We analytically describe the noise properties of a heralded electron source made from a standard electron gun, a weak photonic coupler, a single photon counter, and an electron energy filter. We describe the sub-Poissonian statistics of the source, the engineering requirements for efficient heralding, and several potential applications. We use simple models of electron beam processes to demonstrate advantages which are situational, but potentially significant in electron lithography and scanning electron microscopy.
]]>Heralded photon sources are an elementary resource in quantum optics. Since their first demonstration in 1977 [1], they have found wide-ranging application: from their use in quantum key distribution [2], simulation [3], and sensing [4], to defining the candela [5]. Recently there has been a surge of interest in applying principles of quantum optics to free electrons. This was sparked in part due to various experimental demonstrations of strong coherent coupling between photons and free electrons in electron microscopes [6][7][8][9][10][11], as well as new theoretical works which propose how such interactions could be useful for producing unique optical and free-electron quantum states [12][13][14][15]. In particular, photon-mediated electron heralding in a transmission electron microscope (TEM) was recently demonstrated for the first time [16]. It has also been suggested that heralding could be done using coulomb-correlated electron pairs, which have also recently been produced for the first time [17,18].
In spite of this progress toward heralded electron sources, there has been little discussion of their potential impact. Correlated electron-photon measurements have been shown to increase SNR when the correlations are formed at the sample [16,19], though only when false coincidence rates are sufficiently small [20]. However it remains unclear when heralding by generating electron-photon correlations before the sample can be useful. In optical microscopy, shot noise is often an important limiting effect, giving heralded photon sources clear utility [21]. In high-energy electron microscopy-where optical elements are lossless and detectors can be fast, highly efficient, and low-noise-nearly every electron which interacts with the sample can be recorded. This makes it possible to normalize beam current fluctuations without heralding. Additionally, the proposed heralding schemes involve filters which would substantially reduce beam current.
Here, we analyze an electron heralding system modeled after the one demonstrated by Feist et al [16]. First, we parameterize this system and describe its sub-Poissonian beam statistics in terms of the Klyshko efficiencies. We find that the extra information provided by a heralded source makes it possible to improve the speed, yield, and minimum feature size in electron lithography. With sufficient heralding efficiency, we find an order-of-magnitude increase in write speeds is possible.
We then use the Fisher information (FI) formalism to quantify the increase in information available for various modalities of quantitative electron microscopy, and we identify two limits (high dark counts and low contrast) where signal-to-noise ratio (SNR) enhancement is possible for dose-limited imaging. We argue there is situational improvement to SNR for some high-energy electron microscope applications like bright field scanning TEM (STEM), but not for phase contrast microscopy or for electron-energy-loss spectroscopy (EELS). A heralded source could also be a solution for dose-calibrated in-situ electron microscopy and low background cathodoluminescence (CL) imaging. In addition, we find heralding can improve quantitative SEM when the dose is limited by damage or charging. We calculate the error in surface tilt estimation of a steep-walled feature, finding that heralding can reduce the dose required to reach a prescribed level of measurement error by a factor of more than 2.
A schematic for a photon-heralded electron source is shown in figure 1 (appendix A of the supplementary material contains a table with a more complete set of symbols). Free electrons are generated from a standard electron gun, then focused and steered to pass within the evanescent field of a waveguide, generating photons as they pass. The photons are sent to a single photon detector, and the corresponding electrons to an electron-counting spectrometer. The interaction creates an entangled electron-photon state [22,23]. If the waveguide input is the ground state, then the output will have Poissonian photon-number statistics. Experimental demonstrations of coherent coupling to-date have mean photon number (per electron) g ≲ 0.1, however g > 1 is possible in principle. While electron-photon coupling mediated by a material (e.g. cathodoluminescence) can yield many photons per electron, limited optical collection efficiency reduces correlations.
In photon heralding, a typical setup involves a nonlinear medium which converts a higher energy pump photon to two lower energy photons: the signal and the idler. A photon detected in the idler channel 'heralds' the presence of a photon in the signal channel [24]. The traditional figure of merit for heralded single-photon sources is the Klyshko efficiency [25], which is the conditional probability of measuring a photon in the signal channel given the detection of a photon in the idler channel. The Klyshko efficiency for electron heralding can similarly be defined as the conditional probability of measuring an electron given the detection of a photon. However, this figure of merit does not penalize 'false negative' events, where an electron arrives without a coincident photon. A low false negative rate is critical for applications like dose control.
