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Detector non-uniformity in scanning transmission electron microscopy
Ultramicroscopy 124 (2013) 52–60
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Detector non-uniformity in scanning transmission electron microscopy S.D. Findlay a, J.M. LeBeau b,n a b
School of Physics, Monash University, Victoria 3800, Australia Department of Materials Science and Engineering, North Carolina State University, Raleigh North Carolina 27695, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 25 June 2012 Received in revised form 24 August 2012 Accepted 3 September 2012 Available online 11 September 2012
A non-uniform response across scanning transmission electron microscope annular detectors has been found experimentally, but is seldom incorporated into simulations. Through case study simulations, we establish the nature and scale of the discrepancies which may arise from failing to account for detector non-uniformity. If standard detectors are used at long camera lengths such that the detector is within or near to the bright field region, we find errors in contrast of the order of 10%, sufficiently small for qualitative work but non-trivial as experiments become more quantitative. In cases where the detector has been characterized in advance, we discuss the detector response normalization and how it may be incorporated in simulations. & 2012 Elsevier B.V. All rights reserved.
Keywords: Scanning transmission electron microscopy (STEM) High-angle annular dark field (HAADF) Detector efficiency.
1. Introduction Recent years have seen a rise in the number of atomic-resolution scanning transmission electron microscopy (STEM) investigations involving quantitative comparisons between experimentally recorded and simulated images [1–19]. This requires some assumptions to be made about the structure being investigated. In addition, several microscope parameters need to be adequately characterized. In one approach to quantification, comparisons are based on relative contrast measures such as peak height ratios and profile shape [1–4,6–8,14,17]. Alternatively, quantitative analyses can be made which measure and compare the absolute signal between experiment and simulation [5,9–12,15,16,18,19]. The former approach reduces somewhat the stringency with which the experimental setup needs to be characterized [2,20]. Relying only on the ratios of values at different points in an image, possibly after ‘‘background’’ subtraction, removes any need to quantify or give meaning to the absolute magnitude of the recorded signal. In a homogenous sample, the parameter most readily determined by the absolute signal is the specimen thickness [11], or, equivalently, a count of the number of atoms present [12]. An absolute scale is often abandoned in quantitative conventional transmission electron microscopy. There it is seen as a small concession when compared to the wealth of information in the conventional images [21]. But in STEM, where the qualitative form of the image often has very little thickness dependence, knowing the thickness on a column-by-column basis allows for determination of
n
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the three-dimensional structure from a two-dimensional image. Quantitative analyses which measure and compare the absolute signal between experiment and simulation achieve this by accurately estimating the number of atoms in a column [12,16]. In both approaches to quantification—relative contrast and absolute intensities—comparison with simulation is essential. Accuracy depends, therefore, on adequate characterization of all aspects of the experimental set-up to which images are sensitive. Such parameters include the probe-forming and detector collection aperture angles and the coherent and incoherent aberrations in the lens. In work on a system that was not aberrationcorrected, image contrast depended strongly on the spatial incoherence [5]. There is some suggestion that reliable quantification is even more challenging in aberration-corrected systems [13,15,22]. It has been shown, for instance, that the uncertainties in the coherent aberration coefficients within the standard tolerances for an ‘‘acceptably’’ aberration-corrected probe are large enough to notably reduce the accuracy of quantification attempts [22]. In this paper we will explore another factor which may limit the accuracy of quantification: deviations from uniformity of the detector response. LeBeau and Stemmer’s approach to calibrating the HAADF signal as a fraction of the total incident probe intensity (or current) involves deflecting the entire probe onto the HAADF detector to record a reference signal in the absence of any intervening specimen [20]. Moreover, LeBeau and Stemmer were able to scan a localized probe across the detector surface and so build up an efficiency map of the detector, as reproduced in Fig. 1. The detector is not uniformly sensitive. Most notably, there is a ‘‘weak patch’’ asymmetrically located on one side of the inner rim. This imperfection is highly directional. The detector response is a consequence of
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Fig. 1. (a) Detector response map in units of volts as measured using a dynamic signal analyzer [20]. (b) Circularly averaged detector response. (c) Circularly averaged detector response in the vicinity of the detector inner angle.
the geometry of the nearly universal scintillator-based ADF detector developed by Kirkland and Thomas [23]. While the central detector hole is generally coated with a reflective metal, the additional reflections reduce the total collection efficiency on the side of the detector opposite the photomultiplier tube. If the scattering distribution was approximately rotationally symmetric then all that would matter is the circular average of the detector response. This is plotted in Fig. 1(b), where we see a significant reduction of efficiency near the inner detector edge. The inner region is magnified in Fig. 1(c) where we have additionally imposed a sharp inner edge cutoff at 65 mrad, the tail in Fig. 1(b) being an artifact of the width of the probe used in the scan across the detector. The weaker signal near the inner edge was compensated for in Refs. [5,9,11,12] by taking the reference signal to be the average of the detector efficiency over an inner portion of the detector response map. Ref. [10] incorporated the circularly averaged detector sensitivity directly into simulations. The success of quantitative comparisons in previous papers suggests that the assumption of rotational symmetry is adequate, at least for high angle scattering. However, an analysis of the influence of this assumption on contrast and absolute signal strength has not
been presented. Expecting similar behavior from the detectors used at lower scattering angles (since they are often the same detectors simply at different camera lengths), there is a further need to explore the impact of a non-uniform detector response on images formed from the rotationally non-symmetric scattering distribution expected in the bright field and low angle dark field region. This seems particularly important given the recent rise in annular bright field (ABF) imaging [16,24–27], which can be implemented using an annular detector, like that in Fig. 1(a), at extended camera length. We will explore this question in more generality than the single efficiency map in Fig. 1. The paper is structured as follows. In Section 2 we explore the sensitivity of STEM signals to the precise values of the detector inner and outer angles, extending in Section 3 to issues of rotationally symmetric detector non-uniformity. Section 4 considers simple cases of what can happen with strong directional non-uniformity of the detector, and also explores using the experimentally measured (non-uniform) detector response (Fig. 1(a)). The effect of the mis-centering of a range of STEM detectors is considered in Section 5. Consistent with the hope in previous work, the effect of modest non-uniformity is small. Nevertheless, these effects can and should be included in simulation if precise quantification is sought.
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Fig. 2. Composite image comparing the scale of diffraction pattern features to the detector response map at different camera lengths. Top left: position averaged convergent beam electron diffraction pattern [30] for the aberration-corrected probe and a 390 A˚ thick sample. Bottom left: convergent beam electron diffraction pattern with the probe positioned above the Sr column for the aberration-corrected probe and a 390 A˚ thick sample. Top right: detector response map at camera length such that it spans the range 65–375 mrad (hard outer cutoff applied). Bottom right: detector response map at camera length such that it spans the range 25–125 mrad (hard outer cutoff applied). The fourth root of the diffraction pattern intensities is plotted in order to make weak features more visible.
As usual in high resolution STEM, the range of possible behavior is sufficiently complex that broad generality of results is difficult to establish. We present results of specific case studies with fair confidence in the generality of the trends that arise, and some hopeful optimism for the wide applicability of their quantitative conclusions in an order-of-magnitude sense. All simulations assume the specimen to be SrTiO3 viewed along the [001] zone axis. Two probes, both for 300 keV electrons, are considered Aberration-corrected: C s ¼ 6 mm, Df ¼ 0 A˚ and a 20 mrad probe-forming aperture semiangle. Uncorrected: Cs ¼1.2 mm, Df ¼ 550 A˚ and a 9.8 mrad probeforming aperture semiangle (corresponding to typical conditions for the STEM instrument at the University of California, Santa Barbara). Here, Cs denotes third order spherical aberration and Df denotes defocus with the convention that overfocus is positive. For a sense of scale, Fig. 2 compares convergent beam electron diffraction patterns using the aberration-corrected probe and a 390 A˚ thick sample with the detector span range for two typical camera lengths.
