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Casino de Monte-Carlo 1992The Monte Carlo method is useful to help understand these instruments Joy, b. For various reasons, but principally because of the long simulation time and large computer memory needed, the previous version of CASINO was limited to simple geometry Drouin and others, To apply the Monte Carlo method to more realistic applications with complex sample, three-dimensional 3D Monte Carlo softwares are needed.

Various softwares and code systems were developed to fill this need of a 3D Monte Carlo software Babin and others, ; Ding and Li, ; Gauvin and Michaud, ; Gnieser and others, ; Johnsen and others, ; Kieft and Bosch, ; Ritchie, ; Salvat and others, ; Villarrubia and Ding, ; Villarrubia and others, ; Yan and others, However, either because of their limited availability to the scientific community or their restriction to expert users only, we have extended the software CASINO Drouin and others, to 3D Monte Carlo simulation.

Two main challenges were encountered with the simulation of 3D samples: This paper presents how we responded to these challenges and goals. We also present the new models and simulation features added to this version of CASINO and examples of their applications.

The simulation of electron transport in a 3D sample involves two computational aspects. The first one is the geometry computation or ray tracing of the electron trajectory inside the sample.

For complex geometry, the geometry computation can involve a large effort simulation time , so fast and accurate algorithms are needed.

The second aspect is the physical interaction with the matter inside the sample. Both are needed to successfully simulate the electron trajectory.

Using the electron transport 3D feature, the beam and scanning parameters allow the simulation of realistic line scans and images. From the simulated trajectories, various distributions useful for analysis of the simulation are calculated.

The type of distribution implemented was driven by our research need and various collaborations. Obviously, these distributions will not meet the requirements of all users.

To help these users use CASINO for their research, all the information from the saved electron trajectories, such as each scattering event position and energy, can be exported in a text file for manual processing.

Because of the large amount of information generated, the software allows the filtering of the exported information to meet the user needs.

The main aim of this work was to simulate more realistic samples. Specifically, the Monte Carlo software should be able to build a 3D sample and track the electron trajectory in a 3D geometry.

The 3D sample modeling is done by combining basic 3D shapes and planes. Each shape is defined by a position, dimension and orientation.

Except for trivial cases, 3D structures are difficult to build without visualization aids. The 3D navigation tool rotation, translation and zoom of the camera allows the user to assert the correctness of the sample manually.

In particular, the navigation allows the user to see inside the shape to observe imbedded shape. The first category has only one shape, a finite plane.

The finite plane is useful to define large area of the sample like a homogenous film. However, the user has to be careful that the plane dimension is larger than the electron interaction volume because the plane does not define a closed shape and unrealistic results can happen if the electron travels beyond the lateral dimensions of the plane see Figure 2E and next section.

The second category with two shapes contains 3D shape with only flat surfaces, like a box. The box is often used to define a substrate.

Also available in this category is the truncated pyramid shape which is useful to simulate interconnect line pattern.

The last category is 3D shape with curved surface and contains 4 shapes. For these 3D shapes the curved surface is approximated by small flat triangle surfaces.

The user can specify the number of divisions used to get the required accuracy in the curved surface description for the simulation conditions.

This category includes sphere, cylinder, cone, and rounded box shapes. Schema of the intersection of an electron trajectory and a triangle and the change of region associate with it.

Complex 3D sample can thus be modeled by using these basic shapes as shown in the examples presented in this paper. Each shape is characterized by two sides: A region, which defines the composition of the sample, is associated to each side.

The definition of outside and inside is from the point of view of an incident electron from the top above the shape toward the bottom below the shape.

The outside is the side where the electron will enter the shape. The inside is the side right after the electron crosses the shape surface for the first time and is inside the shape.

The chemical composition of the sample is set by regions. For each region, the composition can be a single element C or multiple elements like a molecule H 2 O or an alloy Au x Cu 1-x.

For multiple elements, either the atomic fraction or the weight fraction can be used to set the concentration of each element. The mass density of the region can be specified by the user or obtained from a database.

For a multiple elements region, the mass density is calculated with this equation. This equation assumes an ideal solution for a homogeneous phase and gives a weight-averaged density of all elements in the sample.

If the true density of the molecule or compound is known, it should be used instead of the value given by this equation. Also the region composition can be added and retrieved from a library of chemical compositions.

For complex samples, a large number of material property regions two per shape have to be specified by the user; to accelerate the sample set-up, the software can merge regions with the same chemical composition into a single region.

The change of region algorithm has been modified to allow the simulation of 3D sample. In the previous version, only horizontal and vertical layers sample were available Drouin and others, ; Hovington and others, An example of a complex sample, an integrated circuit, is shown in Figure 1A.

Top view of the sample with the scan points used to create an image. Electron trajectories of one scan point with trajectory segments of different color for each region.

The sample used is a typical CMOS stack layer for 32 nm technology node with different dielectric layers, copper interconnects and tungsten via.

When the creation of the sample is finished, the software transforms all the shape surfaces into triangles. During the ray tracing of the electron trajectory, the current region is changed each time the electron intersects a triangle.

The new region is the region associated with the triangle side of emerging electron after the intersection.

