 1 ⫻10⫺8 Torr in the source section, 1⫻10⫺10 Torr in the monochromator section, and less than 1⫻10⫺10 Torr in the scattering chamber. All beams were focused into a shielded Faraday cup mounted in the center of the scattering chamber. The Faraday cup, shown in Fig. 1, consists of a faceplate with a 1 mm aperture leading into a copper current trap.10 The trap and faceplate are separated by an aperture which can be biased to prevent secondary electrons produced at the faceplate from reaching the trap. The trap is also enclosed by a shield to keep stray electrons and ions in the chamber from being collected. Current in the trap was measured using a Keithley 617 electrometer.11 Typical beam currents ranged from 0.25 to 2.00 nA. Two-dimensional cross-sectional current profiles of all beams were obtained by sampling the current in the Faraday cup, I(r), over a uniform grid of points r in the plane perpendicular to the beam axis. These grids typically consisted of one large grid of 21⫻21 points 共0.254 mm spacings兲 and one small grid of 21⫻21 points 共0.127 mm spacings兲. The
FIG. 1. A cross-sectional view of the Faraday cup used to obtain the current profiles. The cup is located on a sample manipulator mounted at the center of the UHV chamber.
FIG. 2. 共a兲 A gray-scale plot of the measured current profile of a 50 eV O⫹ beam after linear interpolation has been performed to obtain the profile on a uniform grid. 共b兲 A three-dimensional plot of the current density, j(r), obtained after deconvolution of the current profile shown in 共a兲. The grayscale has an extra white bar to make it easier to see the very low intensity points in 共b兲.
small 21⫻21 grid replaced the inner 11⫻11 points of the large grid, giving a total of 761 grid points over an area of approximately 28.45 mm2 . Linearly interpolating between the remaining points from the large grid, a uniformly spaced grid of 41⫻41 points was obtained, as is shown in Fig. 2共a兲. The current density, j(r), was obtained from the measured current profile using a deconvolution procedure outlined in the Appendix. Figure 2共b兲 shows a typical current density obtained for an O⫹ beam following this procedure. This method of current density determination has been used successfully in recent experiments involving the trapping of hyperthermal Na⫹ and O⫹ beams on Cu共001兲.12–15 For O⫹ beams, it was possible to compare the beam shapes extracted from measured current profiles to Auger
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FIG. 3. Profile of an oxygen distribution on a Cu共001兲 crystal obtained using an Auger electron spectrometer with the transmission energy set on the O共KLL兲 transition at 503 eV. The oxygen was deposited at normal incidence with a 400 eV O⫹ beam. The solid line is a Gaussian fit to the distribution.
electron spectroscopy scans of oxygen deposited on a Cu共001兲 crystal at normal incidence.14,15 The deposited oxygen should retain the shape of the incident current density, j(r), since the amount deposited on the crystal, ⌰(r,t), during any time interval, t, is simply given by16 ⌰ 共 r,t 兲 ⫽ P T 共 E i 兲 j 共 r兲 t,
共1兲
where P T (E i ) is the probability for trapping at a particular incident energy, E i . The intensity of the O共KLL兲 Auger transition at 503 eV is directly proportional to ⌰(r,t). 17 Therefore, by scanning horizontally and vertically across the surface of the Cu共001兲 crystal and monitoring the O共KLL兲 intensity, the shape of the incident beam used during the deposition can be obtained, as shown in Fig. 3. III. RESULTS
All of the deconvolved current densities were Gaussian in shape. Consequently, our analysis involved extracting a width, , from the beam profiles. Following Gilmore and Seah,18 the current density j(r) can be written as j 共 r兲 ⫽
I0 2 2
exp共 ⫺r 2 / 2 兲 ,
共2兲
and the total current within a radius R, obtained from Eq. 共2兲 by integration with respect to r from 0 to R, is I 共 R 兲 ⫽I 0 关 1⫺exp共 ⫺R 2 /2 2 兲兴 .
