Elsevier

Ultramicroscopy

Volume 211, April 2020, 112948
Ultramicroscopy

Source shot noise mitigation in focused ion beam microscopy by time-resolved measurement

https://doi.org/10.1016/j.ultramic.2020.112948Get rights and content

Highlights

  • Compound Poisson models for focused ion beam microscope measurements are introduced.

  • Time-resolved sensing reduces the effect of source shot noise in FIB microscopy.

  • Low-dose measurements have the most Fisher information per incident ion.

  • Dose reduction by factor approximately equal to the secondary electron yield.

Abstract

Focused ion beam microscopy suffers from source shot noise – random variation in the number of incident ions in any fixed dwell time – along with random variation in the number of detected secondary electrons per incident ion. This multiplicity of sources of randomness increases the variance of the measurements and thus worsens the trade-off between incident ion dose and image accuracy. Repeated measurement with low dwell time, without changing the total ion dose, is a way to introduce time resolution to this form of microscopy. Through theoretical analyses and Monte Carlo simulations, we show that three ways to process time-resolved measurements result in mean-squared error (MSE) improvements compared to the conventional method of having no time resolution. In particular, maximum likelihood estimation provides reduction in MSE or reduction in required dose by a multiplicative factor approximately equal to the secondary electron yield. This improvement factor is similar to complete mitigation of source shot noise. Experiments with a helium ion microscope are consistent with the analyses and suggest accuracy improvement for a fixed source dose by a factor of about 4.

Introduction

State-of-the-art techniques for imaging the structure of a sample at near-atomic resolution depend on the use of microscopes that scan the sample with a focused beam of particles. For instance, a focused electron beam is employed in scanning electron microscopy (SEM) [1], laser beams in confocal laser-scanning microscopy [2] and two-photon laser-scanning fluorescence microscopy [3], and focused ion beams in focused ion beam (FIB) microscopy [4]. A fundamental goal with these technologies is to produce the best image quality for a given number of incident particles. This is especially relevant when each incident particle appreciably damages the sample; because helium ions cause such damage, we henceforth concentrate on helium ion microscopy (HIM) [5].

FIB imaging methods have randomness in the number of incident particles (the source shot noise) and in the influence of each incident particle on the device measurement. The goal of the imaging is to infer properties of the sample that are revealed through the number of detected secondary electrons (SEs) per incident ion, and the source shot noise is detrimental to this effort because it is unrelated to the sample. It is intuitive that one would prefer to have a precisely known number of incident ions, and we provide a simple analytical result to demonstrate this in Section 2.2.

The main idea of this work is that time-resolved measurement of SEs can be used to mitigate the effect of source shot noise. Here, time-resolved (TR) measurement means to divide any given pixel dwell time t into n dwell times t/n and to jointly process the n low-dose measurements to produce one pixel of the micrograph. This type of TR measurement requires no change of hardware: it is a data-processing innovation implemented with existing hardware. The main limitation is whether the dose in dwell time t/n is small enough; roughly, the mean number of incident ions in dwell time t/n should be less than 0.5 to attain at least half of the advantages described herein. Though total dose is not increased, total acquisition time may be increased, depending on the data transfer rate and whether the hardware requires raster scanning to be completed n times to implement this conception of TR measurement.

In certain limiting cases, we can completely eliminate the effect of source shot noise, producing estimation performance equivalent to a deterministic incident ion beam. More importantly, for parameters that reasonably model HIM, the improvement is substantial and validated by both simulations and experiments. While our initial modeling and theoretical results assume direct detection of SEs, our experimental demonstration of improved performance is with extensions of the modeling and algorithms for use with instruments without direct SE detection.

We first presented the TR measurement concept for FIB microscopy in abstract form in [6]. Here, we provide theoretical analyses, develop three estimators for use with TR data, compare these three with the conventional estimator in synthetic simulations, and also compare the best of these estimators with the conventional estimator for experimental data.

