SURFACE AND INTERFACE ANALYSIS Surf. Interface Anal. 2005; 37: 912–918 Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/sia.2109
Application of Monte Carlo simulation to SEM image contrast of complex structures Z. J. Ding∗ and H. M. Li Hefei National Laboratory for Physical Sciences at Microscale and Department of Physics, University of Science and Technology of China, Hefei 230026, Anhui, P.R. China Received 3 January 2005; Revised 3 May 2005; Accepted 4 May 2005
A new Monte Carlo technique for the simulation of scanning electron microscopy (SEM) images for an inhomogeneous specimen with a complex geometric structure has been developed. The simulation is based on constructive solid geometry (CSG) modelling, i.e. constructive solid geometry modelling with simple geometric structures, as well as a ray-tracing technique for correction of electron step-length sampling when an electron trajectory crosses the interface of the inhomogeneous structures. This correction is important for the simulation of nano-scale structures whose size is comparable with or even less than the electron-scattering mean free paths. Copyright 2005 John Wiley & Sons, Ltd.
KEYWORDS: secondary electrons; backscattered electrons; SEM images; Monte Carlo; CSG
INTRODUCTION Scanning electron microscopy (SEM) imaging of surface microstructures with secondary electrons (SEs) and backscattered electrons (BSEs) from a surface bombarded by electrons has been very important for the study of many kinds of materials in many scientific and technological fields.1 SE images, formed by secondary electrons of very low energies (several eV) emitted from the surface region, provide mainly topographic information of the specimen surface with subnanometer resolution in a modern scanning electron microscope. BSE images can provide more information about the matrix composition because the signal electrons are transported from the sample interior that is reached by the incident primary electrons (of several keV energy). Simulations of SEM images can help not only for explaining image contrast (for improving the precision of measurements by SEM) but also for understanding the physical processes of electron–solid interaction. Recently, some studies using a Monte Carlo electron-trajectory simulation technique have been carried out for several kinds of simpler geometrical specimens.2 – 13 Gauvin2 performed simulations of X-ray images and backscattered-electron images for a spherical inclusion of homogeneous composition embedded in a matrix, based on use of Mott elastic-scattering cross sections and a modified continuous slowing-down approximation. Their CASINO program is a single-scattering Monte Carlo simulation of electron trajectories in a solid, specially designed for the interactions of low-energy electrons in bulk solids and thin Ł Correspondence to: Z. J. Ding, Hefei National Laboratory for Physical Sciences at Microscale and Department of Physics, University of Science and Technology of China, Hefei 230026, Anhui, P.R. China. E-mail:
[email protected] Contract/grant sponsor: National Natural Science Foundation of China; Contract/grant number: 10025420 and 90206009.
foils, and can be used to generate the usual recorded signals in a SEM (X-rays, SEs, and BSEs) either for point analysis, line-scans, or images.3 – 5 Ly et al. simulated SEM images of spheres of different materials on a substrate surface and at various depths beneath the flat surface.6 Radzimski and Russ performed simulations of BSE images of three-dimensional (3D) multilayer and multielement structures, on the basis of a single-scattering procedure, to study electron beam and detector characteristics.7 Their simulation procedure also took into account the effects of the electrical and angular characteristics of a solid-state detector and the effect of the electron beam size on image quality and certain artifacts. Howell et al. developed a program to illustrate macrotopographies on electron backscattering.8 This program can simulate a target constructed with a choice of a flat surface, a circular filament, or a rough surface simulated by a sine wave. Another Monte Carlo simulation program, MONSEL, has been used to model the interaction of an electron beam with one or two lines lithographically produced on a multilayer substrate.9,10 The simulated signals include transmitted, backscattered and secondary electrons. Another application of this program was concerned with the simulation of twodimensional (2D) SE and BSE images of a simple notch.11 Recently, the MONSEL program has been extended by Seeger et al. to simulate SE and BSE images of a complex structure consisting of many triangles to create a complex specimen surface.12 Certain programming techniques enabled faster calculations. Yan and Gomati developed a 3D Monte Carlo code to simulate images of BSEs and Auger electrons for a more complex specimen.13 Their code required the 3D geometric structure to be described in analytic form. Because of the difficulty of simulating secondary electron generation and emission processes for specimens with complex structures, most previous studies emphasized
Copyright 2005 John Wiley & Sons, Ltd.
