Shot noise-mitigated secondary electron imaging with ion count-aided microscopy

Fig. 1 shows a schematic of our imaging setup. We used a Zeiss Orion Plus helium ion microscope operated at 30 keV beam energy in our experiments. We outcoupled both the SED and the beam scan signals into a 12-bit, 100 MHz digitizer (GaGe RazorExpress 1642), which sampled both signals at 10 ns. Fig. 1 shows a 20 $\mathrm{\mu }\text{s}$ snapshot of the SED voltage signal. The SED is typically an Everhart–Thornley detector (28), which consists of a scintillator followed by a photomultiplier. The signal generated by the SED consists of a series of voltage pulses of varying heights, where each pulse corresponds to a burst of detected SEs (29). The mean full width at half maximum (FWHM) of the pulses is $\sim$160 ns (30). In our experiments, we used beam current $I_b=0.11$ pA (measured with a picoammeter connected to a Faraday cup), which was low enough to make pulse pile-up effects manageable.

To create images, we collected the SED voltage and beam scan signals at pixel dwell times $t_d$ varying between $2\mathrm{\mu }\text{s}$ and $32\mathrm{\mu }\text{s}$ for a 512 $\times$ 512-pixel image. These settings correspond to an incident dose $\lambda =I_b{t}_d/e$ between 1.4 and 22 ions per pixel, where $e$ is the elementary charge. The maximum image size and dwell time were set by the available memory of the digitizer. Next, using custom MATLAB scripts, we extracted the SED voltage pulse heights ${\tilde{U}}_i$ and the number of pulses $\tilde{M}$ for each pixel. Neglecting pulse pile-up, $\tilde{M}$ would be the number of incident ions that produced one or more detected SEs. To account for pile-up, we divide the raw value of $\tilde{M}$ by a correction factor that accounts for the probability of pulse overlap to yield a value ${\tilde{M}}_{\text{corr}}$ (Materials and Methods). The raw datasets used to create all SE images in this work are available at ref. 31, and the MATLAB scripts used to create the images using these datasets are available at ref. 32.

We used the measurements $(\tilde{M},{\tilde{U}}_1,{\tilde{U}}_2,\dots ,{\tilde{U}}_{\tilde{M}})$ to create two pixelwise maps of $\eta$: conventional, which models typical SEI; and ion count-aided (ICA), which implements our source shot noise-mitigated estimator. While typical SEI uses only $V=\sum _i{\tilde{U}}_i$, the performance of ICAM demonstrates the value of using $\tilde{M}$ as well.

From its definition as the number of SEs generated per incident particle, it is natural for an estimate of SE yield $\eta$ to be framed as

In bulk estimates of SE yield, measurements of the beam and sample currents may be used to calculate the numerator and denominator of this expression (33). Our nanoscale mapping of $\eta$ uses this intuitive expression with numerator and denominator estimated from $(\tilde{M},{\tilde{U}}_1,{\tilde{U}}_2,\dots$, ${\tilde{U}}_{\tilde{M}})$.

Fig. 2 shows comparisons between conventional and ICAM images of a scratch on a silicon chip at doses of $\lambda =4.1$, $\lambda =8.2$, and $\lambda =22$ ions per pixel. All images have $288\times 512$ pixels (cropped from $512\times 512$ images) and a $10\mathrm{\mu }\text{m}$ horizontal field of view. SI Appendix, Fig. S9 shows uncropped images of this sample at $\lambda =22$. These images are SE yield maps; instead of an arbitrary scale, gray-scale values correspond to physically meaningful values of $\eta$ as indicated by the color bar. We can see that for each dose, the ICAM images appear less noisy than the conventional images due to source shot noise mitigation. The ICAM image at a dose of 8.2 ions/pixel (Fig. 2B2) appears visually similar to the conventional image at a dose of 22 ions/pixel (Fig. 2A3). Fig. 2C is a plot of the SD measured over all the pixels as a function of the imaging dose for the conventional () and ICAM () images. In addition to the three imaging noise components (source shot noise, target shot noise, and detector noise), this SD has a contribution from variations in the features in the sample. As the dose increases, we expect all three imaging noise components to reduce, so at high doses, the SD should asymptotically approach the inherent feature SD. This is the behavior we observe in Fig. 2C. At low doses, both the conventional and ICAM SDs vary inversely with the dose (straight line on a log–log plot), and they show saturation at higher doses. The horizontal gaps marked in Fig. 2C indicate that ICAM images have the same SDs as conventional images at 2- to 3-times the dose, and the doses selected for Fig. 2 A and B illustrate this as well.

