Beam Cross Sections Create Mixtures: Improving Feature Localization in Secondary Electron Imaging

Abstract

Secondary electron (SE) imaging techniques, such as scanning electron microscopy and helium ion microscopy (HIM), use electrons emitted by a sample in response to a focused beam of charged particles incident at a grid of raster scan positions. Spot size -- the diameter of the incident beam's spatial profile -- is one of the limiting factors for resolution, along with various sources of noise in the SE signal. The effect of the beam spatial profile is commonly understood as convolutional. We show that under a simple and plausible physical abstraction for the beam, though convolution describes the mean of the SE counts, the full distribution of SE counts is a mixture. We demonstrate that this more detailed modeling can enable resolution improvements over conventional estimators through a stylized application inspired by semiconductor inspection: localizing the edge in a two-valued sample. We derive Fisher information about edge location in conventional and time-resolved measurements (TRM) and also derive the maximum likelihood estimate (MLE) from the latter. Empirically, the MLE computed from TRM is approximately efficient except at very low beam diameter, so Fisher information comparisons are predictive of performance and can be used to optimize the beam diameter relative to the raster scan spacing. Monte Carlo simulations provide an example of the MLE giving a 5-fold reduction in root mean-squared error (RMSE) of edge localization as compared to conventional interpolation-based estimation. The RMSE is substantially below both the beam diameter and the raster scan spacing and thus sub-pixel localization is demonstrated. Applied to three real HIM datasets, the average RMSE reduction factor is 5.4.

Figures (9)

• Figure 1: Model of data collection. (a) Depiction of incident particle beam and generation of SEs. (b) Temporally, incident particles arrive as a Poisson process (one representative realization shown). (c) Spatially, deviations from the nominal raster scan location follow a Gaussian distribution.
• Figure 2: A sample with two SE yield values separated by a vertical edge at horizontal position $\gamma$. (a) Depiction of the function $\varphi(s_1,s_2)$. (b) Sample and beam distribution projected to 1D functions of only the horizontal position.
• Figure 3: Variances in SE yield estimation with and without exploiting a mixture model \ref{['eq:X-distribution-two-component']} and knowledge of $(\eta_1,\eta_2)$ as a function of mixing parameter $q$. Without a distributional model, SE yield is estimated with the sample mean, which can be clipped to exploit knowledge of $(\eta_1,\eta_2)$. With the mixture model, an estimate achieving the Cramér--Rao bound would provide an improvement. Plotted for $m=100$, $\eta_1 = 2$, $\eta_2 = 8$.
• Figure 4: Normalized Fisher information $\mathcal{I}^{\mathrm{mix}}_{Y}(q \,;\, \eta_1,\eta_2,\lambda) \, / \, \lambda$ as a function of dose $\lambda$ for $\eta_1=2$, $\eta_2=6$ and $q=0.6$. Also shown are the asymptotes \ref{['eq:NFI_Y_low_lambda']} for low $\lambda$ and \ref{['eq:NFI_Y_high_lambda']} for high $\lambda$. The ratio of the asymptotes is the information gain factor $\beta_{\mathrm{mixture}}$ given in \ref{['eq:TRM-gain-mixture']}. It is the factor by which dose or MSE could be reduced due to the use of time-resolved measurement.
• Figure 5: The information gain factor $\beta_{\mathrm{mixture}}$ from time-resolved measurement is the ratio of the FI in TRM to the FI in Poisson-compounded measurement $Y$ at high dose $\lambda$ as given in \ref{['eq:TRM-gain-mixture']}. With $\eta_1 \approx \eta_2$ as in (a), this gain factor is consistent with the $\beta_{\mathrm{Poisson}}$ factor for Poisson-distributed $X$ given in \ref{['eq:TRM-gain-Poisson']}.