Effects, Determination, and Correction of Count Rate Nonlinearity in Multi-Channel Analog Electron Detectors
EEffects, Determination, and Correction of Count Rate Nonlinearity in Multi-ChannelAnalog Electron Detectors
T. J. Reber, a) N. C. Plumb, b) J. A. Waugh, and D. S. Dessau Dept. of Physics, University of Colorado, Boulder, 80309-0390,USA (Dated: 24 January 2018)
Detector counting rate nonlinearity, though a known problem, is commonly ignored inthe analysis of angle resolved photoemission spectroscopy where modern multichannelelectron detection schemes using analog intensity scales are used. We focus on anearly ubiquitous “inverse saturation” nonlinearity that makes the spectra falselysharp and beautiful. These artificially enhanced spectra limit accurate quantitativeanalysis of the data, leading to mistaken spectral weights, Fermi energies, and peakwidths. We present a method to rapidly detect and correct for this nonlinearity.This algorithm could be applicable for a wide range of nonlinear systems, beyondphotoemission spectroscopy.PACS numbers: 79.60,74.25 a) Now at Condensed Matter Physics and Material Science Department, Brookhaven National Lab, Upton,NY, 11973 b) Now at Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland a r X i v : . [ phy s i c s . i n s - d e t ] M a y . INTRODUCTION As the technology of angle resolved photoemission spectroscopy (ARPES) continuesto advance and new discoveries are made, one must be sure to eliminate all experimentalartifacts from the data. One well-known but commonly ignored effect is the detector non-linearity. The nonlinearity of photo-electron detectors used by Scienta, which dominates theARPES field, was first detected by Fadley et al. during a multi-atom resonant photoemis-sion spectroscopy (MARPES) experiment . As angle resolved photoemission spectroscopy(ARPES) is in a fundamentally different regime, such nonlinearity has generally been ig-nored, with a few exceptions . However, we find the effects of the nonlinearity, thoughsubtle, are pernicious and must be compensated before any analysis beyond the most rudi-mentary can be trusted. Here, we present the first detailed discussion of the effects of thisnonlinearity as well as a new method to quickly detect and correct for this nonlinearity.Detecting the nonlinearity of an photo-emission spectroscopy setup involves varying thephoton flux (the input) over a wide range while simultaneously plotting the photoelectroncounts (the response). Such a plot will be linear if the system has a linear response . Sucha seemingly simple test is actually difficult or impossible in most ARPES systems becausethe light sources that most of them utilize either can not be easily varied over a very widerange, do not have a perfectly calibrated photon counter, or both. Even if such tools areavailable, this characterization is considered a time consuming endeavor and so it is rarelycarried out.Figure 1 briefly presents our new technique for detecting the nonlinearity, which will workin a multichannel setup such as a camera-based MCP/phosphour screen detection setup with“ADC or Gray Scale” analog intensity schemes for signal intensity. Two measurements ofa spectrum with high dynamic range (such as a dispersive peak crossing a Fermi edge) aremade back-to-back in time, with the only change being an alteration of the incident photonflux. The absolute ratio of the photon flux is not critical, though we typically use a ratioof approximately 2. We then go through each of the two images pixel-by-pixel, making ascatter plot of the count rate of each pixel on the high count image against that of the lowcount image. These plots would be fully linear for the ideal detector system, though asshown here the commercial systems rarely are.We note that the count rate scales used in figure 1 are counts per binned pixel. To convert2 .521.51.50 H i gh C oun t I m age I n t en s i t y Low Count Image Intensity
Pulse Counting(Regular Saturation) Analog Counting(Inverse Saturation)
FIG. 1.