The first image of a solid sample based on secondary electrons emitted in response to an electron beam scanner was produced by Knoll in 1935, inspiring the development of a dedicated SEM [1]. Ever since their development, SEMs have been ubiquitous in both research and industrial imaging, as well as in nanometerological applications [7]. Building upon decades of research in focused ion beam microscopy, the first commercial HIM was introduced in 2006 [5], [8], with the promise of producing images with sub-nanometer resolution [9] and reduced charging of the sample, when compared with SEM. However just like SEM, HIM uses a focused particle beam to produce lateral spatial resolution in a ballistic configuration [5]. Both material composition (e.g., atomic number) and shape (topographic yield variations common to SEM as well) contribute to the number of SEs dislodged from the specimen [10]. These properties, along with improved imaging resolution, larger depth-of-field, and reduced sample charging, have enabled superior imaging of insulators without the need for metal coating. Hence HIM is an important imaging technology for semiconductor and nanofabrication research [11].

Notwithstanding the progress in the pursuit of ultra-high resolution, these imaging technologies all have the disadvantage of causing damage to the sample through sputtering [12], [13], [14]. Whilst sample damage can have especially severe impact on biological samples, it also occurs for many other types of materials. It is thus recognized and modeled as a fundamental limit to imaging with focused beams [13], [15]. With a helium ion being 7300 times more massive than an electron, mitigating sample damage in HIM is paramount. One possible approach is imaging using lower ion doses but at the cost of lower image quality [13]. Consequently, studies analyzing the extent of beam damage and establishing safe imaging dose have appeared [16], [17].

In Section 2, we present our baseline Poisson–Poisson measurement model and basic analyses of this model. These analyses provide the foundations for our development, in Section 3, of the advantage provided by dividing any fixed ion dose into small doses through TR measurement. We present both abstract numerical results and image simulations. While a Poisson–Poisson model sufficiently describes direct SE detection, indirect SE detection necessitates additional modeling. Inspired by the indirect detection of SEs in current HIM instruments, Section 4 introduces suitable hierarchical compound models. Section 5 presents experimental results using data from a Zeiss HIM.

Section snippets

Single measurement: Model and analyses

Two main components enable FIB-based imaging: a stable source to generate the FIB and a detector to measure the number of SEs leaving the sample’s surface. Due to ion–sample interaction, SEs become excited and dislodged from the sample’s surface [18], accelerating towards the SE detector. Imaging is achieved by raster scanning the ion beam with some fixed dwell time per pixel. For each pixel, the number of detected SEs is mapped to a grayscale level, hence producing an image of the sample.

Time-resolved measurements

Taken together, the analyses in Section 2 suggest that there may be a way for the conventional estimate from (5) to be improved upon to give a reduction in MSE by the factor in (15). TR measurement indeed achieves this improvement. We examine this first through Fisher information and then through simulated performance of the ML estimator for imaging.

Hierarchical compound models

The model introduced in Section 2.1 assumes direct secondary electron counting, so that the number of SEs is the final readout of the device. In current HIM instruments, the output is more indirect. We now discuss some plausible models for the SE detection process and show that simulations continue to suggest substantial advantages for TR measurement.

Experiment details

Our methods were validated with data from a Zeiss ORION NanoFab HIM used to image a carbon-based defect on a silicon substrate. The instrument was used to collect 128 sub-acquisitions of the sample using a 0.1 pA beam current and 200 ns dwell time, resulting in low ion dose of 0.125 ions per pixel. The image of one typical sub-acquisition is shown in Fig. 6(a). In the first three columns of Fig. 6, the scaling for display maps the range of the data linearly to the full black-to-white range,

Discussion

The main contribution of this paper is to introduce the idea that a set of low-dose focused ion beam microscope measurements can be substantially more informative than a single measurement with the same total dose. We refer to the acquisition of the set of low-dose measurements as “time-resolved measurement” because it can be realized by keeping beam current and total dwell time unchanged, while dividing the dwell time into short time segments.

Our demonstrations of the potential of TR

Declarations of competing interest

The authors declare no competing financial interests.

Author contributions

KKB and VKG conceived of time-resolved measurement in FIB microscopy. MP, JMB, and VKG derived the mathematical results. MP wrote software for image formation and completed all numerical experiments. MP, JMB, and VKG wrote the manuscript. All authors edited the manuscript.

Acknowledgments

The authors thank John Notte and Deying Xia of Carl Zeiss Microscopy LLC for enlightening discussions and experimental data and thank Akshay Agarwal and Emily Toomey for comments on an earlier manuscript.

Funding: This material is based upon work supported in part by the US National Science Foundation under grant no. 1422034 and grant no. 1815896.

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