Further evidence of the reduction in noise is provided by a calculation of Thong’s signal-to-noise ratio (SNR) for both images (34). This metric aims to calculate SNR for a single SEM image (without knowledge of ground truth) by separating the contributions of signal and noise to the image’s autocorrelation. We calculated this SNR to be 5.6 for the ICAM image and 2.8 for the conventional image at $\lambda =22$. SI Appendix, Fig. S11 presents a calculation of imaging resolution using Fourier ring correlation (35–38) at the $1/3$ threshold (39, 40) for both images; the ICAM image shows a 21% improvement in resolution.

Fig. 3 A and B show the conventional and ICAM images of a patterned silicon substrate with $1\mathrm{\mu }\text{m}$ gold squares. The ICAM image again appears to be smoother and less noisy than the conventional image, especially in the brighter regions that correspond to gold. The lower noise in the ICAM image is also reflected in Thong’s SNR: The ICAM image has an SNR of 2.12, while the conventional image has an SNR of 1.61.

Since this sample has two types of pixels, it provides a good platform for further numerical characterization of the advantages of ICA estimation. Fig. 3 C and D are image histograms with bin width 0.1 for two subsets of pixels in this sample—the darker, silicon pixels, and the brighter, gold pixels—at the same imaging dose. The histograms plot the frequency of occurrence of different pixel SE yields. Fig. 3C shows the conventional and ICAM histogram for the dark (silicon) pixels, and Fig. 3D for the bright (gold) pixels. For the silicon pixels, we measured a mean SE yield of 1.82. We can see that the histogram for the ICAM image is narrower than that for the conventional image at the same dose of $\lambda =21$. The width of the ICAM histogram at a lower dose of $\lambda =15.8$ is comparable to that conventional image histogram. In other words, comparable image quality was attained using the ICA estimator at a dose that was lower than that for the conventional estimator by a factor of 1.33. For the gold pixels, we measured a mean SE yield of 2.75. The dose improvement factor in this case was 1.78.

Other performance quantifications for the images in Figs. 2 and 3 also support the conclusion that ICAM improves upon conventional image formation (SI Appendix).

The reduction in imaging noise in Figs. 2 and 3 agrees with Monte Carlo performance predictions computed from our model of the imaging process. Fig. 4A and B compare the theoretical SD as a function of dose with the experimental values we measured for the silicon and gold pixels in Fig. 3, for both conventional and ICA estimators. Unlike the plots in Fig. 2C, we expect no saturation since each square in Fig. 3 is almost featureless; this is exactly what we observe in Fig. 4A and B. The experimental SDs agree closely with the theoretical values for all doses. We also see a bigger gap between the SDs of conventional and ICAM images at higher SE yield, as expected from the histograms in Fig. 3C and D. As SE yield rises, an increasing fraction of incident particles produce detected SEs, making the estimate of $M$ in Eq. 3 more accurate and improving source shot noise mitigation by the ICAM estimator.

The experiments and performance predictions from simulations are consistent with theoretical analysis through Fisher information (FI). FI is a measure of the sensitivity of noisy data to the parameter to be estimated; higher FI indicates proportionately lower imaging noise. Fig. 4C shows the ratio between the FI from ICAM and conventional measurement at $\lambda =21$ as a function of $c_{\sigma }/c_{\mu }$, the SD of the contribution of one SE normalized by its mean, for $\eta =2.75$ (solid black curve) and $\eta =1.82$ (dashed black curve). The ratio $c_{\sigma }/c_{\mu }$ is a measure of the nonideality of the SED—the larger this ratio, the more the SED deviates from ideal SE counting. As $c_{\sigma }/c_{\mu }$ approaches zero, the FI ratio is simply $(\eta +1)(1-\eta e^{-\eta })$, a strictly increasing and approximately linear function of $\eta$ (25). We can see that the ratio of the FIs of ICAM and conventional measurement is greater than 1 for the entire range of $c_{\sigma }/c_{\mu }$, suggesting that ICAM images remain less noisy even for highly nonideal detectors.