Instead, we will describe the heralding efficiency in terms of the fractional reduction of variance (FRV) in the estimate of the number of electrons N (p) e that pass the energy filter, given the detection of N (d) γ photons in a given time interval [t, t + ∆t]. Assuming the source can be described with Poissonian statistics, which is generally valid for typical operating conditions [17,18,26],
where κ e is the Klyshko efficiency for heralding electrons using photons and κ γ is the Klyshko efficiency for heralding photons using electrons
where η γ is the photon detection efficiency, f 1 is the probability that an electron belonging to the chosen sideband passes through the filter, f 0 is the probability that an electron outside of the chosen sideband passes through the filter, Γ (g) e = I/q is the average rate of electron production from the gun with current I (q is the electron charge magnitude), and Γ (bg) γ is the photon detector background count rate. A derivation of both equations is available in the section B in the supplement.
To achieve a high FRV, both κ e and κ γ must be close to 1. This requires a highly efficient photon detector. While a strong electron-photon coupling is not necessarily required, it must be possible for the energy filter to efficiently isolate the chosen sideband, which has intensity proportional to g and may be superposed with the tail of the zero-loss distribution (i.e. we need f 1 ∼ 1 and gf 1 ≳ f 0 ). In practice, this will require the sideband spacing (equivalently, the photon energy) to be much larger than the energy spread of the electron gun.
In the supplement we tabulate the FRV for various types of sources and detectors (table B1) and calculate the maximum energy filter efficiency (appendix B.1). Based on that analysis, we will assume η γ = 0.9, f 1 = 0.9, and f 0 = 0. With g = 0.01, I < 1 nA, and Γ (bg) γ < 1 kHz, the effect of background counts is negligible.
Then κ e ≈ f 1 and κ γ ≈ η γ , simplifications we assume through the rest of this work.
Dose calibration-measuring the mean beam current beyond the beam forming apertures-is difficult to do accurately in most electron-optical systems. Resolving the moment-to-moment Poissonian fluctuations, which ultimately requires non-destructive measurements of the beam, is currently impossible. In microscopy, dose estimation is important for in-situ electron microscopy and for controlling sample damage. In lithography, dose noise limits device yield and minimum feature size.
Without heralding, some noise can be suppressed by sampling
e , the number of primary electrons registered in one time bin (perhaps indirectly, e.g. through secondary electrons (SEs) bursts in SEM). If the probability of registering a primary electron is D, the FRV in the estimation of the dose is FRV(N e ) = 1. In other words, with a perfect electron detector and no loss in the sample, there is no shot noise and so no benefit to heralding. As derived in the supplement, the FRV for dose estimation obtained by combining data from both detectors is FRV
where
e ( fgη γ f 1 )D is the rate of additional background counts due to unheralded electrons. Using the photon detector, the FRV can be close to 1 even when the electron detection efficiency is low, which is the case for lithographic systems where the primary electron is used to expose a resist and SEs are not counted efficiently. When D = 0, the FRV for dose estimation is identical to the FRV for heralding (FRV = κ e κ γ ).
In electron lithography, Poissonian fluctuations in the beam current can cause errors in the written pattern. For a resist with critical areal dose density d c (charge per area required for full exposure), the expected relative dose error for a pixel of area A is 1/ √ d c A/q. The error can be reduced by using larger pixels, which increases the minimum feature size, or using a less-sensitive resist [27]. For a beam-current density J and clock speed C, the maximum write speed is possible for d c ⩽ J/C. For d c > J/C, the writing speed becomes current-limited. As an example, suppose J = 100 A cm -2 and C = 100 MHz. Then the maximum write speed is possible for resists with d c < 1 µC cm -2 or 6 electrons per (10 nm) 2 pixel. To get less than 10% dose error in each pixel, the resist sensitivity would need to be decreased by a factor of 16, slowing write speed proportionally. To achieve less than 10% error using (4 nm) 2 pixels, the write speed would need to be 100 times slower than the maximum clock speed. Often, it is not possible to choose a resist with the optimal critical dose due to issues of availability and process compatibility.