2. Sensitivity to detector inner and outer angles It is well established that the range of the detector can influence the magnitude and contrast of STEM images [28]. We quantify this here through a SrTiO3 [001] specimen case study and assuming small variations in detector inner angle, binner , and detector span angle, bspan bouter binner . This would apply if the detector angles were not well measured. It also provides a first order approximation to the case described in the introduction where the innermost portion of
the detector has a lower collection efficiency. We consider two quantities. One is the peak intensity on the Sr column, which we denote Ipeak. The other is the contrast, which we denote by C and define via C ¼ ðImax Imin Þ=Imean . (Other definitions of contrast were explored, but their behavior proved to be very similar.) To investigate the dependence of these quantities on binner and bspan , we calculate derivatives. The derivatives of Ipeak and C with respect to binner and bspan are plotted in Fig. 3 as a percentage variation. Plots are given for both case study probes and for two different specimen thicknesses. The angular range in these plots is restricted to the dark field region. The range of the plots has been restricted to maximize the visibility of the variations in the regions of greatest interest—the numerical values on these plots cease to be valid in regions where saturation occurs. The simulations were carried out using the Bloch wave method, with the positive contribution of thermally scattered electrons to the recorded STEM signal included via an effective scattering potential [29]. These calculations involved a 5 5 mesh sampling within the first Brillouin zone and included reciprocal space lattice vectors out to 3 A˚ 1 ð 60 mradÞ. Because we use an effective scattering potential for thermal diffuse scattering, the 3 A˚ 1 limit is only a limit for elastic scattering: there is scattering beyond 60 mrad, but it is assumed to all be thermal diffuse. This is an excellent approximation for the parameters considered here. The projection approximation was assumed, i.e. zero order Laue zone only. All other calculations in this manuscript use the frozen phonon model [31,32]. The frozen phonon simulations used a 1024 1024 pixel mesh for a 19.5 A˚ 19.5 A˚ supercell (i.e. also a 5 5 mesh sampling within the first Brillouin zone). The distinct lattice planes were included in distinct phase grating steps that, to a first order approximation, takes higher order Laue zones into account. We assume uncorrelated vibrational motion, i.e. the Einstein model for phonons. As shown by Muller et al., this is sufficient in most cases [33]. This approach has also been referred to as the ‘‘frozen lattice’’ model, to differentiate it from the ‘‘frozen phonon’’ model which can then be reserved strictly for cases where correlated phonon modes are included. The frozen lattice/phonon model is required for accurate quantitative simulations at larger thickness [5]. However, for the thicknesses considered in this section the Bloch wave model should suffice, and moreover will avoid many of the problems of trying to use a square multislice mesh when attempting to calculate derivatives with respect to scattering angle away from the optical axis. Let us first consider the variation in peak intensity, which proves to be fairly intuitive. For a given inner angle, varying the span angle is equivalent to varying the detector outer angle. Increasing the outer angle (for fixed inner angle) will always increase the peak intensity. However, the scattered intensity drops off with increasing scattering angle and the effect of varying bspan is smaller when bspan is itself already large. Thus, a 1 mrad change in the span of the detector near bspan ¼ 10 mrad produces a variation in Ipeak of around 5% to 10%, whereas for bspan 4 40 mrad, small variations of the span produce no perceptible changes in the signal. This behavior is essentially independent of detector inner angle. Given a fixed outer angle, increasing the inner angle decreases the peak intensity. Increasing the inner angle with a fixed span angle primarily leads to a decrease in peak intensity: more signal drops off the detector for a unit increase in inner angle than is gained by that same unit increase in outer angle. For example, for a fixed bspan ¼ 50 mrad, a 1 mrad change in inner angle in the vicinity of binner ¼ 60 mrad changes the peak intensity by over 3%. Some exceptions to this trend are visible in the plots: for binner ¼ 30 mrad and bspan ¼ 5 mrad, there is a slight increase with increasing inner angle but fixed span angle. This implies that the
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Fig. 3. Plots of the percentage differential variation in peak intensity, Ipeak, and contrast, C, with detector inner angle, binner , and detector span angle, bspan bouter binner , for the two different probes and two different thicknesses. The specimen is SrTiO3 [001]. Probe parameters are given in the body text. A uniform detector response was assumed.
scattering intensity is not monotonically decreasing in this area. That behavior, though, is the exception rather than the rule, and for most combinations the peak intensity reduces with increasing inner angle. Let us now consider the contrast. The variation in contrast for small span angles and small inner angles is quite complicated since coherent interference effects dominate. With that extreme in mind, the dependence of the contrast on the span angle is quite weak, and more so as the inner angle increases. The outer angles contribute little to the signal and so have little effect on either its contrast or its magnitude, as in the peak intensity already discussed. Nearly independent of the span angle, the sensitivity of the contrast to varying the inner angle decreases with increasing inner angle. Thus while the
contrast formed with an inner angle in the vicinity of binner ¼ 30 mrad in the aberration-corrected probe case can vary by a few percent for a 1 mrad variation in detector inner angle, this reduces to less than 1% in the vicinity of binner ¼ 60 mrad (not shown). This is consistent with consensus expectations of the robustness of HAADF imaging: the contrast in STEM images at high scattering angles is largely insensitive to the precise values of inner and outer detector angles. There is no great difference in this behavior between the two probes or the two thicknesses considered. It should be appreciated that the percentages given here are for unit (i.e. 1 mrad) variation of the detector angles. These quantities are rarely measured with such precision. Indeed, with the non-uniform sensitivity of the inner angles of the detector seen in Fig. 1, it is questionable
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whether we could define a fixed inner angle value with 1 mrad precision. For binner t 60 mrad we find the error in the differential peak signal to be 3% to 5%, assuming a 1 mrad deviation. Therefore, we would expect a 6% to 10% error for a 2 mrad deviation. Thus we conclude that using simple simulations as a quantitative reference for experiment could easily lead to errors of the order of 10% in attempting to quantify the peak signal, unless the detector has been carefully characterized [20].
3. Rotationally symmetric non-uniformity Let us suppose that the inner angle in the rotationally averaged detector signal in Fig. 1(c) has been accurately characterized, i.e. it is not a source of uncertainty. Nevertheless, the weaker detector response near the inner edge will have a similar effect to that seen in the previous section: a simulation assuming a uniform response will not correctly describe the slight relative suppression of the intensity of scattering near the detector inner edge. The consequences of this depend on how the quantification of the detector response is used to normalize the STEM signal and incorporated in the simulations. Since characterization of the detector is a prerequisite of this process, the circularly averaged detector response can be included in the simulation (as done by Rosenauer et al. [10]). Indeed, as we discuss later, the full 2D detector response can be included in the simulation. LeBeau and coworkers [5,9,11,12] took the simpler route of selecting a single normalization factor to compare the experimental signal against simulations assuming a detector with uniform response. By way of exploring rotationally symmetric but non-uniform detector response, we will reconsider that approach, justifying it as it was applied in earlier work but pointing out its limitations.
Suppose the electron intensity distribution in the diffraction plane is approximately rotationally symmetric, at least within the region corresponding to the ADF detector. Let us denote the intensity by i(u), where u is the radial distance in the diffraction plane, which it will be convenient to express in units of the angle away from the optical axis. For an annular detector spanning the range from uinner to uouter, and assuming the ideal, unit detector response in the simulation, the STEM signal would be Iideal ¼ 2p
Z
uouter
iðuÞu du:
ð1Þ
uinner
Let D(u) denote the circular average of the measured detector response. However, the scale of the measured detector response is not particularly meaningful and so we will further include a scaling factor Z. Assuming the same inner and outer angles, the measured STEM signal so normalized is thus Imeasured ¼ 2p
1
Z
Z
uouter
iðuÞDðuÞu du:
ð2Þ
uinner
The assumption of LeBeau and coworkers [5,9,11,12] is that, regardless of the exact form of D(u), a single normalization factor Z can be chosen to ensure that Imeasured ¼ Iideal regardless of the specimen or thickness involved. If the recorded detector response was constant (up to noisy fluctuations on a scale finer than that over which the scattered intensity i(u) varies), Z would just be the average detector response. For a non-uniform detector it is less clear that this should be true in general. We claim, however, that for high scattering angles (short camera lengths) Z can be given as the average detector response out to some scattering angle ueffective
Fig. 4. Peak intensity on the Sr column in SrTiO3 [001] as a function of specimen thickness for the two probes considered. Effective probe size as described via convolution with a Gaussian of half-width-half-maximum 0.5 A˚ has been incorporated in these simulations. The ‘‘ideal’’ simulations assume an annular detector, with range as per the row label, with unit response at all points. The remaining simulations use the circular average of the measured detector response (i.e. Fig. 1(b,c)) renormalized as per Eq. (3) for the value of ueffective given on the legend. Symbols are shown when complete overlap with the ideal detector curve occurs.