The nonideality of the detector contributes additional noise to the image, leading to degradation in the FI ratio with increasing $c_{\sigma }/c_{\mu }$. For our system, $c_{\sigma }/c_{\mu }=0.6$, and at this value, we get FI ratios of about $2.1$ for $\eta =2.75$ and $1.4$ for $\eta =1.82$. Due to the additive property of Fisher information, we consequently expect a dose reduction by a factor of $2.1$ at $\eta =2.75$ and $1.4$ at $\eta =1.82$. These numbers are close to the experimental dose reductions we obtained for the sample in Fig. 3.

The SE yield values we report here are not absolute but are implicitly multiplied by the SED’s detection quantum efficiency (DQE) (41). We measured the DQE to be about $0.88$ (SI Appendix). Therefore, all SE yield values quoted here should be divided by this number for absolute SE yields. For example, the DQE-corrected mean SE yield for the sample in Fig. 2 is 3.62, which is in agreement with theoretical predictions of the He-ion SE yield for silicon at 30 keV (42).

Conventional estimates of SE yield require precise measurement of beam current. We observed that the beam current value measured by the picoammeter fluctuated by up to 10% for the same nominal setting of 0.11 pA. As seen in Eq. 2, the conventional estimate is inversely proportional to $\lambda$. Therefore, a 10% uncertainty in the value of the beam current results in a 10% uncertainty in ${\widehat{\eta }}_{\mathrm{conv}}$. In contrast, the ICAM SE yield measurement is much less sensitive to precise knowledge of $\lambda$ because it uses the observed count of ions in addition to knowledge of $\lambda$ to estimate $\eta$ as seen in Eq. 3. At $\eta =2.75$, a 10% variation in $\lambda$ causes 0.8% variation in ${\widehat{\eta }}_{\mathrm{ICA}}$ (SI Appendix). This reduced sensitivity to precise knowledge of the beam current could be used for removal of stripe artifacts that result from beam current variations (43).

We used a Zeiss Orion Plus HIM operating at 30 keV and a beam current of 0.11 pA to collect all our data. The beam current was set to the lowest stable value to reduce pulse pile-up. This beam current corresponds to a dose rate of 0.68 ions per $\mathrm{\mu }\text{s}$. Assuming a focused probe diameter of $1{\text{nm}}^2$ and a pixel dwell time of $32\mathrm{\mu }\text{s}$ as in Fig. 2 of the main paper, we get a dose of $2.2\times {10}^{15}\text{ions}/{\text{cm}}^2$. We outcoupled the voltage signal from the SED into a GaGe RazorExpress 1642 CompuScope PCIe Digitizer. In preliminary experiments to characterize the response of the SED, we observed that the SE pulses typically had a FWHM of $\sim$160 ns. We decided to use a sampling rate of 100 MS/s (corresponding to a sampling time of 10 ns) since that results in $\sim$20 samples per pulse, which we considered sufficient to ensure accurate sampling of the pulses’ peak voltages. We typically outcoupled $256\times {10}^6$ samples, corresponding to a total collection time of 2.5 s. This signal was then processed using custom MATLAB scripts that counted the pulses and detected their peak voltages.

In our model, we denote the (random) number of incident ions at a given pixel by $M$. The $i$th ion generates $X_i$ SEs; the total number of SEs generated from the pixel is $Y=\sum _{i=1}^M{X}_i$. Both the number of ions $M$ and the $X_i$s are described as Poisson distributed random variables. The mean of $M$ is known as the dose $\lambda$. The mean of $X_i$ is the SE yield $\eta$ and is the quantity we wish to estimate. Elementary calculations give $\mathrm{E}[Y]=\lambda \eta$ and $\mathrm{var}\left(Y\right)=\lambda \eta (\eta +1)$.

