Febr. B Monte-Carlo-Simulationen der Wechselwirkungen des B. mit der frei zugänglichen Software Casino, „monte CArlo SImulation of electroN. Oct 17, by Monte Carlo simulation in electron-beam lithography. S.-Y. Leea) through simulations using the software CASINO v The rest of the. Gauvin, R., P. Hovongton, D. Drouin, P. Horny, H. Demers und A.R. Couture. CASINO - monte CArlo SImulation of electroN trajectory in sOlids.
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. 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, It is clear for a user to see, line by line, that new layers are being added to the layer and you can use this pattern within a forloop.
This means that you can do many simulations and draw them from a single function. In this pattern it is also relatively easy to assign colors to series of gamblers if they result in a long term profit or loss.
Also, it might make sense to have the length of the random path be an input for our simulation function.
The following functions help the simulation. Notice that I am assuming the probability of getting black p B and the getting p R to be equal to 0.
This is not how a casino would play roulette but we will use it as an upper bound. If I can show that in this optimistic version of roulette the outcome of this tactic is doubtful then it will surely fail in a more realistic roulette game.
Later we will also consider a more realistic roulette game. Ouch, on the long term it does seem like you are making a net profit but you are not without risk.
The probability of getting red 12 times in a row is small but when you are playing this game for a very long time then this event suddenly becomes likely.
And when it happens you need to compensate with 2 12 cash. So the simulations are giving us reasons to be pessimistic about the tactic, but the long term end result does seem to be positive.
What does mathematics tell us? I will be a bit formal here, but thats a given when dealing with math. Suppose that we play the roulette game until we have made a profit of 0 or 1.
Note that a single game G can have multiple bets. Then also let p B be the probability for getting black, p R the probability for getting red, s B the payout for getting the state of black and s R the payout for getting the state of red.
This could feel very curious but should come with no suprise. If we play the game an infinite amount of time we will have an infinite amount of risk.
In laymans terms, if you want to earn an infinite amount of money you will need an infite amount of money. Even if we assume that you have an infinite amount of betting money, the tactic will still fail in real life.
The main reasons is that casinos tend to apply a maximum bet in all their gambling games.