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(less than uouter) R ueffective 2p uinner DðuÞu du R ueffective : Z¼ u du 2p uinner
ð3Þ
At high scattering angles the rate of decay of i(u) is approximately independent of the position of the probe, since the high angle scattering of all elements tends to the Rutherford-like scattering from the nuclei. The conversion factor Z can thus be determined by using iðuÞp1=u4 Rutherford scattering and the measured detector response D(u) to evaluate and compare and evaluating Eqs. (1) and (2). The value of Z having thus been established, Eq. (3) serves as an implicit definition of ueffective, which, consistent with previous work [20], we deem a more informative value to quote than Z. That ueffective is well-defined by Eq. (3) follows because the measured signal is dominated by the inner region of the detector and D(u) varies monotonically there. Broadly speaking, ueffective measures how much of the detector is contributing significantly to the signal. At camera length such that uinner ¼65 mrad, it turns out that ueffective ¼125 mrad for a circularly symmetric detector with the detector response function shown in Fig. 1(b) and (c), assuming a hard outer aperture is applied at 365 mrad. This is evident in Fig. 4, which compares the ideal detector result, Iideal, with Imeasured for various choices of the normalization as labelled by ueffective values. For the 65–365 mrad detector, ueffective ¼125 mrad is seen to be in excellent agreement with the uniform detector results over the full range of thicknesses for both the aberration-corrected and uncorrected probes. In particular, Fig. 4 shows that comparing a simulation with uniform detector response against an experimental signal from a non-uniform detector where normalization is effected by the average response over the whole detector would produce an error of about 15% in the peak intensity of a 200 A˚ thick sample. (Note: while we have opted in Fig. 4 to renormalize simulations using the measured detector response to those using an ideal detector response, we could equally have used the measured detector response – in the somewhat arbitrary units of volts – and considered various forms of simulation normalization against it. Regardless, the same conclusions would be reached.) For the 25–125 mrad detector range, obtained with the same detector response as per the 65–325 mrad range but assuming an adjusted camera length, ueffective ¼165 mrad gives the better agreement. For these lower scattering angles, the intensity drop off no longer has the high scattering angle 1/u4 distribution, changing the appropriate value of ueffective. Fig. 4 shows that the single factor normalization holds over a wide range of thicknesses and probe sizes provided the camera length is fixed. However, the value of ueffective will vary with camera length, especially if this brings the inner angle into the vicinity of significant elastically scattered intensity. This is to be held against the generality of the single normalization factor approach. Contrast measures are not explored here because they involve a ratio which eliminates any absolute intensity scale. If the real detector response is such that the behavior of the signal is proportional to the idealized case of a uniform detector response, it is not necessary to determine the normalization constant if only contrast or other ratio measures are to be used in comparing simulation and experiment. This is true if the detector inner angle is large enough that the scattering profile within the detector is that of Rutherford scattering, independent of probe position (as seen later in Fig. 7). 4. Detectors without circular symmetry The diffraction pattern is only rotationally symmetric when an electron probe with width smaller than the inter-column spacing
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is precisely centered atop a column in the thinnest of crystals. In the high angle dark field regime, the deviations from rotational symmetry are due primarily to the Kikuchi bands. However, these tend to involve a fairly localized redistribution of electrons. Unless the non-uniform detector response varies on the scale of these features, the high angle scattering tends to average to circular symmetry, justifying the treatment in the previous section. However, depending on the probe location, coherent interference effects can give some distinct directionality to the electron scattering distribution at lower scattering angles. Thus, should we change the camera length such that the inner edge of the detector lies within the bright field or in the low angle dark field region, highly directional detector non-uniformity as evident in Fig. 1(a) may have a notable impact on the form of the STEM signal when compared to idealized simulations assuming detector uniformity. Three circularly non-symmetric detectors are shown in Fig. 5: (a) the experimentally measured detector (truncated to give a 5-to-1 ratio between outer and inner angles), (b) a two-level approximate to the measured detector in (a), and (c) a uniform detector with a missing sector. Fig. 5(c) corresponds loosely to a circular detector partially occluded by the support arm of a circular
Fig. 5. Detector response functions: (a) experimentally measured, (b) two-level approximation to that experimentally measured, (c) uniform with missing a segment, and (d) uniform. (e) Contrast as a function of thickness for a SrTiO3 crystal using the aberration-corrected probe and the four detector response functions above at a camera length such that they span the range 25–125 mrad.
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65–375
65–375
45–225
35–175
25–125
10–20
45–225
Segment Ideal Fig. 6. ABF simulations using the three detectors in the righthand column, derived from Fig. 5(a), (c) and (d), for a camera length such that the inner angle is 10 mrad and applying an aperture such that the outer angle is restricted to 20 mrad. Results are shown for two thicknesses. Each image is split: the top half is the result for perfect spatial coherence, while the bottom half includes the effect of spatial incoherence as ˚ described by convolution with a Gaussian of half-width-half-maximum 0.5 A.
35–175
Detector response
125 Å
Measured
62 Å
properly account for the non-uniform detector response changes not just the quantitative scale but also the qualitative shape of the signal. Contrast is a (possibly overly) simple measure of any such change. Fig. 5(e) plots the contrast as a function of thickness for these four detectors taken at a camera length such that they span the range 25–125 mrad, assuming the aberration-corrected probe and including spatial incoherence. It is seen that, at least by the single measure of contrast, the results for detectors (a) and (b) are very similar to one another, suggesting that the two-level approximation captures the most significant variation in uniformity of the measured detector (for this camera length). Interestingly, the contrast results for detectors (c) and (d) are also very similar to one another. Visual impression from the detector maps might have inclined one to say that detectors (b) and (d) were more similar to one another than (c) and (d), but this is not what is ˚ the contrast difference of found. For a thickness of around 50 A, detectors (a) and (b) with that of (c) and (d) is around 10%. These results imply that if the uniformity of detector response of round detectors is better than that of annular detectors, then implementing ABF via a circular detector and a beam stop might be preferable to using an annular detector at suitable camera length if the detector response was not well characterized. The similarity between the contrast from (c) and (d) would be expected if the scattering distribution was circularly symmetric. However, the single, simple measure of contrast does not give the complete story: images with similar contrast are not automatically similar images. STEM images for this low angle annular dark field situation, however, show perceptible distortion only for ˚ and in the absence of significant spatial thin specimens ð o 50 AÞ incoherence. The effect is a little more pronounced for a camera length such that the detector overlaps the bright field region. Fig. 6 considers the detectors from Fig. 5(a), (c) and (d) (‘‘measured’’, ‘‘segment’’ and ‘‘ideal’’, respectively), at a camera length such that the inner angle is 10 mrad and applying a limiting circular aperture such that the effective outer angle is 20 mrad—this is one approach for implementing ABF imaging using an annular detector already present in the column. In the absence of spatial incoherence (top half of each image) we find that the ‘‘segment’’ detector shows perceptible qualitative changes in the form of the STEM images, while differences between the ‘‘ideal’’ and ‘‘measured’’ detectors are small. The changes all but vanish if significant spatial incoherence effects are present (bottom half).
25–125
beam-stop, a common approach to implementing ABF imaging. For reference, we also consider (d) a circularly symmetric, uniform detector. Henceforth we shall concentrate on the contrast measure. This circumvents the question about how to normalize intensities (though this question must be faced if absolute signals from two different detectors are going to be compared). If the STEM signals from the different detectors were identical up to a factor, the problem of correct normalization is the problem of finding that factor. More fundamental, then, is the possibility that the signals differ by more than just a proportionality constant, that failing to
10–20
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Fig. 7. A comparison of the real detector response, the circularly averaged detector response, and the ideal, uniform detector response for a range of different camera lengths for both the aberration-corrected and uncorrected probes. Spatial incoherence is included. For each camera length (giving the span ranges as labelled in the top ˚ plots in mrad units), the horizontal axis runs from zero to 500 A.
