Figure 1: Schematic of an SEM [source: ]

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1 EECI Course: -9 May 1 by R. Sanfelice Hybri Control Systems Eelco van Horssen E.P.v.Horssen@tue.nl Project: Scanning Electron Microscopy Introuction In Scanning Electron Microscopy (SEM) a (bunle) beam of electrons, guie by an electromagnetic lens system, hits a specimen target (sample). This excites the specimen which can be etecte. The specimen may e.g. reflect the electron, can release seconary electrons or x-rays may be inuce as a result. This varies with the material the specimen consists of. Various etectors can be use to measure the effect. A schematic representation of an SEM is epicte in Figure 1. Figure 1: Schematic of an SEM [source: ] The relation between the original energy of the beam an the effect at the etector level can be use to ientify the material of the specimen at a specific spot. This focus spot is where the electron beam is bunle to a small iameter. The ientification is one by integrating the sensor value, accumulating the etector energy, over a specific time, the well time. This gives a value in the image. In SEM the focus spot is move over the area that is of interest of specimen in a (raster) pattern. This raster pattern is often a line by line sweep. The specimen area is grie, such that each gri area correspons to a in the image. While the focus spot is in a specific area of the gri, the etector value will contribute to the value of the corresponing. During a line sweep, the focus spot flows over the specimen. When it reaches the ege of the area that is of interest, it jumps to the start of the next line. When it reaches the en of the last line it 1

2 returns to the start of the first line. (Note that, in general, lines have an equal number of s, but is not necessarily the case) Note: In general the focus spot is not an ieal point, but is a istribution aroun a center point. This allows for an area of the specimen to be excite, the result of this excitation can be etecte in various ways. Assumptions Ieal focus spot: the location of the focus spot on the specimen is represente by a point on the specimen. (This spot coul correspon to the center of the beam istribution) Ieal bunle size: if the beam sweeps in a straight line from mile of the left ege of the area to the right, it will result in a value that is sufficiently representative of the specimen material in the area. Preetermine scan area: bouns for the focus spot are known

3 System moel A schematic of the system is show in in Figure. Figure : Scanning of a specimen in an SEM The following hybri moel H ( C, F, D, G,*) represents the movement of the focus spot p p p p with position ( xy, ) over the specimen, using a line-by-line raster pattern with non-consistent number of s per line. xs j 1 j n x xs 1 j n vsweep x vsweep x y j n Fp( z, u) y, Gpz, u y, z y, u n j n j j xs j 1 j n xe j 1 j n..., n C ( z, u) : ( z, u) x k x x k, j k, k 1,, p s e D ( z, u) : ( z, u) x x k, j k, k 1,,..., n p e Where, vsweep, an n 1 are the with/height (taken equal), the sweep well spee, an the number of lines in the raster, respectively, which are given constants. The bouns 3

4 (eges of the raster) for the focus spot, x an x, are the start an en positions for each line, s respectively. Each element of the array correspons to a line in the raster, where we have that e e x k x k for each k, such that the scan position lies in the positive quarant. Note that, for a well-efine gri, line k. e x k x k m is an integer value representing the number of s in k The system is a hybri system since state x has nontrivial flow (continuous) an jump (iscrete) ynamics. During a line sweep the system flows in positive x irection. At the en of the line the system jumps to the start of the next line. Note: A counter (iscrete ynamics) coul be ae, but it is not very interesting in this project since it oes not a hybri ynamics (only iscrete). Note: The scanning raster coul be rotate, such that the sweep is both in x an y-irection. This can be solve by a coorinate transformation. Extensions: Varying sweep spees per or line coul be use. The system coul jump at the ege of any an to any new ege. Disturbances may affect the x an y position of the beam uring flow. Disturbances may affect the jump map for the x an y position of the beam. Moeling the set point control system that controls the x an y position of the beam as a negative feeback on the error in the (x,y)-position.

5 linecounter j y position x position Simulations The scans will normally start at the beginning of a line. The system is initialize with initial state z x, y, j xs k, 1, k k for some integer1 k n. If the input u is kept constant, then the solutions of the system, the hybri arcs, are non-trivial, complete an compact. Also they are perioic, since the system will return to the initial state after a fixe perio of time, which epens on the inputs an the scan raster. Simulations of the system for initial conition z x, y, j xs 1,,1 Figure 3, Figure an Figure. Parameters chosen for the simulation are are shown in 1 vsweep well. an a curve pattern for the scanning raster: 7 lines by 1 columns with n 7 the three corner s remove. The simulation horizon is taken to be 1 secons or 1 jumps Figure 3: Trajectories of the states of the system

6 x y x Figure : Position of the focus point of the SEM, starting at the re circle in ark blue an ening at the re cross in ark re after just over one scan perio. The gri squares that have a thick line going through them are the s that are scanne t j Figure : Hybri arc of the x-position of the beam.

7 Analysis The system H satisfies the hybri basic conitions (HBCs): I. C an D are close subsets of n : C is a close line an D is a point in x, for each line number j. The number of lines n is iscrete an finite. The union of a finite number of close sets is finite. II. F is single value, F is continuous an C is in the omain of F. III. G is single value, G is continuous an D is in the omain of G. The trajectories of the system are perioic, this is inherent to the repetitive nature of the scanning proceure. No feeback of position errors is implemente. Therefore isturbances an uncertainties in the ynamics will prouce errors that will propagate in the trajectories of the system, giving a perturbe system H. The beam position may go out of the area of interest an prouce poor scan results. Position error feeback uring flow or position reset uring jumps coul cope with this to minimize the scan error. The feeback coul make perturbe trajectories converge to the nominal trajectory that is moele an simulate in this ocument. The nominal trajectory is then a compact set A, that is stable an locally pre-attractive, for the system H. This implies robustness if H satisfies the HBCs if a well-esigne feeback controller is implemente for the position error. Note that in practice it is very ifficult to measure the beam position without affecting the beam itself. Conclusion In this report it is shown that the scanning of a region of interest of a specimen in Scanning Electron Microscopy can be moele as a hybri ynamical system. The sweeping of the electron beam over the specimen can be escribe by a flow map, the scan area is escribe by a flow set, the movement of the beam to the start of a new line is escribe by a jump map, an the en of a line of s is escribe by a jump set. Simulations show the trajectories of the system, the hybri arc, with flows an jumps. A feeback metho for the error in the beam position, yieling robustness for the case where isturbances or uncertainties are affecting the system, is escribe. Suggestions are mae for useful extensions to the moel, which lie beyon the scope of this ocument. 7