Alternatively, the trade-off between dose error and speed can be controlled with heralding. To do so, the exposure is divided into m stages and the dose applied at each stage is chosen based on the number of . The top axis shows the current-limited write time using a 100 A cm -2 electron source. In the grey regions, write time is limited by clock speed. The error can be decreased by increasing line width or, at the cost of write time, by using less sensitive resist. Alternatively, dose error can be reduced using a heralded electron source and applying the dose into m > 1 stages. Here, we assume the source is a FEG (so f 1 = 1) and the photon detector is an SNSPD (so κγ = 0.9). The points A, B, and C are referenced in the text.
remaining stages and the estimated dose applied so far. This multi-pass, heralded electron lithography requires up to m times as many clock cycles, slowing clock-limited write times. The source current is also reduced by a factor of gf, which slows current-limited write times. Despite this, in some circumstances heralding can significantly increase writing speed.
In figure 2, we use the heuristic strategy of selecting the dose at each stage k of the m exposures according to
where dk ′ is the estimate of the dose applied at stage k ′ , and F is a number between 0 and 1. We optimize F, which depends on d c and m, by calculating the average (root mean square) dose error for 10 5 multi-pass exposures for each trial value, terminating when F is within 1% of the optimal value. In general, the optimal value of F (which ranges between 0.11 and 0.85 in the figure) is closer to 1 when d c is large and when m is small.
For a resist with critical dose of 0.6 electrons nm -2 , the dose error for a 20 nm linewidth (4 nm pixels) is 30% (see the point labeled A). Reducing the error below 10% would require increasing the linewidth to more than 60 nm or finding a resist 9 times less sensitive (see points B and C, respectively). Alternatively, the dose error could be reduced by the same amount using a heralded source and a multi-pass exposure with m = 7 (e.g. with a 100 MHz clock). However, the multi-pass strategy has a higher minimum number of clock cycles. This would be a limiting factor for a 10 MHz clock. If a 20 nm linewidth is required for only a small fraction (≪ 1/m) of the pattern, then a targeted multi-pass exposure will not substantially change the total number of required clock cycles.
Information in phase contrast microscopy and EELS is encoded in the relative number of counts at different detector positions. In this sense, these modalities are self-normalized and do not benefit from heralding. However, in amplitude contrast (e.g. bright field) microscopy, the information is encoded in the number of electrons which reach the detector compared to the number which are scattered outside of the collection aperture. Heralding could benefit this modality by disambiguating fluctuations in beam current from variation in the sample's scattering power [28]. Similarly, heralding in SEM could disambiguate current fluctuations from variation in the SE yield. However this extra information comes at the cost of reduced beam current.
A heralded source may improve the SNR for fixed acquisition time if the measurement information increases sufficiently to compensate for the reduced signal rate. Often, the SNR is limited by effects other than acquisition time and beam current, like detector well depth or sample charging and damage. In these cases, an appropriate figure of merit is the information gain per electron. A standard theoretical tool for optimizing measurements is the FI, I(x), which is used in conjunction with the Cramer-Rao bound σ 2
x ⩾ (n
e I(x 0 )) -1 to determine the minimum measurement error σ x when estimating an unknown sample parameter x (e.g. thickness, scattering mean free path, SE yield, etc) near a particular value x = x 0 [29,30]. The quantity n
e is a sample from the random variable
e . The advantage of this formalism is that we can determine the information value of a measurement without constructing an optimal method for estimating x based on n (d) e . See section D of the supplement for a brief review of this topic. We can re-write the Cramer-Rao bound in terms of the SNR = x/σ x ⩽ x √ n (d)
e I(x 0 ). The fractional increase in FI achieved by heralding is
)
)
By examining equation (4), we can see two potential limiting cases where the fractional increase in information is large: when dark count rate is large (Γ
or when the contrast is low (D(x 0 ) ∼ 1). In the supplement (appendix E.1) we apply these results to bright field STEM and find that heralding could substantially improve dose efficiency for thin samples. However some existing measurement techniques, like differential phase contrast, provide a similar advantage.