Both $M$ and the $X_i$s (or, equivalently, $Y$) are not observed directly in the instrument. We refer to the number of detected SEs as ${\tilde{X}}_i$, which is described by a zero-truncated Poisson distribution since it is impossible to distinguish between an incident ion producing zero SEs and there being no incident ion at all. Finally, ${\tilde{X}}_i$ is mapped to a voltage ${\tilde{U}}_i$. We assume that given $X_i=n$, ${\tilde{U}}_i$ is a Gaussian-distributed random variable described by $\mathcal{N}(n{c}_{\mu },n{c}_{\sigma }^2)$. Here, $c_{\mu }$ and $c_{\sigma }^2$ are the mean and variance of the voltage produced by the SED in response to one SE, with such voltages being independent and additive. An earlier work (26) uses the less evocative $(c_1,c_2)$ in place of $(c_{\mu },c_{\sigma }^2)$. Overall, the probability density for ${\tilde{U}}_i$ is given by

where $f_Z(u;j,c_{\mu },c_{\sigma }^2)$ is the probability density function of a $\mathcal{N}(j{c}_{\mu },j{c}_{\sigma }^2)$ random variable. We assume that $\lambda$, $c_{\mu }$, and $c_{\sigma }$ are known.

The assumption of linearity in our detection model will hold as long as pulse pile-up is small. At small levels of pile-up, the correction term to the raw pulse counts works well. At beam currents significantly higher than those used in this study, we expect that more careful consideration will need to be given to the statistics of overlapping pulses, which would lead to improved estimators.

To characterize the detector’s response to SEs and measure $c_{\mu }$, we imaged a featureless silicon chip to ensure that SE emission was uniform over the scan region. Further, we defocused the beam by 5 mm to reduce beam-induced damage and mitigate variations in SE yield $\eta$ due to local contamination or topography. Since $c_{\mu }$ and $c_{\sigma }$ are properties of the detector, we expect them to remain unchanged with variations in the sample; i.e., our measurements of $c_{\mu }$ and $c_{\sigma }$ should be constant for samples with variable $\eta$. Therefore, we found it useful to be able to continuously vary $\eta$ to validate our measurements. This variation of $\eta$ would be difficult to accomplish with physical samples; instead, we varied the collector bias on the Everhart–Thornley SED to change the effective SE yield at the detector. Everhart–Thornley detectors typically have a positively biased metal cage at the front to attract SEs. The default bias on the cage was 500 V; we varied this collector bias between 0 V and 500 V to get different effective values of $\eta$.

In our model, $c_{\mu }$ is the mean voltage produced by the detector per SE. Therefore, the most direct method for measuring $c_{\mu }$ would be to have exactly one SE incident on the detector many times and compute the mean voltage produced by the SED. Unfortunately, such deterministic irradiation of the SED is not possible since the production of SEs is a Poisson process. One could imagine placing the SED under an electron beam produced by an electron gun, as was done in ref. 29. However, such an experiment requires significant modification of the microscope, which was not possible on our tool.

Instead, we measured $c_{\mu }$ by imaging a sample with low SE yield $\eta$. At sufficiently low SE yield, the probability that the incident particle will excite more than one SE is small. Therefore, we can assume that most detected pulses are excited by one SE. Under these conditions, if our model is accurate, we would expect the pulse height distribution (PHD), i.e., the histogram of the peak voltages of the SE pulses, to be approximately Gaussian with a peak at $c_{\mu }$. Using this technique, we extracted $c_{\mu }=0.163$ V from a histogram measured at $\widehat{\eta }=0.58$. In SI Appendix, we describe the use of the probabilistic model to fit experimental PHDs at various values of $\widehat{\eta }$.

As discussed in the paper, ions arriving consecutively in a short time frame can generate overlapping pulses, a phenomenon known as pulse pile-up. Consequently, the number of local maxima in a voltage signal can be an underestimation of $\tilde{M}$. One way to correct for missed detections due to pile-up is by introducing a multiplicative factor:

where

is the probability that two pulses arrive within time $\tau$ of each other, and $\mathrm{\Lambda }=\lambda /t_d$ is the dose per unit time. We chose $\tau =0.13\mathrm{\mu }\text{s}$; more details on the choice of $\tau$ can be found in SI Appendix.

A.A., L.C.F., and V.K.G. designed research; A.A. and L.K. performed research; J.A.S. contributed new reagents/analytical tools; A.A., L.K., X.H., R.K., O.K.H., and M.P. analyzed data; and A.A. and V.K.G. wrote the paper.

Schultz/Ionwerks owns patents on secondary electron detector designs and methods of use in correlated secondary electron imaging and ion scattering time-of-flight measurements.