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Monte Carlo simulation is a way to represent and analyze risk and uncertainty. After repeating the process a number of times typically to 10, , it estimates probability distributions for the uncertain outputs of the model from the random sample of output values.

The larger the sample size, the more accurate the estimation of the output distributions. This is true for simple discrete probability tree or decision tree methods.

The sample size you need is controlled by the degree of precision that you want in the output distributions you care about.

For most models, a few hundred up to a thousand runs are sufficient. You only need a larger sample if you want high precision in your resulting distributions, and a smooth-looking density function.

Given the inherent uncertainty in the inputs, higher precision is usually an aesthetic preference rather than a functional need. Sometimes you can fit a distribution to historical data based on the quantity.

In that case, you may ask an expert on the topic to express his or her best judgment about the quantity in the form of a probability distribution.

There is a well-developed method for expert elicitation of this form. If the quantity is important, you may consult several experts and aggregate their opinions.

Values for some elements and compounds are included, but the user can add or modify these values. We refer the user to the original article of each model for the validity of the models.

CASINO allows the user to choose various microscope and simulation properties to best match his experimental conditions. Some properties greatly affect the simulation time or the amount of memory needed.

These properties can be deactivated if not required. The nominal number of simulated electrons is used to represent the electron dose with beam diameter or beam current and dwell time.

The simulation time is directly proportional to the number of electrons. The shot noise of the electron gun Reimer, is included as an optional feature, which results in the variation of the nominal number of electrons N used for each pixel of an image or line scan.

The number of electrons for a specific pixel N i was obtained from a Poisson distribution P N random number generator with:. The SE feature is very demanding on computing resource.

For example, each 20 keV primary incident electron can generate a few thousands of SE electrons. Three types of scan point distributions can be used in the simulations: For all types, the positions are specified in 3D and a display is used to set-up and draw the scan points, see Figure 1B , or alternatively they can be imported from a text file.

To manage the memory used in the simulation, the user can choose to keep or not the data enabled distributions, displayed trajectories for each simulated scan point.

The cost of keeping all the data is the large amount of memory needed during the simulation and the large file size. The main advantage is to have access to all the results for each scan point which allows further post-processing.

For example, the energy absorption results presented in Figure 7 needed 4 GB of memory during the simulation. Simulation of the electron dose effect on electron beam lithography.

Two experimental secondary electron images, after electron beam lithography, where the pattern was A: The number of electrons per scan point was: The energy absorbed is normalized and displayed on a logarithmic scale.

The beam parameters now include the semi-angle and focal point, the energy range of the physical models are extended up to keV and the transmitted electrons are detected by an annular dark field detector ADF.

These changes are described in detail elsewhere Demers and others, The user should note that the rotation is applied around the Y axis first, when values are given for both directions.

For the first, distributions are calculated for each scan point independently of the other scan points. For the second type, the distributions are obtained from the contribution of all scan points either as line scan or area scan image.

The primary electron PE which is incident on the sample is either at the end of the trajectory simulation: Secondary electron SE and PE that exit the sample with energy less than 50 eV are used to calculate the secondary yield.

The following distributions are used to understand the complex interaction between incident electron and the sample. The maximum penetration depth in the sample of the primary and backscattered electrons, the energy of BSEs when escaping the surface of the sample, the energy of the transmitted electrons when leaving the bottom of the thin film sample, the radial position of BSEs calculated from the primary beam landing position on the sample, and the energy of BSE escaping area as a function of radial distance from the primary beam landing position are distributions available in CASINO and described in detail elsewhere Drouin and others, A new distribution calculated for each scan point is the energy absorbed in a 3D volume.

The volume can be described in Cartesian, cylindrical, or spherical coordinate. The 3D volume options are the position relative to the scan point, the size and number of bins for each axis.

To help choosing the 3D volume setting, a display shows the distribution volume position and size relative to the sample.

Care must be taken when choosing the number of bins as the memory needed grows quickly. A typical simulation of energy absorbed can use 2 GB of memory for one scan point.

The following distributions either sum the contribution of all scan points or compare the information obtained from each scan point. The total absorbed energy distribution is the sum of energy absorbed for all the electron trajectories of all scan points for a preset 3D volume.

In this case, the 3D volume position is absolute, i. Intensity distributions related to line scan and image are also calculated. The intensities calculated are the backscattered electrons, secondary electrons, absorbed energy, and transmitted electrons.

The absorbed energy intensity is defined by the sum of all energies deposited by the electron trajectories in the selected region for a given scan point.