In the remainder of this section, we will examine quantitative SEM as another use case for heralded microscopy. The interpretation of SEM images can change dramatically based on the beam energy and sample composition and morphology. For example, more SEs escape from the interaction volume near the edges of nanostructures, causing a bright halo and a reduction in contrast. This so-called edge effect [31,32] can be partially ameliorated using a heralded electron source. To show this we model the tilt-dependent SE yield as
where θ is the surface tilt, δ 0 is the SE yield for a horizontal surface (θ = 0), and δ 1 is the SE yield at the edge of a tall vertical step [33]. We assume the detector clicks at most once for each primary electron, which is true when a single SE saturates the detector for longer than the spread in SE arrival times, and so counting SEs is not possible [34]. We also assume that the SE yield is Poissonian (with mean δ). Then we write the probability of detection as
From equation ( 4), the increase in information from heralding is proportional to e ηeδ when the background count rate is small and heralding efficiencies are high. Heralding is especially advantageous when the SE yield is much larger than 1 (i.e. where θ ∼ π/2), so contrast is low due to detector saturation. In such cases, the most information-rich events are primary electrons which fail to produce a SE. Without heralding, these events cannot be recognized.
Figure 3 shows a simulation of surface tilt and SE yield estimation of a 450 nm bump with a parabolic profile in an SEM. The simulation neglects some aspects of image formation in SEM (e.g. shadowing and the point spread function), but it provides an approximate visual comparison of quantitative SEM with and without heralding. At each pixel, the number of incident primary electrons is drawn from a Poisson distribution with mean 10. Then numbers are drawn from a multinomial distribution to determine the number of coincident and non-coincident events at the electron and photon detectors. Without heralding the estimates of the surface tilt and SE yield are θ and δ. With heralding the estimates are θh and δh . See section E.2 of the supplement for more details.
Prior probability distributions for θ and δ must be specified. We assume a uniform distribution for θ (which induces a non-uniform distribution for δ). As the true surface tilt is not well-represented by a uniform distribution, these estimators are biased (e.g. |⟨ δ -δ⟩| > 0 at finite dose). As seen in the radial plots on the right of the figure, heralding reduces this bias (an effect not captured by the FI analysis).
The root mean square errors ε( X) for these estimates averaged over the circular footprint of the feature are ε( θ) = 0.31, ε( θh ) = 0.19, ε( δ) = 1.6, and ε( δh ) = 1.2. Heralding reduces the error for the estimation of θ and δ by a factor of 1.6 and 1.3, respectively. Equivalently, the dose required to reach a prescribed estimate error for δ and θ is reduced by a factor of 2.6 and 1.8 respectively. Compositional analysis could also be improved by heralding, but the advantage would only be exist when SE yields are high enough to saturate the detector.
We have described a realistic heralded electron source, calculated its statistics, and estimated its effectiveness in specific applications of STEM, SEM, and electron lithography. A general figure of merit for describing the effectiveness of a heralded source is the FRV, which is proportional to the Klyshko efficiencies for both electron and photon heralding. To achieve a high FRV in practice will not necessarily require additional advances in sources, detectors, energy filters, or electron-photon coupling structures, but it will require sophisticated engineering to integrate state-of-the-art implementations of each of these systems.
The goal of this analysis was to identify realistic circumstances where electron beam technologies may be improved by a heralded source. We chose to examine a series of minimal models which may motivate more detailed investigations of each potential application in the future.
Multi-pass heralded electron lithography would have reduced beam current and may require more clock cycles. However, it enables dynamic control of the trade-off between speed and noise. Using a single layer of resist, heralded electron lithography could quickly expose regions of low detail, then apply a low-noise multi-stage exposure to areas where noise could limit device yield.
Not all forms of microscopy benefit from heralding. For example, in phase contrast electron microscopy, information is extracted from the relative brightness of electron detector pixels. However, for amplitude (i.e. bright field) microscopy, heralding makes it possible to detect events where primary electrons fail to arrive at the electron detector. These events are particularly informative when they are rare (i.e. low-contrast imaging). For BF STEM of thin samples, a heralded electron source can more-than double the SNR at constant dose. Similarly, heralding is beneficial in SEM for low-contrast imaging conditions. An electron microscope equipped with a heralding system would likely be operated in a high-current, unheralded mode for alignment, focusing, and feature-finding, then switched to a low-current, heralded mode for dose calibration and enhanced performance.