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BF: 0–5 mrad t = 50 Å
8 mrad
Imax=0.06 C=0.34
Imax=0.06 C=0.35
Imax=0.09 C=1.34
Imax=0.08 C=1.07
Imax=0.07 C=0.75
Imax=0.71 C=0.24
Imax=0.61 C=0.20
Imax=0.50 C=0.18
Imax=0.70 C=0.43
Imax=0.60 C=0.34
Imax=0.47 C=0.19
Imax=0.13 C=2.11
Imax=0.14 C=2.06
Imax=0.15 C=1.89
Imax=0.08 C=2.48
Imax=0.08 C=2.48
Imax=0.08 C=2.48
ABF: 10–20 mrad t = 50 Å
BF: 0–5 mrad t = 200 Å
Imax=0.06 C=0.35
ABF: 10–20 mrad t = 200 Å
Though not a deviation from uniformity, a complete characterization of the detector geometry calls for considering the precision of detector centering. Fig. 8 shows simulations of bright field (BF), ABF, low angle annular dark field (LAADF) and HAADF images of SrTiO3 viewed along the [001] axis for the aberration-corrected probe for detectors of the ranges specified in the row labels and with the detector off-centered by the amount shown in the column labels. These images include spatial incoherence via convolution with a Gaussian of 0.5 A˚ half-width-half-maximum. The image maximum intensity, Imax, and image contrast, C, are given beneath each image. The effects on the images are more pronounced for the detectors in the bright field region. One reason for this is that the coherent interference in the bright field region makes for a more rapidly varying diffraction pattern intensity. For the BF and ABF detectors, results are shown for two thicknesses, and the patterns and behaviors vary with thickness due to dynamical channelling effects. But the main reason is that the shifts considered amount to a more significant fraction of the detector span. Consequently, mis-centering of these detectors is more likely to show up as an asymmetry in the images. In the BF and ABF images shown here, the image maximum intensity and the image contrast tend to decrease with increased detector offset, although there are exceptions. For the LAADF detector, the gain in signal in the inner edge of the detector which moves closer to the optical axis tends to slightly outweigh the loss from the inner edge which moves further away: the image maximum intensity increases slightly for an offset LAADF detector. It turns out that this effect is more pronounced for the off-column contribution, and as such we find an appreciable contrast decrease for appreciably off-centered LAADF detectors. We might expect such effects to be true of the HAADF detector, but the magnitude of the effect is greatly reduced: mis-centering the HAADF detector has a very weak effect on both image maximum intensity and, especially, on contrast. (In the absence of spatial incoherence,
4 mrad
LAADF: 30–80 mrad t = 200 Å
5. Off-centred detectors
0 mrad
HAADF: 60–160 mrad t = 200 Å
Fig. 7 compares the contrast as a function of thickness for the real detector response of Fig. 1(a) with its circular average and with an ideal, uniform detector for a number of different camera lengths for both the aberration-corrected and uncorrected probes. The discrepancies are largest for the smaller detector inner angles and also small thicknesses. For the aberration-corrected probe, the discrepancies with the uniform detector are around the 5% level, while those with the circularly averaged detector are below the 3% level. The discrepancies are larger for the uncorrected probe. The effect reduces as the detector inner angle increases. For the aberration-corrected probe at small thickness, the ideal detector in the 35–175 range can be in error by 3%, whereas for the 45–225 case this is down to 1%. The relative error between the real detector and circular average in this range is essentially zero, supporting our earlier assertions of effective rotational symmetry at higher scattering angles given the scale of the detector nonuniformity. Though the specimens considered differ, for ABF imaging with the aberration-corrected probe, errors at the 5% to 10% level occur for the ˚ of previous atom counting by ABF thicknesses range ð 15250 AÞ experiments [16]. Thus, detector non-uniformity can have a significant effect when seeking to quantify ABF images. In addition, it is important to note that rotations of the diffraction information can be introduced by the post specimen projection system, leading to an additional parameter to take into consideration. If the alignment of the experimental CBED pattern relative to detector is not measured for low angle or ABF imaging, then it is advisable to introduce the rotationally averaged detector response in the simulations.
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Fig. 8. BF, ABF, LAADF and HAADF simulations for SrTiO3 using the aberrationcorrected probe and assuming a finite effective source size of 0.5 A˚ half-widthhalf-maximum. The left column assumes perfectly centered detectors, while the other two columns assume off-centering by angular displacements of 4 mrad and 8 mrad, directed down the page.
simulations show a 3% increase in Imax in 400 A˚ of SrTiO3 for the conditions in Fig. 8 for a detector shift of 8 mrad.) Thus precise centering of the HAADF detector, which is challenging, is also unnecessary in most cases. Some of our colleagues have asked us whether modest deviations from ideal centering can lead to an apparent shift in the column locations in simultaneously recorded ABF and HAADF images, and moreover, since these two detectors could be off-centered in different
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directions, whether there might be any apparent relative shift of the column locations between the two modes. Reasons for suspecting this include the asymmetry in the images for significantly misaligned detectors and recent work which has demonstrated small column shifts in the HAADF range for small detectors as a function of off-axis position [34]. However, the present simulations show that for modest detector shifts – ‘‘modest’’ meaning that image distortion is sufficiently small that detector misalignment is not automatically suspected – any relative displacements of column centres are imperceptible.
6. Summary and conclusion Even if the detector response can be assumed to be uniform, an uncertainty in the detector inner angle of a few milliradian could produce a 10% error in the prediction/interpretation of the peak intensity. When detectors with a non-uniform detector response comparable to that seen in Fig. 1(a) are used at camera lengths placing them in high angle dark field region, the intensity can be correctly normalized against simulations which assume a uniform detector response by a judicious choice of normalization factor. For longer camera lengths, such that this detector sits in the lower angle dark field and/or bright field regions, the directional nature of the non-uniformity produces variations in contrast of the order of 5% relative to the assumption on a uniform detector. When in the bright field region, directional non-uniformity of the detector response can moreover lead to visible qualitative variations in the form of the image (especially if the spatial coherence of the effective source is high). The problems discussed here are hardware problems, which can likewise ultimately be overcome through hardware. It has long been appreciated that the ultimate goal in flexible detector geometry is to record the whole two-dimensional diffraction pattern for each probe position [35]. Proof-of-principle experiments collecting such data sets exist in the literature [34], though improvements in high-speed electronic readout remain necessary before this imaging mode becomes routine. Alternatively, as implicit in the simulations presented here, accurate theoretical modeling is always possible for a sufficiently well-characterized detector by including the measured, non-uniform detector response into the simulations. The similarity between the detector response of Rosenauer et al. [10] and that of LeBeau and Stemmer [20] suggests that areas of notably reduced detector response correspond closely to the design geometry of the detector. There is no evidence of significant fluctuations in detector response at arbitrary locations. Nevertheless, we conclude that studies seeking to quantity absolute intensity to within 10% level or contrast to within the 5% level would do well to characterize their detector response and incorporate it into supporting simulations, especially when using annular detectors at lower scattering angles.
Acknowledgments S.D.F. acknowledges support by the Australian Research Council. J.M.L. acknowledges Susanne Stemmer and the UC Santa Barbara
MRL Central Facilities for the detector map. The MRL Central Facilities are supported by the MRSEC Program of the NSF under Award No. DMR 1121053; a member of the NSF-funded Materials Research Facilities Network (www.mrfn.org).
References [1] D.O. Klenov, S. Stemmer, Ultramicroscopy 106 (2006) 889. [2] D.O. Klenov, S.D. Findlay, L.J. Allen, S. Stemmer, Physical Review B 76 (2007) 014111. [3] Z.Y. Li, N.P. Young, M. Di Vece, S. Palomba, R.E. Palmer, A.L. Bleloch, B.C. Curley, R.L. Johnston, J. Jiang, J. Yuan, Nature 451 (2008) 46. [4] K. Kuramochi, K. Suzuki, T. Yamazaki, K. Mitsuishi, K. Furuya, I. Hashimoto, K. Watanabe, Ultramicroscopy 109 (2008) 96. [5] J.M. LeBeau, S.D. Findlay, L.J. Allen, S. Stemmer, Physical Review Letters 100 (2008) 206101. [6] P.D. Robb, A.J. Craven, Ultramicroscopy 109 (2008) 61. [7] S.I. Molina, D.L. Sales, P.L. Galindo, D. Fuster, Y. Gonza´lez, B. Ale´n, L. Gonza´lez, M. Varela, S.J. Pennycook, Ultramicroscopy 109 (2009) 172. [8] H. Inada, J. Wu, J. Wall, D. Su, Y. Zhu, Journal of Electron Microscopy 58 (2009) 111. [9] J.M. LeBeau, S.D. Findlay, X. Wang, A.J. Jacobson, L.J. Allen, S. Stemmer, Physical Review B 79 (2009) 214110. ¨ [10] A. Rosenauer, K. Gries, K. Muller, A. Pretorius, M. Schowalter, A. Avramescu, K. Engl, S. Lutgen, Ultramicroscopy 109 (2009) 1171. [11] J.M. LeBeau, A.J. D’Alfonso, S.D. Findlay, S. Stemmer, L.J. Allen, Physical Review B 80 (2009) 174106. [12] J.M. LeBeau, S.D. Findlay, L.J. Allen, S. Stemmer, Nano Letters 10 (2010) 4405. [13] Y. Kotaka, Ultramicroscopy 110 (2010) 555. [14] Z.W. Wang, Z.Y. Li, S.J. Park, A. Abdela, D. Tang, R.E. Palmer, Physical Review B 84 (2011) 073408. [15] S. Kim, Y. Oshima, H. Sawada, T. Kaneyama, Y. Kondo, M. Takeguchi, Y. Nakayama, Y. Tanishiro, K. Takayanagi, Journal of Electron Microscopy 60 (2011) 109. [16] S. Lee, Y. Oshima, H. Sawada, F. Hosokawa, E. Okunishi, T. Kaneyama, Y. Kondo, S. Niitaka, H. Takagi, Y. Tanishiro, K. Takayanagi, Journal of Applied Physics 109 (2011) 113530. [17] P.D. Robb, M. Finnie, P. Longo, A.J. Craven, Ultramicroscopy 114 (2012) 11. [18] C. Dwyer, C. Maunders, C.L. Zheng, M. Weyland, P.C. Tiemeijer, J. Etheridge, Applied Physics Letters 100 (2012) 191915. [19] H.L Xin, Y. Zhu, D.A. Muller, Microscopy and Microanalysis 18 (2012) 720. [20] J.M. LeBeau, S. Stemmer, Ultramicroscopy 108 (2008) 1653. ¨ ¨ [21] G. Mobus, R. Schweinfest, T. Gemming, T. Wagner, M. Ruhle, Journal of Microscopy 190 (1998) 109. [22] E.J. Kirkland, Ultramicroscopy 111 (2011) 1523. [23] E.J. Kirkland, M.G. Thomas, Ultramicroscopy 62 (1996) 79. [24] S.D. Findlay, N. Shibata, H. Sawada, E. Okunishi, Y. Kondo, T. Yamamoto, Y. Ikuhara, Applied Physics Letters 95 (2009) 191913. [25] G. Radtke, M. Couillard, G.A. Botton, D. Zhu, C.J. Humpreys, Applied Physics Letters 97 (2010) 251901. [26] R. Ishikawa, E. Okunishi, H. Sawada, Y. Kondo, F. Hosokawa, E. Abe, Nature Materials 10 (2011) 278. [27] H. Schmid, E. Okunishi, T. Oikawa, W. Mader, Micron 43 (2012) 49. [28] S. Hillyard, J. Silcox, Ultramicroscopy 58 (1995) 6. [29] L.J. Allen, S.D. Findlay, M.P. Oxley, C.J. Rossouw, Ultramicroscopy 96 (2003) 47. [30] J.M. LeBeau, S.D. Findlay, L.J. Allen, S. Stemmer, Ultramicroscopy 110 (2010) 118. [31] R.F. Loane, P. Xu, J. Silcox, Acta Crystallographica A 47 (1991) 267. [32] E.J. Kirkland, Advanced Computing in Electron Microscopy, Plenum Press, 1998. [33] D.A. Muller, B. Edwards, E.J. Kirkland, J. Silcox, Ultramicroscopy 86 (2001) 371. [34] K. Kimoto, K. Ishizuka, Ultramicroscopy 111 (2011) 1111. [35] J.M. Rodenburg, R.H.T. Bates, Philosophical Transactions of the Royal Society London A 339 (1992) 521.