Comparison of “Normal” and “Inverse” Saturation . Scatter plot of count ratesper pixel of a high photon flux images (vertical) vs. low photon flux images (horizontal) (see textfor an explanation of the units). Most detectors show saturation at high count rates resulting in aflattening when a high count image is plotted versus a low count image at high absolute count rates(panel a). The analog detection schemes in modern multi-channel plate detectors have a non-linearthe unusual effect of enhancement at high count rates causing a steeper slope at high count rates(panel b), i.e. an “inverse” saturation effect. The count rate units of the two plots are in differentunits and so can’t be directly compared. to the total flow of information onto the detector per second we multiply the average countsper pixel in the figures (of order 1-10) by the number of total binned pixels across thedetector (980 in angle and 173 in energy for the plots used here) and the number of framesper second (15) to get a total information flow for these plots of order ten MHz.Figure 1a shows a scatter plot from the pulse counting mode of a Scienta detector, showingsaturation behavior at high relative count rates (upper right part of image). In practice,this saturation effect occurs at such low absolute count rates that the pulse counting modeof the Scienta systems is very rarely used. In its place the analog mode is typically used asthis gives a more linear dependence in the range of count rates that are readily achievable.However, as shown in figure 1b this mode is not fully linear and in fact displays an “inverse”3aturation effect consistent with previous findings using the standard method of detectingnonlinearity (varying the photon flux over a wide dynamic range) . We have observedthis inverse saturation effect in at least 5 individual Scienta detectors, including SES100,SES2002, and R4000 models, as well as on a Specs Phoibos 225 spectrometer. Thus thisappears to be a ubiquitous problem, likely affecting all modern camera-based ARPES setups.This effect may seem minor but can significantly alter the spectra as we will show. Laterwe will use scatter plots of the type shown in figure 1 to accurately correct for the observednonlinearity.
II. SIMULATING THE EFFECTS OF NONLINEARITY
In Fig. 2, we detail the effects of the “inverse” nonlinearity on a simple ARPES spectrum.We show the effect of a linear detector response (blue), and two nonlinear responses: onewith a discrete change in slope (red), which makes the effects more obvious and a smoothlyvarying one (green) which makes the effects less obvious but is closer to what is observed.In 2a we show the nonlinearity of the detector in measured counts vs true counts (as willbe shown later this is not exactly the same as the high count vs low count image). Toelucidate how the actual spectra are affected, we depict the two simple ARPES spectra, alinear one and a continuously nonlinear one, side by side in figure 2b. We assumed a linearbare band and Marginal Fermi liquid peak broadening appropriate for near-optimally dopedcuprate samples . In 2c we show the effects of the nonlinearity on a sample momentumdistribution curve(MDC) . While the deviation from a Lorentzian is obvious in the discretecase, the smoothed one is decently well described by a Lorentzian. Consequently, detectingnonlinearity from a line-shape is difficult. The peak of the Lorentzian does not shift when thenonlinearity is applied, so analysis based on peak locations (e.g. band mapping, dispersions,Re(Σ)) are robust against the nonlinearity (2d). In the case of an asymmetric peak inmomentum or two overlapping peaks, extracted peak positions could clearly be impactedby the nonlinearities. The peak enhancement also raises the half max level, effectivelynarrowing the peak width. Consequently, these widths, a common measure of the electronscattering rate, can be significantly sharpened by the detector nonlinearity (2e). Howeveras the intensity above E F is rapidly suppressed by the Fermi edge, the distorted nonlinearwidths quickly return to the linear values. This creates a noticeable asymmetry in the4idths that is roughly centered at E F , which could be incorrectly interpreted as electron-hole asymmetry. Finally, the spectral weight, determined by integrating the MDC’s showsa clear enhancement due to the nonlinearity (Fig. 2f). However, with no reference thisenhancement can be hard to detect in a single spectrum.Since the asymmetry and E F drift is an effect of the spectral intensity change at theFermi edge, it is strongly temperature dependent. To illustrate this behavior, we show atemperature dependence of the widths for a simulation of Marginal Fermi Liquid (hyperbolicenergy dependence ) in figure 3a and the corresponding nonlinear ones