共3兲
For R⫽0.5 mm, the radius of the Faraday cup aperture, I(R) corresponds to the current measured with the Faraday cup at the center of the beam profile. Taking the extracted current density at the center of the beam, j(0), and dividing it by I(R), we obtain an expression that is dependent only on the width, , of the beam
⫹ FIG. 4. j(0)/I(R) for beams of Na⫹ , O⫹ , Ne⫹ , O⫹ 2 , and K . The beam energies and number of spatial grids obtained are listed in Table I. The decrease in j(0)/I(R) for energies less than approximately 75 eV is due to an increase in the beamwidth from space charge spreading. The dashed line corresponds to Eq. 共4兲 with ⫽0.42 mm.
j共0 兲 ⫽ 兵 2 2 关 1⫺exp共 ⫺R 2 /2 2 兲兴 其 ⫺1 . I共 R 兲
共4兲
In Fig. 4 we have plotted measured values of j(0)/I(R) for a large number of beams in the incident energy range from 5 to 600 eV 共see Table I兲. It is clear that our results are species independent and constant for incident energies greater than approximately 75 eV. Therefore, we conclude that our beams, independent of energy and species, have a constant width above approximately 75 eV. Extracting this width from Eq. 共4兲, we get a value of ⫽0.42⫾0.02 mm. The fact that j(0)/I(R) begins to decrease below approximately 75 eV is evidence that the beam width is increasing at these lower energies due to space charge spreading. This is expected since low energy beams, which are transported through the beamline at 400 eV, are decelerated to their final energy approximately 20 cm in front of the Faraday cup. Space charge spreading begins to play a significant role in determining the final beam width over this short flight path at E i ⬍75 eV.19 For example, for Na⫹ at 50 and 10 eV, the beam current density is still Gaussian in shape; however, space charge spreading increases the beam width at these energies by 29% 共0.54 mm兲 and 76% 共0.74 mm兲, respectively. The beamwidth extracted from this simple analysis was checked against on-sample distributions of oxygen created with a normal incidence O⫹ beam on a Cu共001兲 sample at 100 and 400 eV, such as the one shown in Fig. 3. The widths extracted were 0.46⫾0.02 mm and 0.44⫾0.02 mm, respectively. These values compare well with that extracted from the deconvolved current densities using Eq. 共4兲. IV. DISCUSSION
The results presented in Fig. 4 demonstrate that above approximately 75 eV, the current density of our beams is
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NSF-DMR-9722771兲. Additional support was provided by the Air Force Office of Scientific Research 共AFOSR-910137兲. APPENDIX: DECONVOLUTION OF MEASURED CURRENT PROFILES
FIG. 5. j(0) plotted as a function of the current measured in the Faraday cup, I(R). The dashed line shows the current density one obtains using I(R)/ R 2 .
Gaussian in shape with a constant width. This is rather surprising, since the methods of ion production for the various beams is quite different. One might expect a different shape for the current density of beams extracted from a gas source 共e.g., O⫹ and Ne⫹ ) compared to those extracted from a solid state source 共e.g., Na⫹ and K⫹ ). Instead, our results reveal that the design of our beamline is such that it is possible to produce beams whose current density is independent of species, energy, or method of ion production. Furthermore, because the current density can be described by a Gaussian with a constant width, which makes the left hand side of Eq. 共4兲 a constant, it is clear that we can invert Eq. 共4兲 and use measurements of I(R) to solve for j(0). Figure 5 shows j(0) plotted versus I(R) for the same data that is shown in Fig. 4. Using the clear linear dependence between j(0) and I(R), we see that one can compile a relatively small set of deconvolved current grids, determine the beamwidth 共or equivalently, the slope of the line in Fig. 5兲, and then easily determine j(0) by making one Faraday cup current measurement. This linear dependence was used successfully in the analysis of Refs. 14 and 15 to determine j(0) values without the need for performing a full current grid measurement and deconvolution for every beam used. Also plotted in Fig. 5 is a line corresponding to the current density obtained by taking the measured values of I(R) and dividing by the area of the Faraday cup, R 2 . It is clear that this method gives current densities which are smaller by a factor of three from the true values determined from the deconvolution data. This clearly illustrates the need for using a deconvolution procedure to accurately determine the current density.