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Ultramicroscopy journal homepage: www.elsevier.com/locate/ultramic
Detector non-uniformity in scanning transmission electron microscopy S.D. Findlay a, J.M. LeBeau b,n a b
School of Physics, Monash University, Victoria 3800, Australia Department of Materials Science and Engineering, North Carolina State University, Raleigh North Carolina 27695, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 25 June 2012 Received in revised form 24 August 2012 Accepted 3 September 2012 Available online 11 September 2012
A non-uniform response across scanning transmission electron microscope annular detectors has been found experimentally, but is seldom incorporated into simulations. Through case study simulations, we establish the nature and scale of the discrepancies which may arise from failing to account for detector non-uniformity. If standard detectors are used at long camera lengths such that the detector is within or near to the bright field region, we find errors in contrast of the order of 10%, sufficiently small for qualitative work but non-trivial as experiments become more quantitative. In cases where the detector has been characterized in advance, we discuss the detector response normalization and how it may be incorporated in simulations. & 2012 Elsevier B.V. All rights reserved.
Keywords: Scanning transmission electron microscopy (STEM) High-angle annular dark field (HAADF) Detector efficiency.
1. Introduction Recent years have seen a rise in the number of atomic-resolution scanning transmission electron microscopy (STEM) investigations involving quantitative comparisons between experimentally recorded and simulated images [1–19]. This requires some assumptions to be made about the structure being investigated. In addition, several microscope parameters need to be adequately characterized. In one approach to quantification, comparisons are based on relative contrast measures such as peak height ratios and profile shape [1–4,6–8,14,17]. Alternatively, quantitative analyses can be made which measure and compare the absolute signal between experiment and simulation [5,9–12,15,16,18,19]. The former approach reduces somewhat the stringency with which the experimental setup needs to be characterized [2,20]. Relying only on the ratios of values at different points in an image, possibly after ‘‘background’’ subtraction, removes any need to quantify or give meaning to the absolute magnitude of the recorded signal. In a homogenous sample, the parameter most readily determined by the absolute signal is the specimen thickness [11], or, equivalently, a count of the number of atoms present [12]. An absolute scale is often abandoned in quantitative conventional transmission electron microscopy. There it is seen as a small concession when compared to the wealth of information in the conventional images [21]. But in STEM, where the qualitative form of the image often has very little thickness dependence, knowing the thickness on a column-by-column basis allows for determination of
n
Corresponding author. Tel.: þ1 919 515 5049. E-mail address: [email protected] (J.M. LeBeau).
0304-3991/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ultramic.2012.09.001
the three-dimensional structure from a two-dimensional image. Quantitative analyses which measure and compare the absolute signal between experiment and simulation achieve this by accurately estimating the number of atoms in a column [12,16]. In both approaches to quantification—relative contrast and absolute intensities—comparison with simulation is essential. Accuracy depends, therefore, on adequate characterization of all aspects of the experimental set-up to which images are sensitive. Such parameters include the probe-forming and detector collection aperture angles and the coherent and incoherent aberrations in the lens. In work on a system that was not aberrationcorrected, image contrast depended strongly on the spatial incoherence [5]. There is some suggestion that reliable quantification is even more challenging in aberration-corrected systems [13,15,22]. It has been shown, for instance, that the uncertainties in the coherent aberration coefficients within the standard tolerances for an ‘‘acceptably’’ aberration-corrected probe are large enough to notably reduce the accuracy of quantification attempts [22]. In this paper we will explore another factor which may limit the accuracy of quantification: deviations from uniformity of the detector response. LeBeau and Stemmer’s approach to calibrating the HAADF signal as a fraction of the total incident probe intensity (or current) involves deflecting the entire probe onto the HAADF detector to record a reference signal in the absence of any intervening specimen [20]. Moreover, LeBeau and Stemmer were able to scan a localized probe across the detector surface and so build up an efficiency map of the detector, as reproduced in Fig. 1. The detector is not uniformly sensitive. Most notably, there is a ‘‘weak patch’’ asymmetrically located on one side of the inner rim. This imperfection is highly directional. The detector response is a consequence of
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Fig. 1. (a) Detector response map in units of volts as measured using a dynamic signal analyzer [20]. (b) Circularly averaged detector response. (c) Circularly averaged detector response in the vicinity of the detector inner angle.
the geometry of the nearly universal scintillator-based ADF detector developed by Kirkland and Thomas [23]. While the central detector hole is generally coated with a reflective metal, the additional reflections reduce the total collection efficiency on the side of the detector opposite the photomultiplier tube. If the scattering distribution was approximately rotationally symmetric then all that would matter is the circular average of the detector response. This is plotted in Fig. 1(b), where we see a significant reduction of efficiency near the inner detector edge. The inner region is magnified in Fig. 1(c) where we have additionally imposed a sharp inner edge cutoff at 65 mrad, the tail in Fig. 1(b) being an artifact of the width of the probe used in the scan across the detector. The weaker signal near the inner edge was compensated for in Refs. [5,9,11,12] by taking the reference signal to be the average of the detector efficiency over an inner portion of the detector response map. Ref. [10] incorporated the circularly averaged detector sensitivity directly into simulations. The success of quantitative comparisons in previous papers suggests that the assumption of rotational symmetry is adequate, at least for high angle scattering. However, an analysis of the influence of this assumption on contrast and absolute signal strength has not
been presented. Expecting similar behavior from the detectors used at lower scattering angles (since they are often the same detectors simply at different camera lengths), there is a further need to explore the impact of a non-uniform detector response on images formed from the rotationally non-symmetric scattering distribution expected in the bright field and low angle dark field region. This seems particularly important given the recent rise in annular bright field (ABF) imaging [16,24–27], which can be implemented using an annular detector, like that in Fig. 1(a), at extended camera length. We will explore this question in more generality than the single efficiency map in Fig. 1. The paper is structured as follows. In Section 2 we explore the sensitivity of STEM signals to the precise values of the detector inner and outer angles, extending in Section 3 to issues of rotationally symmetric detector non-uniformity. Section 4 considers simple cases of what can happen with strong directional non-uniformity of the detector, and also explores using the experimentally measured (non-uniform) detector response (Fig. 1(a)). The effect of the mis-centering of a range of STEM detectors is considered in Section 5. Consistent with the hope in previous work, the effect of modest non-uniformity is small. Nevertheless, these effects can and should be included in simulation if precise quantification is sought.