in 3b. Note thatthe asymmetry is strongest in the coldest sample but the other effect of the nonlinearityis a softening (shifting to higher binding energy) of the width minimum with decreasingtemperature. This softening is unphysical in that the minimum of the scattering rate shouldbe pegged to E F , which can be understood by considering that the allowable phase spacefor decay channels is minimized at the E F . This softening is more easily observed than theasymmetry so it is a clear sign of nonlinearity in the spectra.The temperature dependence of the linear and nonlinear spectral weights show anothersymptom of nonlinearity (Fig. 3c and 3d). Namely the isosbestic point (point of constantspectral weight) for the linear term is centered at E F in energy and half filling in weightas expected for particle conservation. However in the nonlinear case the isosbestic pointdeviates slightly from E F (here it is a very subtle effect) and its filling is less than half ofthe max value so apparent particle conservation is broken. For the simulations already de-scribed we show the temperature dependence of both in Fig. 3e. If these edges were utilizedto determine the experimental E F we would obtain the false appearance of a temperaturedependent E F and minimum width location. Worse, if the reference spectra (say a polycrys-talline Au) had a different count rate than the sample that science was being carried outon (say a superconductor whose gap was being measured), then each spectrum would havedifferent shifts from the true Fermi edge location. Such a drifting Fermi edge calibrationwould have deleterious effects on procedures which require highly accurate determination ofthe experimental E F , particularly gap measurements and spectra where the Fermi functionis divided out in order to extract information about thermally occupied states above E F .5 M DC I n t en s i t y .44.43.42.41.4.39 Momentum (1/¯)403020100 M ea s u r ed C oun t s M DC W i d t h ( / ¯ ) -.04 -.02 0Energy (eV) .42.41.4.39 M o m en t u m ( / ¯ ) -.05 -.04 -.03 -.02 -.01 0Energy (eV)1.2.8.40 S pe c t r a l W e i gh t -.02 -.01 0 .01 .02Energy (eV).45.4.35 M o m en t u m ( / ¯ ) -.12 -.08 -.04 0Energy (eV).45.4.35 (a) (b)(c) (d)(e) (f)Lin NL Smooth NL Discrete Lin Linear NL Normed NL
LinearNonlinear
FIG. 2.
Effects of a Nonlinear Detector on Typical ARPES Spectrum (a)
Exampledetector nonlinearities showing both a smooth (green) and a discrete (red) deviation from a linearresponse (blue) (b)
Spectra before (top) and after (bottom) nonlinearity inclusion (c)
Sample MDCwidths showing that the nonlinearity is one of the few experimental artifacts that make spectrasharper rather than broader (d)
As the nonlinearity is monotonic the peaks remain the peaks, sothe dispersion is unaffected by the nonlinearity (e)
The energy dependence of the MDC widthsshow the narrowing expected below E F , but above E F the falling spectral intensity shifts the entireMDC into the low count linear regime causing the MDC widths to return to the intrinsic value.The resulting asymmetry in the widths should not be confused for true electron-hole asymmetry. (f ) The spectral weights for the linear and nonlinearity spectra, showing that the asymmetricenhancement around the E F results in an apparent shifting of the Fermi energy. M DC W i d t h ( / ¯ ) -.03 -.02 -.01 0 .01.05.04.03.02.01 -.03 -.02 -.01 0 .011.75.5.250 S pe c t r a l W e i gh t -.02 -.01 0 .01 .02Energy (eV)1.75.5.250 -.02 -.01 0 .01 .02-12-10-8-6-4-20 E ne r g y ( m e V ) (a) (b)(c) (d)(e) Lin
30 K 50 K 70 K 90 K 110 K 130 K
Apparent E F Width Minimum
FIG. 3.
Effects on Nonlinearity on Temperature Dependence Studies (a)
Example tem-perature dependence of MDC widths (b)
Temperature dependence of widths after addition ofnonlinearity, showing formation of asymmetry and shifting minimum width (c)
Example of tem-perature dependence of spectral weight with isosbestic point centered at E F and half filling (d) Temperature dependence of spectral weights after addition of nonlinearity showing the isosbesticpoint holds at E F though shifted away from half filling (e) Temperature dependence of the widthminimum and the apparent Fermi edge location after addition of nonlinearity II. CORRECTING FOR THE NONLINEARITY