ACKNOWLEDGMENTS
The authors would like to thank G. V. Chester, J. R. Hampton, and D. C. Ralph for their help in the preparation of this manuscript. This research was supported by the National Science Foundation 共Grants Nos. NSF-DMR-9313818 and
In order to determine the current density, j(r), we took a large two-dimensional grid of current readings with the Faraday cup, as described in Sec. II. Each current reading taken at position r is a convolution of the Faraday cup detector function, d(r), with the desired current density at r. To obtain the full current density, we must deconvolve out the detector function. This is done using both continuous and finite-length discrete Fourier transforms as well as Wiener Optimized Filtering. The deconvolution process is automated using the computer program MATLAB.20,21 The measured current readings can be written as I 共 r兲 ⫽
冕
d 共 rÀr⬘ 兲 j 共 r⬘ 兲 dr⬘ ,
共A1兲
where the integral is taken over the plane perpendicular to the beam direction. The detector function is d 共 r兲 ⫽
再
1
if r⭐R
0
if r⬎R
R⫽0.5 mm,
共A2兲
where R corresponds to the radius of our Faraday cup. By linearly interpolating between points in the measured grid we obtain a square grid with 41 points on a side that covers an area of approximately 28.45 mm2 . This linearly interpolated grid is padded with zeros to form a larger grid of 137 points on a side. The padding is performed to avoid the creation of artifacts by the processes of convolution and deconvolution. Padding changes our results by only about 1%. Because we have finely spaced grids which fall to zero outside of a finitesized region, we are able to use continuous and finite-length discrete Fourier transforms interchangeably.22 Therefore, using a continuous Fourier transform we obtain the detector function, D(k), from d(r), D 共 k兲 ⫽
冕
d 共 r兲 e ik"rdr,
共A3兲
and using a finite-length discrete Fourier transform, we obtain I(k) from the measured current profile, I(r), where I(k) is written as I 共 k兲 ⫽D 共 k兲 J 共 k兲 .
共A4兲
Dividing both sides of Eq. 共A4兲 by D(0) to eliminate the units of D(k), we obtain I n 共 k兲 ⫽D n 共 k兲 J 共 k兲 , where I n 共 k兲 ⫽I 共 k兲 /D 共 0 兲 共A5兲
and D n 共 k兲 ⫽D 共 k兲 /D 共 0 兲 .
In our case the quantity D(0)⫽ R 2 , and we find, upon taking the Fourier transform of the detector function and divid-
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ing by R 2 , that D n (k)⫽2 J 1 (kR)/kR, where J 1 is a Bessel function of the first kind of order one, and k⫽ 兩 k兩 . This detector function tends to zero at high frequencies, which is a standard result for aperture detectors like our Faraday cup. Therefore, in our measured current grids, the signal predominates over noise at low frequencies. At high frequencies, however, the detector function is still nonzero and finite, so the noise predominates. To filter out unwanted high frequency noise, we perform Wiener optimized filtering, a standard deconvolution method.23,24 To perform Wiener optimized filtering, we begin by rewriting our measured current, I(r), in terms of the signal s(r) and noise n(r) as I 共 r兲 ⫽s 共 r兲 ⫹n 共 r兲 ⫽d 共 r兲 丢 j 共 r兲 ⫹n 共 r兲 ,
共A6兲
where n(r) is independent of s(r), and the symbol 丢 denotes convolution. Also, we define the function n (k), which is the noise to signal power ratio, by
n 共 k兲 ⬅
1
兩 N 共 k兲 兩 2
兩 D 共 0 兲 兩 2 兩 J 共 k兲 兩 2
,
共A7兲
where N(k) is the Fourier transform of the noise n(r). Using n (k), Wiener optimized filtering gives an estimate for J(k) in which the noise has been filtered. Denoting this estimate by ˜J (k), it can be written as ˜J 共 k兲 ⫽