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Fig. 2. Composite image comparing the scale of diffraction pattern features to the detector response map at different camera lengths. Top left: position averaged convergent beam electron diffraction pattern [30] for the aberration-corrected probe and a 390 A˚ thick sample. Bottom left: convergent beam electron diffraction pattern with the probe positioned above the Sr column for the aberration-corrected probe and a 390 A˚ thick sample. Top right: detector response map at camera length such that it spans the range 65–375 mrad (hard outer cutoff applied). Bottom right: detector response map at camera length such that it spans the range 25–125 mrad (hard outer cutoff applied). The fourth root of the diffraction pattern intensities is plotted in order to make weak features more visible.
As usual in high resolution STEM, the range of possible behavior is sufficiently complex that broad generality of results is difficult to establish. We present results of specific case studies with fair confidence in the generality of the trends that arise, and some hopeful optimism for the wide applicability of their quantitative conclusions in an order-of-magnitude sense. All simulations assume the specimen to be SrTiO3 viewed along the [001] zone axis. Two probes, both for 300 keV electrons, are considered Aberration-corrected: C s ¼ 6 mm, Df ¼ 0 A˚ and a 20 mrad probe-forming aperture semiangle. Uncorrected: Cs ¼1.2 mm, Df ¼ 550 A˚ and a 9.8 mrad probeforming aperture semiangle (corresponding to typical conditions for the STEM instrument at the University of California, Santa Barbara). Here, Cs denotes third order spherical aberration and Df denotes defocus with the convention that overfocus is positive. For a sense of scale, Fig. 2 compares convergent beam electron diffraction patterns using the aberration-corrected probe and a 390 A˚ thick sample with the detector span range for two typical camera lengths.
2. Sensitivity to detector inner and outer angles It is well established that the range of the detector can influence the magnitude and contrast of STEM images [28]. We quantify this here through a SrTiO3 [001] specimen case study and assuming small variations in detector inner angle, binner , and detector span angle, bspan bouter binner . This would apply if the detector angles were not well measured. It also provides a first order approximation to the case described in the introduction where the innermost portion of
the detector has a lower collection efficiency. We consider two quantities. One is the peak intensity on the Sr column, which we denote Ipeak. The other is the contrast, which we denote by C and define via C ¼ ðImax Imin Þ=Imean . (Other definitions of contrast were explored, but their behavior proved to be very similar.) To investigate the dependence of these quantities on binner and bspan , we calculate derivatives. The derivatives of Ipeak and C with respect to binner and bspan are plotted in Fig. 3 as a percentage variation. Plots are given for both case study probes and for two different specimen thicknesses. The angular range in these plots is restricted to the dark field region. The range of the plots has been restricted to maximize the visibility of the variations in the regions of greatest interest—the numerical values on these plots cease to be valid in regions where saturation occurs. The simulations were carried out using the Bloch wave method, with the positive contribution of thermally scattered electrons to the recorded STEM signal included via an effective scattering potential [29]. These calculations involved a 5 5 mesh sampling within the first Brillouin zone and included reciprocal space lattice vectors out to 3 A˚ 1 ð 60 mradÞ. Because we use an effective scattering potential for thermal diffuse scattering, the 3 A˚ 1 limit is only a limit for elastic scattering: there is scattering beyond 60 mrad, but it is assumed to all be thermal diffuse. This is an excellent approximation for the parameters considered here. The projection approximation was assumed, i.e. zero order Laue zone only. All other calculations in this manuscript use the frozen phonon model [31,32]. The frozen phonon simulations used a 1024 1024 pixel mesh for a 19.5 A˚ 19.5 A˚ supercell (i.e. also a 5 5 mesh sampling within the first Brillouin zone). The distinct lattice planes were included in distinct phase grating steps that, to a first order approximation, takes higher order Laue zones into account. We assume uncorrelated vibrational motion, i.e. the Einstein model for phonons. As shown by Muller et al., this is sufficient in most cases [33]. This approach has also been referred to as the ‘‘frozen lattice’’ model, to differentiate it from the ‘‘frozen phonon’’ model which can then be reserved strictly for cases where correlated phonon modes are included. The frozen lattice/phonon model is required for accurate quantitative simulations at larger thickness [5]. However, for the thicknesses considered in this section the Bloch wave model should suffice, and moreover will avoid many of the problems of trying to use a square multislice mesh when attempting to calculate derivatives with respect to scattering angle away from the optical axis. Let us first consider the variation in peak intensity, which proves to be fairly intuitive. For a given inner angle, varying the span angle is equivalent to varying the detector outer angle. Increasing the outer angle (for fixed inner angle) will always increase the peak intensity. However, the scattered intensity drops off with increasing scattering angle and the effect of varying bspan is smaller when bspan is itself already large. Thus, a 1 mrad change in the span of the detector near bspan ¼ 10 mrad produces a variation in Ipeak of around 5% to 10%, whereas for bspan 4 40 mrad, small variations of the span produce no perceptible changes in the signal. This behavior is essentially independent of detector inner angle. Given a fixed outer angle, increasing the inner angle decreases the peak intensity. Increasing the inner angle with a fixed span angle primarily leads to a decrease in peak intensity: more signal drops off the detector for a unit increase in inner angle than is gained by that same unit increase in outer angle. For example, for a fixed bspan ¼ 50 mrad, a 1 mrad change in inner angle in the vicinity of binner ¼ 60 mrad changes the peak intensity by over 3%. Some exceptions to this trend are visible in the plots: for binner ¼ 30 mrad and bspan ¼ 5 mrad, there is a slight increase with increasing inner angle but fixed span angle. This implies that the
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Fig. 3. Plots of the percentage differential variation in peak intensity, Ipeak, and contrast, C, with detector inner angle, binner , and detector span angle, bspan bouter binner , for the two different probes and two different thicknesses. The specimen is SrTiO3 [001]. Probe parameters are given in the body text. A uniform detector response was assumed.
scattering intensity is not monotonically decreasing in this area. That behavior, though, is the exception rather than the rule, and for most combinations the peak intensity reduces with increasing inner angle. Let us now consider the contrast. The variation in contrast for small span angles and small inner angles is quite complicated since coherent interference effects dominate. With that extreme in mind, the dependence of the contrast on the span angle is quite weak, and more so as the inner angle increases. The outer angles contribute little to the signal and so have little effect on either its contrast or its magnitude, as in the peak intensity already discussed. Nearly independent of the span angle, the sensitivity of the contrast to varying the inner angle decreases with increasing inner angle. Thus while the
contrast formed with an inner angle in the vicinity of binner ¼ 30 mrad in the aberration-corrected probe case can vary by a few percent for a 1 mrad variation in detector inner angle, this reduces to less than 1% in the vicinity of binner ¼ 60 mrad (not shown). This is consistent with consensus expectations of the robustness of HAADF imaging: the contrast in STEM images at high scattering angles is largely insensitive to the precise values of inner and outer detector angles. There is no great difference in this behavior between the two probes or the two thicknesses considered. It should be appreciated that the percentages given here are for unit (i.e. 1 mrad) variation of the detector angles. These quantities are rarely measured with such precision. Indeed, with the non-uniform sensitivity of the inner angles of the detector seen in Fig. 1, it is questionable
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whether we could define a fixed inner angle value with 1 mrad precision. For binner t 60 mrad we find the error in the differential peak signal to be 3% to 5%, assuming a 1 mrad deviation. Therefore, we would expect a 6% to 10% error for a 2 mrad deviation. Thus we conclude that using simple simulations as a quantitative reference for experiment could easily lead to errors of the order of 10% in attempting to quantify the peak signal, unless the detector has been carefully characterized [20].
3. Rotationally symmetric non-uniformity Let us suppose that the inner angle in the rotationally averaged detector signal in Fig. 1(c) has been accurately characterized, i.e. it is not a source of uncertainty. Nevertheless, the weaker detector response near the inner edge will have a similar effect to that seen in the previous section: a simulation assuming a uniform response will not correctly describe the slight relative suppression of the intensity of scattering near the detector inner edge. The consequences of this depend on how the quantification of the detector response is used to normalize the STEM signal and incorporated in the simulations. Since characterization of the detector is a prerequisite of this process, the circularly averaged detector response can be included in the simulation (as done by Rosenauer et al. [10]). Indeed, as we discuss later, the full 2D detector response can be included in the simulation. LeBeau and coworkers [5,9,11,12] took the simpler route of selecting a single normalization factor to compare the experimental signal against simulations assuming a detector with uniform response. By way of exploring rotationally symmetric but non-uniform detector response, we will reconsider that approach, justifying it as it was applied in earlier work but pointing out its limitations.