This work clarifies the specific conditions in which improvement from heralding can be expected. It also motivates more detailed analysis and design of heralded electron sources, which have the potential to become indispensable to next-generation electron-optical systems. Probability that an electron outside of the chosen sideband passes through the filter f 1
Probability that an electron belonging to the chosen sideband passes through the filter f
Total probability that an electron passes through the filter D
Probably the electron detector (secondary or primary) registers a count given that a primary electron passed the filter κe Klyshko efficiency for heralding electrons using photons κγ Klyshko efficiency for heralding photons using electrons I Fisher information FRV Fractional reduction in variance
In this section, we calculate the fractional reduction in variance (FRV) achievable with the heralded source described in the main text. In general, the superscripts (g), (p), and (d) will distinguish between quantities relating to the electron gun, the exit of the energy filter, and the electron/photon detectors, respectively.
To begin, we assume the photon detector is capable of resolving arrival times into bins of size ∆t with ∆tΓ (g) e = ϵ ≪ 1, where Γ (g) e = I/q is the average rate of electron production, I is the beam current, and q is the electron charge. With this assumption, each of the events in the system can be described as Bernoulli (binary) random variables. Let N (g) e be a random variable describing the number of electrons produced by the gun in the time interval [t, t + ∆t]. While space charge and Fermionic particle statistics can in principle result in an anti-bunched electron beam, this effect is typically very small and has only recently been observed [17,18,26]. Therefore we will assume electron arrival times are uncorrelated so N
and p
where g is the average number of photons produced per electron (we will assume g ≪ 1), η γ is the photon system detection efficiency, and Γ (bg) γ is the photon detector background count rate. The term proportional to ϵ 2 comes from simultaneous signal and background events and is small as long as
e . We will assume this is true. Based on the definitions of g and η γ , we can also write the conditional probability p(γ d |e g ) = gη γ .
After interacting with the coupler, the electron beam is energy-filtered. We will assume the filter uses an energy-selecting slit which passes electrons with energy E 0 < E < E 1 . If S(E) is the energy distribution of the electron beam at the source and E γ is the energy of a signal photon (we will assume the energy distribution of the photons is very narrow compared to S(E)), then let
We can interpret f 0 as the probability that an electron passes the filter given that it did not produce a photon and f 1 as the probability that an electron passes the filter given that it did lose energy E γ to a photon. Then
is the total probability that an electron which emerges from the gun passes the filter.
In any given time bin, there are four possible outcomes
true negative with probabilities p ( e p |γ d ) true positive p ( ¬e p |γ d ) false positive p ( e p |¬γ d ) false negative p ( ¬e p |¬γ d ) true negative.
We can also condense these outputs into a single figure of merit in the form of the conditional variance Var
p) e |γ d ) + p (¬γ d ) Var ( N (p) e |¬γ d ) with Var ( N (p) e |γ d ) = p ( e p |γ d ) ( 1 -p ( e p |γ d )) Var ( N (p) e |¬γ d ) = p ( e p |¬γ d ) ( 1 -p ( e p |¬γ d ))
.
In order to calculate these conditional probabilities, we use the above relations to obtain
) .
Table B1. Dimensionless parameters describing potential heralding systems using either a field emission gun (FEG) with ∆E = 0.3 eV or a schottky gun (SG) with ∆E = 0.7 eV; and using either a superconducting nanowire single photon detector (SNSPD) with ηγ = 0.9 or a single photon Avalanch detector (SPAD) with ηγ = 0.7. In all cases, we assume the electron-photon coupling is c = 0.01, the beam current is I = 1 nA, the photon detector background count rate is Γ (bg) γ = 1 kHz, and the photon energy is
SNSPD+FEG 0.9 0 1 1 0.9 0.9 SNSPD+SG 0.9 0 0.9 0.9 0.9 0.8 SPAD+FEG 0.7 0 1 1 0.7 0.7 SPAD+SG 0.7 0 0.9 0.8 0.7 0.6
The intrinsic variance of the source (the variance without information from the photon detector) is Var
) .
Incorporating the information from the photon detector, the FRV is, to first order in ϵ,
where Γ (eγ) , Γ (e) , and Γ (γ) are the rates of coincident events, electron detector events, and photon detector events, respectively. Without heralding, the FRV is 0. With perfect heralding, the FRV is 1. In terms of the system parameters described above,
where Γ (bg) γ and η γ are the photon detector background count rate and detection efficiency, respectively.
Here we justify f 1 = 1 for a field emission gun and f 1 = 0.9 for a Schottky emitter. First we assume the tip current is low enough such that the energy distribution of the emitted electrons is not distorted by space charge effects. Adapting this from Riemer [35] we have
By setting the tip energy spread to 0.7 eV (for a Schottky emitter, which corresponds to a 3320 K tip temperature), we find the overlap of a single 1.1 eV photon sideband with the zero-loss distribution is 0.1, as shown in figure 4 below. An otherwise ideal energy filter could then have f 1 = 0.9.