MC simulation and SEM image contrast of complex structures
simulation of BSE images with very simple structures, and only a few were capable of simulating SE images for specimens with complex structures. Also, inhomogeneous distribution of chemical composition inside a sample has not been considered generally. The purpose of the present study is, therefore, to develop a Monte Carlo simulation program that enables the calculation of secondary electron and backscattered electron signals for a complex structure with different atomic compositions distributed in particular spatial zones. The present program is based on the physical model of Ding and Shimizu for calculation of secondary electron generation, transport and emission.14 We have combined this approach with constructive solid geometry (CSG) modelling of the sample structure by using some simple and basic 3D objects that can be analytically described with a finite number of parameters. A ray-tracing technique has been employed to obtain corrections in the sampling of step lengths for electrons travelling inside such an inhomogeneous solid. A parallel Fortran90 program has been developed and used to simulate SE and BSE images of specimens containing nanoscale complex structures.
SIMULATION PROCEDURE
s
s0 ds0 D ln
2
0
provided that s is positive for all s > 0, and
1
s ds D 1
3
0
so that f s is normalized appropriately. If an inhomogeneous specimen comprises spatial regions that are each homogeneous, Eqn (2) can be reduced to the following summations, i Ti D ln i 4 sD Ti i
Sampling of step lengths The present Monte Carlo model uses Mott cross sections for describing elastic-scattering, a dielectric-function approach for inelastic-scattering (including plasmon excitations, interband transitions and inner-shell ionizations) that uses compiled optical data for metals, and considers the production of cascade secondary electrons. Our previous studies have demonstrated that simulations based on this model can yield excellent agreements with the measurements of the energy distribution of BSEs (from the elastic peak down to the peak of low-energy secondary electrons) and with measured backscattering coefficients and secondary electron yields.15 – 18 Because the specimens considered here are made of complex structural objects differing not only in size and shape but also in atomic composition, the electron-scattering mean free path is no longer independent of position. The conventional sampling procedure for electron-trajectory step lengths is inappropriate for such samples and must be modified. Here we have to consider the problem of determining step lengths when an electron crosses a boundary between two materials:19 (i) the particle is stopped at the boundary and a new trajectory step started using the mean free path for the material that the particle is entering; (ii) the trajectory continues by converting from units of mean free path to actual distance travelled in the new material; (iii) the actual distances travelled in every possible material are determined before starting a trajectory step. We adopt the third method in this work. Starting from the general expression for the distribution function for step lengths by assuming an exponential law:20 s f s D s exp s0 ds0 0
Copyright 2005 John Wiley & Sons, Ltd.
where f s is the probability of an electron travelling a distance s in one step in the specimen, s is the total scattering cross section of electrons in a solid, which is in units of cm1 and is obtained by multiplying scattering cross section for an atom with the atomic number density of the solid, or the inverse electron-scattering mean free path. A conventional Monte Carlo sampling technique for f s requires solving an accumulation function with a uniform random number 2 [0, 1]. This requirement reduces to
1
where Ti denotes a segment of an electron step length s within the i-th zone of a material with scattering cross section i . The problem is then reduced to the calculation of Ti .
Constructive solid geometry modelling There are two main approaches for describing the geometrical structure, one being surface- or boundary-representation modelling and the other being constructive solid geometry (CSG) or combinatorial geometry modelling.21 CSG modelling of a solid modelling is a method that uses simple solid bodies to build a complex structure using Boolean operations: union, difference and intersection, etc. We can then use some basic and simple 3D bodies as elements for constructing geometrical structures. The elements are defined with a few parameters; for example, a cube can be defined by the coordinates of one of its eight vertexes, the corresponding three orientations and three side lengths. Only one kind of material is assumed to be present in each 3D structure. First, a semi-infinite space with a flat top surface is used as the specimen matrix. Basic structures are then used to construct inhomogeneous zones inside the matrix or on the surface of the matrix, as illustrated in Fig. 1. The basic objects are chosen to be convex only, and include spheres, ellipsoids, cylinders, cones, cubes, tetrahedrons and polyhedra. For each such object, a corresponding subroutine was developed to calculate the distance between the last scattering point and the intersection of the electron velocity vector with the surface of an object. There are two reasons for choosing only convex 3D bodies. First, all kinds of concave 3D bodies can be constructed from convex structures. Second, for convex bodies, there are two and only two intersections on the body surface (one is for an electron incoming and the other is for the electron outgoing). The concave bodies would be much
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Figure 1. Sketch for constructing specimen geometric structures by CSG modelling.
Figure 3. Flow chart of the do-loop for calculating the step length.
Parallelization more complex owing to the larger number of intersection points.