Figure 2A illustrates schematically the electron and triangle interaction and the resulting change of region. For correct simulation results, only one region should be possible after an intersection with a triangle.

This condition is not respected if, for example, two triangle surfaces overlap Figure 2B or intersect Figure 2C. In that case, two regions are possible when the electron intersects the triangle and if these two regions are different, incorrect results can occur.

The software does not verify that this condition is valid for all triangles when the sample is created. The best approach is to always use a small gap 0.

No overlapping triangles are possible with the small gap approach and the correct region will always been selected when the electron intersects a triangle.

The small gap is a lot smaller than the electron mean free path, i. Another type of ambiguity in the determination of the new region is shown in Figure 2E when an electron reaches another region without crossing any triangle boundaries.

As illustrated in Figure 2E , the region associated with an electron inside the Au region define by the finite plane the dash lines define the lateral limit and going out of the dimension define by the plane, either on the side or top, does not change and the electron continue his trajectory as inside a Au region.

A typical 3D sample will generate a large number of triangles, for example , triangles triangles per sphere are required to model accurately the tin balls sample studied in the application section.

For each new trajectory segment, the simulation algorithm needs to find if the electron intersects a triangle by individually testing each triangle using a vector product.

This process can be very intensive on computing power and thus time. To accelerate this process, the software minimizes the number of triangles to be tested by organizing the triangles in a 3D partition tree, an octree Mark de Berg, , where each partition a box that contains ten triangles.

The search inside the partitions tree is very efficient to find neighbour partitions and their associated triangles. The engine generated a new segment from the new event coordinate, see electron trajectory calculation section.

The 10 triangles in the current partition are tested for interception with the new segment. If not, the program found the nearness partition that contains the new segment from the 8 neighbour partitions and created a node intersection event at the boundary between the two partition boxes.

From this new coordinate, a new segment is generated from the new event coordinate as described in the electron trajectory calculation section.

The octree algorithm allows fast geometry calculation during the simulation by testing only 10 triangles of the total number of triangles in the sample , triangles for the tin balls sample and 8 partitions; and generating the minimum of number of new segments.

The detailed description of the Monte Carlo simulation method used in the software is given in these references. In this section, a brief description of the Monte Carlo method is given and the physical models added or modified to extend the energy range of the software are presented.

The Monte Carlo method uses random numbers and probability distributions, which represent the physical interactions between the electron and the sample, to calculate electron trajectories.

An electron trajectory is described by discrete elastic scattering events and the inelastic events are approximated by mean energy loss model between two elastic scattering events Joy and Luo, It is also possible to use a hybrid model for the inelastic scattering where plasmon and binary electron-electron scattering events are treated as discrete events, i.

The calculation of each electron trajectory is done as follow. The initial position and energy of the electron are calculated from the user specified electron beam parameters of the electron microscope.

Then, from the initial position, the electron will impinge the sample, which is described using a group of triangle surfaces see previous section.

The distance between two successive collisions is obtained from the total elastic cross section and a random number is used to distribute the distance following a probability distribution.

When the electron trajectory intercept a triangle, the segment is terminated at the boundary and a new segment is generated randomly from the properties of the new region as described previously.

The only difference is that the electron direction does not change at the boundary. This simple method to handle region boundary is based on the assumption that the electron transport is a Markov process Salvat and others, and past events does not affect the future events Ritchie, These steps are repeated until the electron either leaves the sample or is trapped inside the sample, which happens when the energy of the electron is below a threshold value 50 eV.

If the secondary electrons are simulated, the region work function is used as threshold value. Also, CASINO keeps track of the coordinate when a change of region event occurs during the simulation of the electron trajectory.

This EECS model involves the calculation of the relativistic Dirac partial-wave for scattering by a local central interaction potential. The calculations of the cross sections used the default parameters suggested by the authors of the software ELSEPA Salvat and others, These pre-calculated values were then tabulated and included in CASINO to allow accurate simulation of the electron scattering.

The energy grid used for each element tabulated data was chosen to give an interpolation error less than one percent when a linear interpolation is used.

A more accurate algorithm, using the rotation matrix, was added for the calculation of the direction cosines. The slow secondary electrons SSE are generated from the plasmon theory Kotera and others, To generate SE in a region, two parameters, the work function and the plasmon energies, are needed.

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. 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.

If the quantity is discrete, or you find it easier to treat it as discrete, you can ask the expert for probabilities of each discrete value.

There are variants of Monte Carlo simulation that can be more efficient than simple random sampling - they converge faster, reaching higher accuracy, with a smaller sample size.

When generating its n runs, it samples exactly once from each interval. In so doing, it achieves a more uniform sampling over each input distribution than standard Monte Carlo, where the natural randomness usually results in more clumped sampling.

Median LHS is not guaranteed to be unbiased, but in the vast majority of real applications it is unbiased and it usually converges faster than simple Monte Carlo or random LHS.

Microsoft Excel and other spreadsheets do not support Monte Carlo simulation directly. But, there are a number of software products that are add-ins to Excel that let you perform Monte Carlo simulation.

You can define any variable, or any cell in an array, as a discrete or continuous distribution. You can view the probability distribution for any resulting variable as a set of probability bands selected percentiles , as a probability density function, cumulative distribution function, or even view the underlying random sample.

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