For the field emitter, the energy spread is so small that this integral is practically zero, and so f 1 = 1. 1 eV separation, then there is a 10% chance that an electron that lost a photon will be identified as one that did not. This is found by integrating the curve above the decision threshold line, shown in purple.
Here we calculate the reduction in the variance of N (p) e achievable using both a photon detector and an electron detector. We will assume the electron detectors are fast enough to count primary electrons. This assumption may not be appropriate for the pixelated detectors used, for example, in transmission electron microscopy, but it is possible for STEM [36] and SEM [34] detectors with careful calibration.
First, to find the reduction in variance using just the electron detector, we first need Var
) e |N (d) e ) = p (e d ) [ p ( e p |e d ) ( 1 -p ( e p |e d ))] + p (¬e d ) [ p ( e p |¬e d ) ( 1 -p ( e p |¬e d ))] = p (e d ) [ p ( e p |e d ) ( 1 -p ( e p |e d ))] + p ( e p |¬e d ) + O ( ∆t 2 ) . And since p ( e p |e d ) = [ p ( e p ) p (e d ) p ( e d |e p ) ] = Γ (g) e f Γ (g) e fD + Γ (dc) e D + O (∆t) = 1 1 + r e where r e = Γ (dc) e Γ (g) e fD (C1) and p ( e p |¬e d ) = [ p ( e p ) p (¬e d ) p ( ¬e d |e p ) ] = ∆tΓ (e) g f (1 -D) + O ( ∆t 2 ) therefore Var ( N (p) e |N (d) e ) = ∆tΓ (e) g f ( 1 -D 1 + r e ) + O ( ∆t 2 ) . At this point, we can find the fractional reduction of N (p) e using N (d) e : FRV ( N (p) e |N (d) e ) = 1 -Var ( N (p) e |N (d) e ) Var ( N (p) e ) = D 1 + r e . Now we will calculate the additional variance reduction achievable with a heralding system. First, Var ( N (p) e |N (d) e , N (d) γ ) = ∑ n (d) e ,n (d) γ p ( n (d) e , n (d) γ ) Var ( N (p) e |n (d) e , n (d) γ ) = p (e d , γ d ) ( p ( e p |e d , γ d ) ( 1 -p ( e p |e d , γ d ))) + p (e d , ¬γ d ) ( p ( e p |e d , ¬γ d ) ( 1 -p ( e p |e d , ¬γ d ))) + p (¬e d , γ d ) ( p(e p |¬e d , γ d )(1 -p(e p |¬e d , γ d )) ) + p(¬e d , ¬γ d ) ( p(e p |¬e d , ¬γ d )(1 -p(e p |¬e d , ¬γ d )) ) . Since p ( e p |e d , γ d ) = 1 + O ( ∆t 2 ) and O (1 -p (¬e d , ¬γ d )) = O ( p ( e p |¬e d , ¬γ d )) = O (∆t) we can further simplify Var ( N (p) e |N (d) e , N (d) γ ) = p (e d , ¬γ d ) ( p ( e p |e d , ¬γ d ) ( 1 -p ( e p |e d , ¬γ d ))) + p (¬e d , γ d ) ( p ( e p |¬e d , γ d ) ( 1 -p ( e p |¬e d , γ d ))) + p ( e p |¬e d , ¬γ d ) + O ( ∆t 2 ) so we will need to find five probabilities: p(e d , ¬γ d ), p(¬e d , γ d ), and three permutations of p(e p |(¬)e d , (¬)γ d ). Let's start with p(e d , ¬γ d ).
The expression should contain one term representing a dark count at the electron detector and another where a real electron strikes the detector but either did not create a photon or the photon was lost.
p (e d , ¬γ d ) = ∆tΓ (dc) e + ∆tΓ (g) e ( fgf
) .
Using similar reasoning,
) .
Now the conditional probabilities. First, we have
) .