Calculation of path length Knowing the last scattering position and electron moving direction, we have to decide the path length and therefore the next scattering position. Because the constructing basic shapes are spatially located in any possible position, it is necessary to judge all the possible intersecting points of one electron trajectory step, viewed as a ray, with every basic shape specified. This could be done by solving the simultaneous equations for the ray and shapes. The distance T0 -pairs (one T0 for electron incoming and the other for electron outgoing) are thus obtained, where T0 denotes the distance from the starting point to a particular intersection point. Having obtained all possible intersection points for an electron trajectory crossing the surfaces of 3D construction bodies along the direction of motion, we have to calculate the distance Ti that the electron travelled within the i-th material block. The right i-sequence should first be derived. A sorting subroutine is used to obtain the distance between two adjacent intersection points; a two-dimensional array saves both the length Ti and the corresponding scattering cross section i , as shown in Fig. 2. A do-loop then derives the correct step length from Eqn (4), as indicated in Fig. 3. Because the subroutines for calculating the intersection points for each kind of basic body shape are modular, this program can be expanded easily by adding new kinds of basic 3D bodies as necessary. A detailed description of the algorithm has been given elsewhere.22 T ′1 T ′2
T ′3
Monte Carlo simulation requires the calculation of a large number of electron trajectories to reduce the statistical uncertainty for each experimental condition, e.g. beam energy and incidence angle, sample specification (feature size, elemental composition, etc.). Also, one has to perform the calculation for many pixels by scanning the incident beam over the sample surface in order to obtain a simulated image. After including cascade secondary electrons of very low energy, the simulation is quite time consuming. The development of appropriate software and hardware has enabled parallel computing to become more popular. We have used message passing interface (MPI) to parallelize the Fortran90 program. With this programming, each computing node reads in the input parameters and then executes the calculation for one image pixel. Therefore, only the position of the primary electron beam on the surface is varied among the computing nodes, and other conditions (e.g. beam energy, number of primary electrons, sample specification) are kept the same. By the independence of the simulation procedure on the incident beam location, the computing nodes have almost no communication during a simulation. The acceleration is, therefore, almost linear with the number of nodes. A delay only occurs because of the varying simulation time among the sample locations, but the simulation time for nearby incident points is nearly equal. After finishing all trajectory simulations within each cycle on every node, the master computing node controlling the whole simulation process gathers results from all slave nodes and sends outputs to a file. The synchronization time for gathering results is very small. All the nodes then begin a new cycle for other pixels, as illustrated in the schematic diagram of Fig. 4. The typical calculation time for obtaining an image of 300 ð 300 pixels with a good signal-to-noise ratio is around one day when running on a 400 node parallel computer (every node is a 500 MHz CPU), and the parallel efficiency is as high as over 90%.
Sort
RESULTS T 1, s1
T 2, s2
T 3, s3
Figure 2. Schematic diagram for sorting a length Ti that an electron travelled within a particular 3D body. The Ti0 values are the distances from the previous scattering position to an intersection point. By the sorting procedure, the correct T-sequence is obtained with corresponding scattering cross section i . In this sample, T1 D T20 , T2 D T10 T20 , T3 D T30 T10 .
Copyright 2005 John Wiley & Sons, Ltd.
SE and BSE image contrast for SEM is obtained here by calculating the SE and BSE emission intensity for an incident electron beam bombarding different positions on a surface, without considering the characteristics of the electron detector and the magnetic/electric field in front of the detector. We have simulated artificial structures made of gold. A particular sample structure that may contain
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e−
New Circle
Barrier Output Result
Figure 4. Schematic diagram for distributing parallel computing nodes to a set of pixels corresponding to incident beam positions.
empty interior zones is constructed on the nanoscale to show possible changes of image contrast on this scale. Previous
(a)
calculations have shown that the visibility of Pt nanoparticles embedded in a carbon matrix changes with their depth.23 The simulated image contrast agrees reasonably well with the experimental results. The simulation can, of course, be applied to structures on larger length scales. Here, we assume for simplicity that the incident electron beam is infinitely narrow (for a finite probe size, the image could be easily obtained by convoluting the ideal image with a Gaussian distribution24 ). We present here some of our simulated images with 300 ð 300 pixels. For each image pixel, we have tracked 104 trajectories of 3 keV primary electrons and more than ten times that number of secondary electrons inside the solid. Figure 5 shows simulated images for a sample constructed with four vacant spheres in an Au matrix. The
(b)
Figure 5. The simulated SE image (a) and BSE image (b) for four compact vacant spheres in an Au matrix. The center positions of the spheres are: (5.4, 0, 0), (2.9, 5, 0), (2.9, 5, 0), (0, 0, 8.7) in nm. The sphere radius is 5 nm.
(a)
(b)
Figure 6. The simulated SE image (a) and BSE image (b) for four compact solid Au spheres on an Au matrix surface. The center positions of the spheres are: (5.4, 0, 0), (2.9, 5, 0), (2.9, 5, 0), (0, 0, 8.7) in nm. The sphere radius is 5 nm.
Copyright 2005 John Wiley & Sons, Ltd.