The absorbed energy intensity signal will extend the scan point position and will be limited by the interaction volume. The intensity is either for the total number of electrons simulated or normalized by the number of electrons simulated.

The intensity variation between scan points is a combination of the shot noise effect, if selected, and sample interaction. For the analysis of the distributions presented previously it is useful to visualize the data directly in a graphic user interface before doing further processing using other software.

Figure 1A shows the user interface to create and visualize the sample in a 3D display. Figure 1C shows an example of electron trajectories simulated on sample shown in Figure 1A.

Through this interface one can visualize the electrons interaction with the sample. The color of the trajectories can be used to represent the type of trajectory: Another color scheme available allows to follow the regions in which the electron go through, as shown in Figure 1C , by selecting the color of the electron trajectory segment according to the region that contains it.

Another option for the visualization of the trajectory is to represent the energy of the electron by different colors. Also the collision elastic, inelastic and change of region events that occurs along the trajectory can be displayed with the help of small green sphere at the location of the collision.

The distributions obtained for all scan points are displayed as 2D graphic if the scan points form a straight line. In the case that the scan points form an image, an intensity image is displayed with a color bar mapped to the intensity value.

The color scale and minimum and maximum of the scale can be specified by the user. The signals or results obtained from the electron simulation of all scan points that can be used to form a line scan or an image are: For TE signal, the user can choose to see the effect of the detector on the intensity by using an ADF detector with user specified semi-angles and detector quantum efficiency DQE.

For most of the displays, the mouse allows to change the zoom, translate, or rotate the information presented. In addition, the intensity image can be saved as a high intensity resolution TIFF image bit float per pixel.

The simulation of an image needs a large number of scan points. Naturally the total simulation time increases with the number of scan points.

On a bit system there is no memory limitation, so the software can use all memory available. For the more advanced user requiring to investigate the parameterization effect of one or a few simulating parameters a console version of CASINO is available with a basic scripting language.

This feature allows the user to avoid to manually create a large numbers of simulation setting using the graphical user interface which can be time consuming when one requires a specific results such as the evolution of the backscattered electron coefficient with the incident energy shown in Figure 3 for example.

This feature allows the batch simulation of many simulations and to change one or more parameters for each simulation. The following examples illustrate the application of the simulation tool in relation to backscattered electron BSE and secondary electron SE imaging, electron gun shot noise, and electron beam lithography.

Figure 3 compares the simulation of backscattered electron coefficient for the electron incident energy lower than 5 keV with experimental values Bronstein and Fraiman, ; Joy, a for a silicon sample.

The simulated values are in agreement with the measured values except at very low energy less than eV where the simulation and experimental values do not follow the same trend.

It is difficult to assert the accuracy at very low energy of the simulation models from this difference. The experimental values at these energies strongly depend on the contamination or oxidation of the sample surface, which results in large variation of the values obtained experimentally Joy, a.

The linear interpolation problem reported in some Monte Carlo softwares El Gomati and others, was not observed in Figure 3. The interpolated energy grid for the elastic electron cross section data was chosen for each element to produce an interpolation error less than one percent when a linear interpolation model is used.

In a similar manner, the evolution of secondary electron yield with the incident electron energy was used to validate the secondary electron generation implementation in CASINO.

Figure 4 compares the simulation of secondary electron yields for the electron incident energy lower than 5 keV with experimental values Bronstein and Fraiman, ; Joy, a for a silicon sample.

The generation of secondary electrons in the simulation increases the simulation time drastically. For example, in bulk Si sample at 1 keV, the generation of SE increase the simulation time by a factor of 17, and 44 at 5 keV.

For each primary electron trajectory, a large amount of secondary electron trajectories are generated and simulated. For example at 1 keV, secondary electron trajectories are generated for each primary electron.

The amount of SE trajectories increases with more energetic primary electron, e. The increase of the simulation time is not directly proportional to the number of SE trajectories, because, most of these new electron trajectories are low energy electron slow secondary electron and will have few scattering events, which take less time to simulate than a primary electron.

The sample consists of Sn balls with different diameters on a carbon substrate. Two different incident electron energies were used 1 keV and 10 keV.

Simulated images of tin balls on a carbon substrate. The tin ball diameters are 20, 10, 5, and 2 nm. The field of view is 40 nm with a pixel size of 0.

The nominal number of electrons for each scan point was 1, For each image, the contrast range was maximized to the minimum and maximum intensity of the image.

The contrast C was calculated to compare the images using the following definition Goldstein and others, These three quantities are reported in Table I for each image.