I n 共 k兲 D n 共 k兲 * . 兩 D n 共 k兲 兩 2 ⫹ n 共 k兲
共A8兲
n (k) determines the degree to which the noise is filtered in Eq. 共A8兲. To obtain n (k) it is necessary to estimate 兩 N(k) 兩 2
and 兩 J(k) 兩 2 by doing the following: 共1兲 兩 J(k) 兩 2 is estimated by performing Wiener optimized filtering 关Eq. 共A8兲兴 with n (k)⫽ , where ⫽0.01. This choice of values minimizes high frequency noise in J(k) and retains as much low frequency signal as possible. 共2兲 兩 N(k) 兩 2 is assumed to be a constant. Combining the estimates for 兩 J(k) 兩 2 and 兩 N(k) 兩 2 we obtain n (k). However, by assuming 兩 N(k) 兩 2 to be constant, we only obtain the k dependence of n (k). Therefore, n (k) is normalized to n (k)⫽1 at k⫽0, and Wiener optimized filtering is performed with this normalized n (k) multiplied by a scaling factor between 10⫺2 and 10⫺5 . By definition,
the scaling factor is the noise to signal power ratio at k⫽0. So, by examining an azimuthally averaged power spectrum of our Fourier-transformed current profiles, 具 兩 I n (k) 兩 2 典 , we are able to determine an order of magnitude estimate for the n (k) scaling factor. For Na⫹ and O⫹ beams the scaling factors are 4⫻10⫺4 and 2.5⫻10⫺3 , respectively. Using the estimate of J(k), obtained from Eq. 共A8兲 and the scaled n (k), an inverse Fourier transform is performed to obtain j(r). S. Oswald and T. Nestler, Meas. Sci. Technol. 1, 255 共1990兲. J. B. Wang and Y. L. Wang, Appl. Phys. Lett. 69, 2764 共1996兲. B. Assayag et al., J. Vac. Sci. Technol. B 11, 2420 共1993兲. 4 R. L. Kubena and J. W. Ward, Appl. Phys. Lett. 51, 1960 共1987兲. 5 R. L. McEachern et al., Rev. Sci. Instrum. 59, 2560 共1988兲. 6 D. L. Adler and B. H. Cooper, Rev. Sci. Instrum. 59, 137 共1988兲. 7 D. L. Adler, B. H. Cooper, and D. R. Peale, J. Vac. Sci. Technol. A 6, 804 共1988兲. 8 M. Menzinger and L. Wa˚hlin, Rev. Sci. Instrum. 40, 102 共1969兲. 9 D. R. Peale, D. L. Adler, B. R. Litt, and B. H. Cooper, Rev. Sci. Instrum. 60, 730 共1989兲. 10 D. L. Adler, Ph.D. thesis, Cornell University, 1989. 11 Keithley Instruments, Inc., Cleveland, OH, 44139. 12 D. M. Goodstein, E. B. Dahl, C. A. DiRubio, and B. H. Cooper, Phys. Rev. Lett. 78, 3213 共1997兲. 13 E. B. Dahl, D. M. Goodstein, C. A. DiRubio, and B. H. Cooper, Nucl. Instrum. Methods Phys. Res. B 125, 237 共1997兲. 14 A. C. Lavery, C. E. Sosolik, and B. H. Cooper, Phys. Rev. Lett. 83, 5286 共1999兲. 15 A. C. Lavery, C. E. Sosolik, and B. H. Cooper, Nucl. Instrum. Methods Phys. Res. B 157, 214 共1999兲. 16 Equation 共1兲 is valid in the low coverage limit, where the density of oxygen atoms already trapped on the surface is low enough that they do not yet affect the probability for further trapping. This condition was met in the work of Refs. 14 and 15 and in the present work. 17 D. Briggs and M. P. Seah, Practical Surface Analysis 共Wiley, New York, 1990兲. 18 I. S. Gilmore and M. P. Seah, Surf. Interface Anal. 23, 248 共1995兲. 19 R. G. Wilson and G. R. Brewer, Ion Beams With Applications to Ion Implantation 共Krieger, Malabar, FL, 1973兲. 20 MATLAB Version 4.2c, The Math Works, Inc., Natick, Massachusetts, 1994. 21 To obtain a copy of the linear interpolation and deconvolution routines, contact C. E. Sosolik. 22 E. Oran Brigham, The Fast Fourier Transform and its Applications 共Prentice Hall, Englewood Cliffs, NJ, 1988兲. 23 K. R. Castleman, Digital Image Processing 共Prentice Hall, Englewood Cliffs, NJ, 1996兲. 24 W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. 共Cambridge University Press, Cambridge, 1992兲. 1 2 3