Then,
Putting everything together, we have Var
As described in the main text, a multi-pass procedure can be used to implement dose control. In our, strategy the dose applied at stage k is
where d c is the target dose and dk is the estimated dose at stage k. Without any information from the photon detector, we use dk = n γ,k /κ γ where n γ is the number of photons detected at stage k. For the purpose of simulation, we draw n γ from a binomial distribution B(n e ; κ e ) where n e is the number of electrons actually delivered to the sample, which itself is a random number drawn from a Poisson distribution with mean d k .
The optimal value of constant F depends on the total number of dose stages m and on d c .
The goal of a quantitative measurement is to estimate an unknown sample parameter, traditionally labeled θ.
In STEM, θ could represent sample thickness or scattering cross-section. In SEM, θ could represent the secondary electron yield or surface tilt. A function θ(X) which uses measurement data X to produce an estimate of θ is called an estimator. We consider X to be a random variable and the variance Var
⟩ is the square of the measurement error.
It is possible to place an upper bound on the measurement error without formulating θ using the Cramer-Rao bound Var
( θ) ⩽ (NI (θ 0 )) -1 (D1)
where
is the Fisher information (FI). The Cramer-Rao bound applies only to unbiased estimator (with zero expected error).
As an example, a Bernoulli random variable with probability mass function p(k
This means that after N samples of the distribution, the error of the best unbiased estimator is √ Nθ(1 -θ). Similarly, for a random variable with a Poisson distribution p(k; θ) = e -θ θ k /k!,
so after N samples of the distribution, the error of the best unbiased estimator is √ Nθ. Notice that when estimating the parameter of a Bernoulli distribution, the FI diverges for large and small values of θ. But when estimating the parameter of a Poisson distribution, the FI diverges only when θ = 0.
We now to estimate an unknown parameter x, which determines D(x), the probability that an electron produced by the heralded source triggers the electron detector. The Fisher information (FI), I(x), added per time bin is
As above, we simplify the calculation of I by assuming that the electron detector operates in a counting regime (it has a binary response to each primary electron) with D(x) the probability that an electron which is produced by the heralded source is counted by the electron detector. Without heralding, the FI for values of x near x 0 gained from sampling only N (d) e (i.e. without heralding) is
where Γ
(bg) e
is the rate at which unheralded electrons arrive at the detector. Using data from the photon detector, we have
where
and
) .
The first term in equation (E1) represents the information associated with photon detections (both with and without a coincident electron detection) and is similar in form to the FI associated with a binomial random variable with success probability D(x). The second term represents the information associated with electron detections which were not heralded and is similar in form to the FI associated with a Poisson random variable. Combining these two expressions above, the fractional gain in information is
) .
Notice this expression diverges for κ e = 1 and D(x 0 ) → 1. This is the same divergence we get by taking a ratio of the Bernoulli and Poisson FI: I B /I P = 1/1 -θ (see previous section).
In bright field (BF) STEM, an aperture at the backfocal plane of the objective lens removes electrons which scatter to large angles as they pass through the sample. One possible motivation for such a measurement is to estimate the thickness t of a sample with a known, material-dependent scattering mean-free-path λ. If the electron detector has quantum efficiency η e , then the detection probability is D(t) = η e (1 -e -λ/t ). In figure 5, we show the fractional increase in SNR (the square root of equation (E4)) relative to the unheralded case as a function of t for η e = 0.95. With an SNSPD, heralding doubles the SNR ratio at constant dose for samples with λ = 100 nm(200 nm) and t < 20 nm(42 nm). The information added by the heralding system in this case is equivalent to the information collected by a high-efficiency annular dark field detector covering all scattering angles excluded by the bright field detector.
Using FI, we can estimate the SNR improvement for quantitative measurement in an SEM. To show this, we will model the tilt-dependent SE yield as
where θ is the surface tilt, δ 0 is the SE yield for a horizontal surface (θ = 0), and δ 1 is the SE yield at the edge of a tall vertical step [33].
In order to estimate the secondary electron yield δ and surface tilt θ, we need to construct estimators δ and θ which are functions of the measurement data. The measurement data consists of tallies of the number of coincident events, N eγ , electron-only events, N e , and photon-only events, N γ . For applied dose d, we use the estimators and p(δ) is a probability distribution describing our expectations or prior knowledge of δ. We use analogous expressions for θ. For simplicity, we will assume p(θ) is uniform: Finally, according to the model described above, we have
p e (θ) = ( fgη γ f 1 ) D (θ)
with D(θ) = 1 -e -ηeδ(θ) .