Suppose the electron intensity distribution in the diffraction plane is approximately rotationally symmetric, at least within the region corresponding to the ADF detector. Let us denote the intensity by i(u), where u is the radial distance in the diffraction plane, which it will be convenient to express in units of the angle away from the optical axis. For an annular detector spanning the range from uinner to uouter, and assuming the ideal, unit detector response in the simulation, the STEM signal would be Iideal ¼ 2p
Z
uouter
iðuÞu du:
ð1Þ
uinner
Let D(u) denote the circular average of the measured detector response. However, the scale of the measured detector response is not particularly meaningful and so we will further include a scaling factor Z. Assuming the same inner and outer angles, the measured STEM signal so normalized is thus Imeasured ¼ 2p
1
Z
Z
uouter
iðuÞDðuÞu du:
ð2Þ
uinner
The assumption of LeBeau and coworkers [5,9,11,12] is that, regardless of the exact form of D(u), a single normalization factor Z can be chosen to ensure that Imeasured ¼ Iideal regardless of the specimen or thickness involved. If the recorded detector response was constant (up to noisy fluctuations on a scale finer than that over which the scattered intensity i(u) varies), Z would just be the average detector response. For a non-uniform detector it is less clear that this should be true in general. We claim, however, that for high scattering angles (short camera lengths) Z can be given as the average detector response out to some scattering angle ueffective
Fig. 4. Peak intensity on the Sr column in SrTiO3 [001] as a function of specimen thickness for the two probes considered. Effective probe size as described via convolution with a Gaussian of half-width-half-maximum 0.5 A˚ has been incorporated in these simulations. The ‘‘ideal’’ simulations assume an annular detector, with range as per the row label, with unit response at all points. The remaining simulations use the circular average of the measured detector response (i.e. Fig. 1(b,c)) renormalized as per Eq. (3) for the value of ueffective given on the legend. Symbols are shown when complete overlap with the ideal detector curve occurs.
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(less than uouter) R ueffective 2p uinner DðuÞu du R ueffective : Z¼ u du 2p uinner
ð3Þ
At high scattering angles the rate of decay of i(u) is approximately independent of the position of the probe, since the high angle scattering of all elements tends to the Rutherford-like scattering from the nuclei. The conversion factor Z can thus be determined by using iðuÞp1=u4 Rutherford scattering and the measured detector response D(u) to evaluate and compare and evaluating Eqs. (1) and (2). The value of Z having thus been established, Eq. (3) serves as an implicit definition of ueffective, which, consistent with previous work [20], we deem a more informative value to quote than Z. That ueffective is well-defined by Eq. (3) follows because the measured signal is dominated by the inner region of the detector and D(u) varies monotonically there. Broadly speaking, ueffective measures how much of the detector is contributing significantly to the signal. At camera length such that uinner ¼65 mrad, it turns out that ueffective ¼125 mrad for a circularly symmetric detector with the detector response function shown in Fig. 1(b) and (c), assuming a hard outer aperture is applied at 365 mrad. This is evident in Fig. 4, which compares the ideal detector result, Iideal, with Imeasured for various choices of the normalization as labelled by ueffective values. For the 65–365 mrad detector, ueffective ¼125 mrad is seen to be in excellent agreement with the uniform detector results over the full range of thicknesses for both the aberration-corrected and uncorrected probes. In particular, Fig. 4 shows that comparing a simulation with uniform detector response against an experimental signal from a non-uniform detector where normalization is effected by the average response over the whole detector would produce an error of about 15% in the peak intensity of a 200 A˚ thick sample. (Note: while we have opted in Fig. 4 to renormalize simulations using the measured detector response to those using an ideal detector response, we could equally have used the measured detector response – in the somewhat arbitrary units of volts – and considered various forms of simulation normalization against it. Regardless, the same conclusions would be reached.) For the 25–125 mrad detector range, obtained with the same detector response as per the 65–325 mrad range but assuming an adjusted camera length, ueffective ¼165 mrad gives the better agreement. For these lower scattering angles, the intensity drop off no longer has the high scattering angle 1/u4 distribution, changing the appropriate value of ueffective. Fig. 4 shows that the single factor normalization holds over a wide range of thicknesses and probe sizes provided the camera length is fixed. However, the value of ueffective will vary with camera length, especially if this brings the inner angle into the vicinity of significant elastically scattered intensity. This is to be held against the generality of the single normalization factor approach. Contrast measures are not explored here because they involve a ratio which eliminates any absolute intensity scale. If the real detector response is such that the behavior of the signal is proportional to the idealized case of a uniform detector response, it is not necessary to determine the normalization constant if only contrast or other ratio measures are to be used in comparing simulation and experiment. This is true if the detector inner angle is large enough that the scattering profile within the detector is that of Rutherford scattering, independent of probe position (as seen later in Fig. 7). 4. Detectors without circular symmetry The diffraction pattern is only rotationally symmetric when an electron probe with width smaller than the inter-column spacing
57
is precisely centered atop a column in the thinnest of crystals. In the high angle dark field regime, the deviations from rotational symmetry are due primarily to the Kikuchi bands. However, these tend to involve a fairly localized redistribution of electrons. Unless the non-uniform detector response varies on the scale of these features, the high angle scattering tends to average to circular symmetry, justifying the treatment in the previous section. However, depending on the probe location, coherent interference effects can give some distinct directionality to the electron scattering distribution at lower scattering angles. Thus, should we change the camera length such that the inner edge of the detector lies within the bright field or in the low angle dark field region, highly directional detector non-uniformity as evident in Fig. 1(a) may have a notable impact on the form of the STEM signal when compared to idealized simulations assuming detector uniformity. Three circularly non-symmetric detectors are shown in Fig. 5: (a) the experimentally measured detector (truncated to give a 5-to-1 ratio between outer and inner angles), (b) a two-level approximate to the measured detector in (a), and (c) a uniform detector with a missing sector. Fig. 5(c) corresponds loosely to a circular detector partially occluded by the support arm of a circular
Fig. 5. Detector response functions: (a) experimentally measured, (b) two-level approximation to that experimentally measured, (c) uniform with missing a segment, and (d) uniform. (e) Contrast as a function of thickness for a SrTiO3 crystal using the aberration-corrected probe and the four detector response functions above at a camera length such that they span the range 25–125 mrad.
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65–375
65–375
45–225
35–175
25–125
10–20
45–225
Segment Ideal Fig. 6. ABF simulations using the three detectors in the righthand column, derived from Fig. 5(a), (c) and (d), for a camera length such that the inner angle is 10 mrad and applying an aperture such that the outer angle is restricted to 20 mrad. Results are shown for two thicknesses. Each image is split: the top half is the result for perfect spatial coherence, while the bottom half includes the effect of spatial incoherence as ˚ described by convolution with a Gaussian of half-width-half-maximum 0.5 A.
35–175
Detector response
125 Å
Measured
62 Å
properly account for the non-uniform detector response changes not just the quantitative scale but also the qualitative shape of the signal. Contrast is a (possibly overly) simple measure of any such change. Fig. 5(e) plots the contrast as a function of thickness for these four detectors taken at a camera length such that they span the range 25–125 mrad, assuming the aberration-corrected probe and including spatial incoherence. It is seen that, at least by the single measure of contrast, the results for detectors (a) and (b) are very similar to one another, suggesting that the two-level approximation captures the most significant variation in uniformity of the measured detector (for this camera length). Interestingly, the contrast results for detectors (c) and (d) are also very similar to one another. Visual impression from the detector maps might have inclined one to say that detectors (b) and (d) were more similar to one another than (c) and (d), but this is not what is ˚ the contrast difference of found. For a thickness of around 50 A, detectors (a) and (b) with that of (c) and (d) is around 10%. These results imply that if the uniformity of detector response of round detectors is better than that of annular detectors, then implementing ABF via a circular detector and a beam stop might be preferable to using an annular detector at suitable camera length if the detector response was not well characterized. The similarity between the contrast from (c) and (d) would be expected if the scattering distribution was circularly symmetric. However, the single, simple measure of contrast does not give the complete story: images with similar contrast are not automatically similar images. STEM images for this low angle annular dark field situation, however, show perceptible distortion only for ˚ and in the absence of significant spatial thin specimens ð o 50 AÞ incoherence. The effect is a little more pronounced for a camera length such that the detector overlaps the bright field region. Fig. 6 considers the detectors from Fig. 5(a), (c) and (d) (‘‘measured’’, ‘‘segment’’ and ‘‘ideal’’, respectively), at a camera length such that the inner angle is 10 mrad and applying a limiting circular aperture such that the effective outer angle is 20 mrad—this is one approach for implementing ABF imaging using an annular detector already present in the column. In the absence of spatial incoherence (top half of each image) we find that the ‘‘segment’’ detector shows perceptible qualitative changes in the form of the STEM images, while differences between the ‘‘ideal’’ and ‘‘measured’’ detectors are small. The changes all but vanish if significant spatial incoherence effects are present (bottom half).