Comparison of the contrast values calculated from backscattered electron and secondary electron images shown in Figure 5 for 1 and 10 keV incident electron energies.

For both signals, the smaller Sn nanoparticles are visible, because the interaction volume at 1 keV is of the order of few nanometers for both BSE and SE signals.

For BSE images, the contrast decreases with the increase of incident energy. The larger interaction volume decreases the signal from Sn nanoparticles as less electron interaction occurs in the particle.

The decrease of the contrast at 10 keV increases the importance of the noise on the image resolution. The resolution changes drastically between the two energies.

At 10 keV, the smaller tin balls 2 nm diameter are not visible and the 5 nm diameter balls are barely visible.

Similar change in resolution are observed on the SE images for the smaller tin balls 2 nm diameter , but the 5 nm diameter balls are easier to see than on the BSE image.

The SE emission decreases by a factor 10 when the incident energy is increase from 1 to 10 keV. The decrease does not change the contrast as both the carbon substrate and Sn nanoparticle are similarly affected.

Again the decrease of the signal, increases the effect of the noise on the image resolution. The topographic information from the SE signal is clearly observed in the large ball where the edges are brighter than the center.

These images are used to understand the impact of microscope parameters on image resolution and features visibility.

The number of electrons emitted by the electron gun is not constant, but oscillates around an average value. Figure 6 shows the effect of two different numbers of electrons on the BSE image quality.

The sample is a typical microelectronic integrated circuit shown in Figure 1A. At 20 keV, the interaction volume reaches the copper interconnects, which are buried nm depth from the sample surface and increase the BSE emission.

The presence of the tungsten via will increase the BSE emission. The increase by the W via was barely observed in Figure 6B with 10, electrons, but not visible in Figure 6A with 1, electrons.

The decrease of the nominal number of electrons from 10, to 1, illustrates the impact of the electron source noise on image quality.

The shot noise feature in CASINO is useful to calculate the visibility of feature of interest with different instrument parameters and feature size.

Effect of the shot noise on the backscattered electron images of integrated circuit ST sample. The nominal number of electrons for each scan point was A: The incident electron energy was 20 keV.

The field of view is nm with a pixel size of 10 nm. Under certain conditions, two close line patterns, separated only by 50 nm, are connected after the development of the resist.

Monte Carlo simulations of the sample and pattern were done for two different electron doses number of electrons: The expected patterns are clearly observed by their dark red color.

The absorbed energy in the pattern mainly comes from the incident beam. At 20 keV the electrons pass through the 50 nm resist film and nm dielectric film with little deviation.

Most of the elastic collisions occur in the Si substrate. With a pattern composed of more than , scan points, the contribution of the BSE on the absorbed energy cannot be neglected.

This is the background energy observed between patterns in Figure 7C and 7D. The long range combined with the random nature of the BSE exit position created a uniform and noisy background signal.

The average value of the absorbed energy background is proportional to the electron dose. We suspect that for a specific value of the electron dose, the absorbed energy background reaches the threshold value for the breakdown of the PMMA molecule and development of the resist occurs outside the expected patterns as observed in Figure 7B.

However, this is just one possible explanation of the failure. The electron exposure is only the first step of electron beam lithography. The resist development and profile evolution could be the source of the problem as well.

Improved simulation software for modeling signals generation in electron microscope from electron — sample interactions, which include a full 3D sample geometry and efficient 3D simulation model, has been developed.

All features are available through a graphical user interface. The software features like scan points and shot noise allowing for the simulation and study of realistic experimental conditions.

With the improved energy range, this software can be used for SEM and STEM applications, but with the limitation that the sample is considered as amorphous by the models and the simulation scheme used.

The software can be downloaded at this website: The software is in constant development for our research need and from user comments.

For obvious reason, the name of the program is not enough to find it. National Center for Biotechnology Information , U. Author manuscript; available in PMC Jul Find articles by Hendrix Demers.

Find articles by Nicolas Poirier-Demers. Find articles by Dany Joly. Find articles by Marc Guilmain. Find articles by Niels de Jonge. Find articles by Dominique Drouin.

Author information Copyright and License information Disclaimer. See other articles in PMC that cite the published article. Abstract Monte Carlo softwares are widely used to understand the capabilities of electron microscopes.

Monte Carlo simulation, scanning electron microscopy, secondary electron, three-dimensional 3D , scanning transmission electron microscopy.

Introduction Electron microscopes are useful instruments used to observe and characterize various types of samples: Features and Structure The simulation of electron transport in a 3D sample involves two computational aspects.

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