Now that we have discussed a method to detect the nonlinearity as well as its many effectson the measured spectra, we here discuss a method to process the data so as to remove themajor effects of the nonlinearity. This technique has two implicit assumptions. First, weassume the nonlinearity is uniform across the detector, which is reasonable as the standardmethod of taking data in ARPES involves sweeping the spectrum across the entire detectoreffectively averaging out any inhomogeneity. Second, we assume that the very lowest countsregion is representative of the true counts, which is justified as the slope of the high countvs low count is comparable to the change in the photon flux.We begin with the high count vs. low count scatter plots such as those shown in figure 1.While these are not the actual nonlinearity curves (measured counts vs. true counts) they docontain all the information necessary to extract the nonlinearity correction. To remove thestatistical spread, we fit the high count vs. low count plot (red in Fig. 4b) with a high-ordermonotonic polynomial fit (green) from which the nonlinear correction will be extracted.The algorithm to extract the nonlinearity is composed of two steps which allow us to firstiteratively reach the linear low count regime and then extrapolate back to the underlyingtrue counts. The method is shown schematically in fig 4b. We start with a given point onthe green fit and determine the ratio of measured high counts to the measured low counts,knowing that the actual change in the true counts is the ratio of photon fluxes. Then weshift down the green curve (following the gold arrows) until the high counts now equal theold low count value and again find the ratio of high counts to low counts for that new point.This process is iterated until we enter the linear regime. In the linear regime, the measuredcounts are the true counts, and we know the number of iterations and thus the number offlux ratios we traverse, so it is simple extrapolation back up to find the underlying truecounts for the original high count value. We repeat the process for every high count valueand we can build up the detector’s nonlinear response curve (red in Fig. 4c). The responseclearly deviates from linear (blue).The nonlinearity extraction algorithm is shown in the next few lines.Ω = HC ( x ) LC ( x ) HC ( x ) LC ( x ) HC ( x ) LC ( x ) · · · HC ( x n ) LC ( x n ) (1)Which if we express in terms of the nonlinearity function acting on the original true count8 M ea s u r ed C oun t R a t e ( c t s / s ) H i gh C oun t R a t e ( C t s / s ) K i ne t i c E ne r g y ( e V ) -4 4Angle ((cid:176))-4 4 20151050 True Count Rate (Cts/s) HC LC(a) (b)(c)
FIG. 4.
Extracting the Nonlinearity (a)
High count image and low count image (b)
HighCount vs Low Count scatter plot (red), high order polynomial fit (green) and low count linearextrapolation (blue) and gold arrows tracing the nonlinear extraction method’s iterations. (c)
Extracted nonlinear curve (red) and low count linear extrapolation (blue). A similar plot as panelc has been obtained by varying the photon flux over a wide dynamic range . rate at the x . Ω = N L ( I ) N L ( I/R F ) N L ( I/R F ) N L ( I/R F ) N L ( I/R F ) N L ( I/R F ) · · · N L ( I/R n − F ) N L ( I/R nF ) (2)This can be simplified to: Ω = N L ( I ) N L ( I/R nF ) (3)Since we stop the iteration in the linear regime N L ( I/R nF ) = I/R nF (4)which can be simplified to: Ω = N L ( I ) I/R nF (5)9ince we know the values of Ω, N L ( I ), R F and n , it is simple to extract I . Repeating thisprocedure for each point on the high count vs low count fit, we can extract the full nonlinearcurve. This method is more general than that proposed by Kordyuk et al. as it does notpresume a form for the nonlinearity. In fact this algorithm is general enough to be used infields outside of ARPES that have uniform nonlinearity across a two-dimensional detector.This algorithm does fail when the assumption of linearity in the low count region is notvalid. For instance, if the detector had a quadratic response with no linear dependence thenthe HC/LC ratio would be linear even though the response is not. For an arbitrary power n : HC = N L ( I ) = I n (6)and LC = N L ( IR F ) = I n R nF (7)So HC ( LC ) = R nF ∗ LC (8)Consequently, while the high count vs low count curve may appear linear the slope revealsif the low count linearity assumption is valid or not. For the detectors we’ve studied thatassumption is valid. As the extracted nonlinearity from this method closely matches thatmeasured by the much more laborious flux variation method , we do not expect it to bean inherent error of this new extraction method.Because of the proprietary nature of the detection schemes used in these analyzers it ishard to exactly determine the origin of the observed nonlinearity. However, a few likelycandidates exist. First, phosphor has a well known “inverse saturation” type nonlinearitywith kinetic energy of impacting electrons (necessitating the gamma correction on cathoderay tubes.) It is not unreasonable that the phosphor might have a nonlinear response tothe electron flux as well. Second, the background subtraction or thresholding must be doneto remove the very low signal strengths associated with electronic noise or camera read-out noise - a problem that is compounded by the widely varying signal strengths per eventcoming out of the micro-channel plates. If the thresholding is too aggressive, larger fractionsof signal would be removed from the low count regions than the high count regions, creatinga nonlinear response. Third, the output data from these systems undergoes significantproprietary processing with the built-in software and firmware. This nonlinearity could be10n unforseen consequence of that processing. IV. TESTING THE CORRECTED DATA