25–125
beam-stop, a common approach to implementing ABF imaging. For reference, we also consider (d) a circularly symmetric, uniform detector. Henceforth we shall concentrate on the contrast measure. This circumvents the question about how to normalize intensities (though this question must be faced if absolute signals from two different detectors are going to be compared). If the STEM signals from the different detectors were identical up to a factor, the problem of correct normalization is the problem of finding that factor. More fundamental, then, is the possibility that the signals differ by more than just a proportionality constant, that failing to
10–20
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Fig. 7. A comparison of the real detector response, the circularly averaged detector response, and the ideal, uniform detector response for a range of different camera lengths for both the aberration-corrected and uncorrected probes. Spatial incoherence is included. For each camera length (giving the span ranges as labelled in the top ˚ plots in mrad units), the horizontal axis runs from zero to 500 A.
S.D. Findlay, J.M. LeBeau / Ultramicroscopy 124 (2013) 52–60
BF: 0–5 mrad t = 50 Å
8 mrad
Imax=0.06 C=0.34
Imax=0.06 C=0.35
Imax=0.09 C=1.34
Imax=0.08 C=1.07
Imax=0.07 C=0.75
Imax=0.71 C=0.24
Imax=0.61 C=0.20
Imax=0.50 C=0.18
Imax=0.70 C=0.43
Imax=0.60 C=0.34
Imax=0.47 C=0.19
Imax=0.13 C=2.11
Imax=0.14 C=2.06
Imax=0.15 C=1.89
Imax=0.08 C=2.48
Imax=0.08 C=2.48
Imax=0.08 C=2.48
ABF: 10–20 mrad t = 50 Å
BF: 0–5 mrad t = 200 Å
Imax=0.06 C=0.35
ABF: 10–20 mrad t = 200 Å
Though not a deviation from uniformity, a complete characterization of the detector geometry calls for considering the precision of detector centering. Fig. 8 shows simulations of bright field (BF), ABF, low angle annular dark field (LAADF) and HAADF images of SrTiO3 viewed along the [001] axis for the aberration-corrected probe for detectors of the ranges specified in the row labels and with the detector off-centered by the amount shown in the column labels. These images include spatial incoherence via convolution with a Gaussian of 0.5 A˚ half-width-half-maximum. The image maximum intensity, Imax, and image contrast, C, are given beneath each image. The effects on the images are more pronounced for the detectors in the bright field region. One reason for this is that the coherent interference in the bright field region makes for a more rapidly varying diffraction pattern intensity. For the BF and ABF detectors, results are shown for two thicknesses, and the patterns and behaviors vary with thickness due to dynamical channelling effects. But the main reason is that the shifts considered amount to a more significant fraction of the detector span. Consequently, mis-centering of these detectors is more likely to show up as an asymmetry in the images. In the BF and ABF images shown here, the image maximum intensity and the image contrast tend to decrease with increased detector offset, although there are exceptions. For the LAADF detector, the gain in signal in the inner edge of the detector which moves closer to the optical axis tends to slightly outweigh the loss from the inner edge which moves further away: the image maximum intensity increases slightly for an offset LAADF detector. It turns out that this effect is more pronounced for the off-column contribution, and as such we find an appreciable contrast decrease for appreciably off-centered LAADF detectors. We might expect such effects to be true of the HAADF detector, but the magnitude of the effect is greatly reduced: mis-centering the HAADF detector has a very weak effect on both image maximum intensity and, especially, on contrast. (In the absence of spatial incoherence,
4 mrad
LAADF: 30–80 mrad t = 200 Å
5. Off-centred detectors
0 mrad
HAADF: 60–160 mrad t = 200 Å
Fig. 7 compares the contrast as a function of thickness for the real detector response of Fig. 1(a) with its circular average and with an ideal, uniform detector for a number of different camera lengths for both the aberration-corrected and uncorrected probes. The discrepancies are largest for the smaller detector inner angles and also small thicknesses. For the aberration-corrected probe, the discrepancies with the uniform detector are around the 5% level, while those with the circularly averaged detector are below the 3% level. The discrepancies are larger for the uncorrected probe. The effect reduces as the detector inner angle increases. For the aberration-corrected probe at small thickness, the ideal detector in the 35–175 range can be in error by 3%, whereas for the 45–225 case this is down to 1%. The relative error between the real detector and circular average in this range is essentially zero, supporting our earlier assertions of effective rotational symmetry at higher scattering angles given the scale of the detector nonuniformity. Though the specimens considered differ, for ABF imaging with the aberration-corrected probe, errors at the 5% to 10% level occur for the ˚ of previous atom counting by ABF thicknesses range ð 15250 AÞ experiments [16]. Thus, detector non-uniformity can have a significant effect when seeking to quantify ABF images. In addition, it is important to note that rotations of the diffraction information can be introduced by the post specimen projection system, leading to an additional parameter to take into consideration. If the alignment of the experimental CBED pattern relative to detector is not measured for low angle or ABF imaging, then it is advisable to introduce the rotationally averaged detector response in the simulations.
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Fig. 8. BF, ABF, LAADF and HAADF simulations for SrTiO3 using the aberrationcorrected probe and assuming a finite effective source size of 0.5 A˚ half-widthhalf-maximum. The left column assumes perfectly centered detectors, while the other two columns assume off-centering by angular displacements of 4 mrad and 8 mrad, directed down the page.
simulations show a 3% increase in Imax in 400 A˚ of SrTiO3 for the conditions in Fig. 8 for a detector shift of 8 mrad.) Thus precise centering of the HAADF detector, which is challenging, is also unnecessary in most cases. Some of our colleagues have asked us whether modest deviations from ideal centering can lead to an apparent shift in the column locations in simultaneously recorded ABF and HAADF images, and moreover, since these two detectors could be off-centered in different
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directions, whether there might be any apparent relative shift of the column locations between the two modes. Reasons for suspecting this include the asymmetry in the images for significantly misaligned detectors and recent work which has demonstrated small column shifts in the HAADF range for small detectors as a function of off-axis position [34]. However, the present simulations show that for modest detector shifts – ‘‘modest’’ meaning that image distortion is sufficiently small that detector misalignment is not automatically suspected – any relative displacements of column centres are imperceptible.
6. Summary and conclusion Even if the detector response can be assumed to be uniform, an uncertainty in the detector inner angle of a few milliradian could produce a 10% error in the prediction/interpretation of the peak intensity. When detectors with a non-uniform detector response comparable to that seen in Fig. 1(a) are used at camera lengths placing them in high angle dark field region, the intensity can be correctly normalized against simulations which assume a uniform detector response by a judicious choice of normalization factor. For longer camera lengths, such that this detector sits in the lower angle dark field and/or bright field regions, the directional nature of the non-uniformity produces variations in contrast of the order of 5% relative to the assumption on a uniform detector. When in the bright field region, directional non-uniformity of the detector response can moreover lead to visible qualitative variations in the form of the image (especially if the spatial coherence of the effective source is high). The problems discussed here are hardware problems, which can likewise ultimately be overcome through hardware. It has long been appreciated that the ultimate goal in flexible detector geometry is to record the whole two-dimensional diffraction pattern for each probe position [35]. Proof-of-principle experiments collecting such data sets exist in the literature [34], though improvements in high-speed electronic readout remain necessary before this imaging mode becomes routine. Alternatively, as implicit in the simulations presented here, accurate theoretical modeling is always possible for a sufficiently well-characterized detector by including the measured, non-uniform detector response into the simulations. The similarity between the detector response of Rosenauer et al. [10] and that of LeBeau and Stemmer [20] suggests that areas of notably reduced detector response correspond closely to the design geometry of the detector. There is no evidence of significant fluctuations in detector response at arbitrary locations. Nevertheless, we conclude that studies seeking to quantity absolute intensity to within 10% level or contrast to within the 5% level would do well to characterize their detector response and incorporate it into supporting simulations, especially when using annular detectors at lower scattering angles.
Acknowledgments S.D.F. acknowledges support by the Australian Research Council. J.M.L. acknowledges Susanne Stemmer and the UC Santa Barbara
MRL Central Facilities for the detector map. The MRL Central Facilities are supported by the MRSEC Program of the NSF under Award No. DMR 1121053; a member of the NSF-funded Materials Research Facilities Network (www.mrfn.org).
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