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centers of the top three spheres are on the matrix surface so that they actually form hemispherical pits; the other is just under them and in contact (i.e. totally buried). The primary electron beam is at normal incidence. The black region for the bottom sphere varies from a circular shape because of intense emission from the nearby spherical edge region, and the low brightness is obviously due to the difficulty of signal electron emission where primary electrons can enter deep inside the bulk directly. Figure 6 demonstrates simulated images for a sample constructed with four Au-contacted solid spheres on an Au matrix surface. The centers of the bottom three spheres are on the matrix surface; they form hemispherical protrusions. The primary electrons have a 45° angle of incidence. The conventional bright-edge contrast on the microscale is much
(a)
weakened at this feature size. We can also find that the top sphere is brighter than the bottom ones because the signal electrons generated in the top sphere can be more easily emitted into the vacuum. Figure 7 shows simulated SE and BSE images for a sample constructed with six crossed Au cuboids. The bottom three cuboids are on an Au matrix surface while the other three are above them. The angle of primary-beam incidence was 45° . The SE and BSE images show a little different contrast and the brightness changes with the specimen structure in SE image more sensitively. Owing to the influence of the bottom cuboids, the crossing region of the top cuboids with the bottom cuboids is darker in the SE image than in the BSE image. This is because BSEs have deeper signal-generation depths compared to SEs.
(b)
Figure 7. The simulated SE image (a) and BSE image (b) for six crossed solid Au cuboids on an Au matrix surface. The size of a cuboid is: width D 50 nm, thickness D 10 nm, length D 300 nm.
(a)
(b)
Figure 8. The simulated SE image (a) and BSE image (b) for an Au tube on an Au matrix surface. The size of the tube is: inner radius D 3 nm, outer radius D 5 nm, height D 10 nm.
Copyright 2005 John Wiley & Sons, Ltd.
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Figure 8 shows simulated SE and BSE images for an Au tube placed on an Au matrix surface. The sample was constructed with an empty cylinder inside a coaxial solid cylinder. The angle of incidence was 15° . Similar contrast was observed for each image except for the brightness. Figure 9 illustrates simulated SE and BSE images for a hollow Au sphere half-embedded in an Au matrix surface. The sample was constructed with an empty sphere inside a concentric solid sphere. The angle of incidence was 30° . The black region is due to the absence of Au in the hemisphere.
(a)
This simulation shows an extraordinary contrast for such a thin shell. Figure 10 shows simulated SE and BSE images for an Au octahedron on an Au matrix surface. The angle of incidence was 60° . The sample was constructed as a single polyhedron on the surface. In this example, different contrast is found for the SE and BSE images. Figures 5–10 indicate that the SE signals change to a greater extent with specimen morphology than the BSE signals. In addition, the contrast on the nanoscale may be different from that on the micrometer scale.25,26
(b)
Figure 9. The simulated SE image (a) and BSE image (b) for a thick Au bubble on an Au matrix surface. The parameters of the concentric spheres are: inner radius D 4 nm, outer radius D 5 nm, center position is at (0, 0, 0).
(a)
(b)
Figure 10. The simulated SE image (a) and BSE image (b) for an Au octahedron on an Au matrix surface. The vertexes coordinates are: (4.3, 2.5, 5), (4.3, 2.5, 5), (2.5, 4.3, 5), (2.5, 4.3, 5), (0, 0, 10), (0, 0, 0) in nm.
Copyright 2005 John Wiley & Sons, Ltd.
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CONCLUSION Using CSG modelling, complex sample structures can be constructed with some basic shapes having simple surfaces and containing different materials. This technique enables the simulation of SE and BSE images in SEM for complex morphologies of an inhomogeneous sample material. Illustrative calculations have demonstrated reasonable results for the image contrast. Since the material can be inhomogeneous, a correction to conventional calculations of step lengths is necessary. The distances along each trajectory between successive scattering events are obtained by subroutines. The module for calculation of trajectory length can be easily expanded to include arbitrary structural objects. The present Monte Carlo simulation program is thus very flexible and can be applied to a sample of arbitrary complexity. Nevertheless, when a sample comprises many 3D shapes, the computation can be quite time consuming. One method for accelerating the calculation is to perform parallelized calculations with a parallel computer. The present simple parallelization of the simulation code with MPI for distributing computation nodes to image pixels has achieved very high performance. For a very complex structure that would need too many basic objects with CSG modelling, we can further improve the calculation by use of the oct-tree partition of space, which is a hierarchical technique used to subdivide 3D space into 8 sectors (octants).27 The entire spatial region is then divided into some number of blocks, each containing a number of basic objects. The calculation of Eqn (4) is carried out block by block to minimize counting of every possible object along an electron-moving direction. Our program can be expanded easily to this partition method for the independence of modules for finding intersections, while still maintaining the independence of the modules.
Copyright 2005 John Wiley & Sons, Ltd.
Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 10025420 and 90206009).
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