One of the simpler tests for the detector nonlinearity is the temperature dependence ofan amorphous gold sample. Amorphous (non-crystalline) gold is an ideal reference whentaking ARPES data. The non-reactive nature of gold makes it resistant to aging, and theamorphous nature averages over all the bands such that the spectra are uniform in anglebut still show the Fermi edge at E F . Consequently, gold is regularly used to correct fordetector inhomogeneity, as well as to empirically determine both the Fermi energy as well asthe resolution of the instrument. Even this simplest of ARPES data manifests the shiftingFermi edges due to the nonlinearity, but after correction with the curve extracted fromBi2212 spectra the Fermi edges no longer show any sort of thermal drift as expected (Fig.5). Furthermore, we show on experimental data the difference between nonlinear and lin-earized Bi2212 results (Fig. 6), showing many unusual features: drifting minimum widths,electron-hole asymmetric widths, low isosbestic points, are all absent or significantly lesspronounced in the linearized data. The remnant oddities are likely due to an imperfectlinearization rather than representative of true features. V. CONCLUSION
While the effects of nonlinearity are greatly mitigated by the procedure outlined above,it is impossible to be completely certain that the nonlinearity is fully removed for the verylow count rate portions of the spectra, which is where the nonlinearity comes into play forthe analog counting modes (fig 1b). The best option to ensure full-linearity for the lowcount portions is to utilize a pulse counting scheme (fig 1a), except that presently availablecommercial schemes for this then suffer from nonlinearity at higher count rates. In thisregard it is helpful to note that as long as the “regular” saturation is not too severe thescheme presented here can also be used to correct for this form of saturation.We have presented a detailed study of the effects of the typical detector counting ratenonlinearity on a simple ARPES spectrum. While studies that have focused on peak posi-11 .7522.752.7482.7462.7442.742 K i ne t i c E ne r g y ( e V ) -10 -5 0 5 10Angle ((cid:176))2.7522.752.7482.7462.7442.742 K i ne t i c E ne r g y ( e V ) -10 -5 0 5 10Angle ((cid:176))
10K 50K 100K 150K
Raw Linearized (a) (b)
FIG. 5.
Linearizing Amorphous Gold
The shifting Fermi edge with temperature from nonlin-earity is evident in amorphous gold and can be corrected with the nonlinearity extraction. Thecurvature in angle is a known effect of straight slits at the entrance of the curved hemisphericalanalyzer, and is readily corrected. tions are almost fully unaffected by this experimental artifact, studies of the peak widthsand spectral weight can be significantly distorted. Additionally, any report whose findingis critically sensitive to the accurate determination of E F could be negatively influencedby this detector nonlinearity. We also present a simple method to rapidly detect and thenlargely correct for this counting rate nonlinearity. ACKNOWLEDGMENTS
We thank D. H. Lu and R. G. Moore for help at SSRL and M. Arita and H. Iwasawa atHiSOR. SSRL is operated by the DOE, Office of Basic Energy Sciences. Funding for thisresearch was provided by DOE Grant No. DE-FG02-03ER46066.12 .2.8.40 S pe c t r a l W e i gh t -.02 0 .02 12840 M DC I n t en s i t y .44.42.4.38 Momentum (1/¯) .41.4.39 D i s pe r s i on ( / ¯ ) -.04 -.02 0.03.025.02.015.01 M DC W i d t h ( / ¯ ) -.05 0 .024.02.016.012 -.05 0 .03.025.02.015.01 -.05 01.8.6.4.20 -.04 0Energy (eV) 1.8.6.4.20 -.04 0-.1-.08-.06-.04-.020 E ne r g y ( e V ) .4.3 Momentum (1/¯).4.3 NL(a) (b) (c)(d) (e) (f)(g) (h) (i)Lin
NL Lin NL Lin Norm NL 10 K 30 K 50 K 70 K 90 K 110 K 130 K
FIG. 6.
Effects of Linearizing Data (a)
Example nodal spectra of Bi Sr CaCu O δ beforeand after linearization (b) Effects of linearization on sample MDC (c)
Effects of linearization ondispersion (d)
Effects of linearization on MDC widths (e)
Temperature dependence of raw MDCwidths (f )
Temperature dependence of linearized MDC widths (g)
Effects of linearization onspectral weight (h)
Temperature dependence of raw spectral weights with isosbestic point wellbelow half filling (i)
Linearized spectral weights
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