Kwabena Arthur. at the. June2017. Certified by: George Barbastathis Professor Thesis Supervisor

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1 Simulation of X-Ray Phase Imaging on Integrated Circuits by Kwabena Arthur Submitted to the Department of Mechanical Engineering in Partial Fulfillment of the Requirements for the Degree of Bachelor of Science in Mechanical Engineering at the Massachusetts Institute of Technology June2017 MASSACHUSETTS INSTITUTE OF TECHNOLOGY JUL LIBRARIES ARCHIVES C 2017 Massachusetts Institute of Technology. All rights reserved. Signature of Author: ' I Department of Mechanical Engineering May 12, 2017 Certified by: George Barbastathis Professor Thesis Supervisor Accepted by: Rohit Karnik Associate Professor of Mechanical Engineering Undergraduate Officer 1

2 MITibraries 77 Massachusetts Avenue Cambridge, MA DISCLAIMER NOTICE The pagination in this thesis reflects how it was delivered to the Institute Archives and Special Collections. * The images contained in this document are of the best quality available.

3 Simulation of Tomography on Integrated Circuits by Kwabena Arthur Submitted to the Department of Mechanical Engineering on May 12, 2017 in Partial Fulfillment of the Requirements for the Degree of Bachelor of Science in Mechanical Engineering ABSTRACT A study on the simulation of the X-Ray phase imaging on multi-layered integrated circuits (IC) is presented. Model IC's were created with random nanoscale features. First order Rytov approximation was then used in order to accurately and rapidly create diffraction images. This study lays the foundation for future use as a forward model in limited-angle tomography or other inverse problems approaches (e.g neural networks) to reconstruct IC layout from x-ray diffraction images. In particular, it is hoped that the simulation presented here can be used to train neural networks that will carry out the inverse problem in experimental situations. The results of the study show that the first Rytov method is promising for use in this application of IC reconstruction. Thesis Supervisor: George Barbastathis Title: Professor 2

4 Table of Contents Abstract 2 Table of Contents 3 List of Figures 4 1. Introduction Phase-contrast X-ray imaging IC Imaging 5 2. Diffraction Simulation Full Wave Method First Rytov Approximation Accuracy of First Rytov Approximation 7 3. IC Generations General Principles Material Refractive Indices Assumptions on layers and layouts 8 4. Results and Discussion Diffraction Images of IC's Limited-Angle Tomography 9 Conclusion 10 References 11 3

5 List of Figures Figure 1: Coordinate system used in analysis 5 Figure 2 - A: Phase delay of component single layer refractive index Figure 2 - B: Attenuation of single layer refractive index Figure 3 - A: Subsection of an IC layer Figure 4 - A: Diffraction image of a 4-layer IC Figure 4 - B: Diffraction image of an 8-layer IC Figure 4 - C: Diffraction image of a 16-layer IC Figure 5 - A: Diffraction image on a 16 layer IC Figure 5 - B: Diffraction image on an 8' rotated 16 layer IC Figure 5 - B: Difference in diffraction images of perpendicular and 80 rotated 16 layer IC Figure 6 - A: Diffraction image on a 16 layer IC Figure 6 - B: Diffraction image on an 1 rotated 16 layer IC Figure 6 - B: Difference in diffraction images of perpendicular and 1 rotated 16 layer IC 4

6 1. Introduction: Phase-contrast X-ray imaging (PCI) allows for a non-destructive method of three-dimensional scanning and analysis of thin, or volumetric phase objects. Thin objects may be characterized directly, whereas for volumetric objects, an additional step of tomographic reconstruction is required. This study seeks to simulate the diffraction images created by performing PCI on ICs. The purpose of these images is to train a neural network enabling it to complete the inverse problem: generating the three-dimensional structure from diffraction images on real IC's. X-ray diffraction, like other electromagnetic radiation, is governed by the wave equation (1). A rigorous full-wave method for the diffraction is, however, computationally intensive. In this study, first order Rytov approximations were used to generate the diffraction images. The Rytov approximation assumes a more gradual change in refractive index over the length of the material. As such, the thickness and refractive indices of the material and medium are important parameters in accuracy of this explicit approximation. Simple realistic-looking IC-like layouts were created with nanometer scale structures. The features were randomized in order to create several examples necessary for training the neural network in the future. In this study, limited-angle projections were further used to generate in order to improve the results of structure reconstruction. 2. Diffraction Simulation 2.1 Full Wave Equation Method Figure 1 shows the geometry used in the analysis. An x-ray source is located a distance behind the object being imaged. The propagating wave interacting with an object can be fully described by the wave equation assuming correct boundary conditions: 5

7 Ij x Incoming wave front X-ray --- >-3 source Distorted wave front Z Detector Figure 1: Diagram of geometry used in analysis. Wave fronts are distorted by the object. (V 2 +k()2)l(?) = 0, (1) where k is given by the kon(r) where ko is the wavenumber in the medium and n() is complex refractive index of the object. The perturbation of the complex refractive index can be modeled as k( = k (1 - Q(r)). 2.2 Rytov Approximation Suppose the complex wave can be written in the form: P(i) = 0o(r)exp {(r)}, (2) where 0o(r) is the slow-varying wave envelope and q(r) the phase of the complex wave amplitude. The real and imaginary parts of 5 capture the distortion and attenuation of the wavefront, respectively. Using this expression for the wave leads to: V (Vd/) 2 + k 2 [1 - Q(r)]2 = 0, (3) The phase is then expanded in a perturbation approach as 00 = EM(m (4) m=o where e is a small parameter denoting the strength of the dielectric index perturbation. Using the expression for 0 and the scalar wave equation, the first three perturbation orders can be expressed as: 6

8 V 2 o0 + (Vo 0 ) 2 + k 2 = 0 (5) V 2 ol + 2Vo 0 -Vo 1 + 2k 2 Q(r) 2 = 0 (6) V Vob -V12+ 2k 2 Q(r) 2 + (Vo 1 ) 2 = 0 (7) The 0th order equation is the wave equation (1) and represents the solution for the incident wave in absence of the object. The first order equation is the first-order Rytov equation that will be used in this study. It can further be shown that: #1(r) = G(r - r') P 0 (r'){(vo 1 (r')) 2 - Q(r')}d 3 r' (8) where G(r) = exp(ikr) and Q (r) = k02(1 - n(r) 2 ) is the scattering potential of the object. The first Rytov approximation greatly simplifies the equation by assuming Equation (8) can now be simplified as: (Vol)2 < Q(9) 1 - f -(r) G(r - r') V5 0 (r')q(r')d 3 r' (10) - Vo (r)f By using the convolution principle of Fourier transform, we obtain a more elegant equation for # 3( )= 3(G) - 3(@OQ) (11) o 1 ( = 1(3(G) -3(o - Q)) 0 (12) (r) These three-dimensional Fourier transforms can be quickly be computed numerically and used to model wave interaction with an object. 2.3 Accuracy of First-Order Rytov Approximation The first-order Rytov approximation is valid under Equation (9). This is condition is especially strong for x-ray imaging since the refractive indices of metals have a 5 ~ 10-6 and fl 10-'. The refractive index measures the phase velocity of light, in contrast to its group velocity. Since its phase velocity carries no information, it can be much faster than the speed of light and refractive indices can be less than 1. 7

9 <I 3. IC Generation 3.1 General Principles IC layouts and architecture are proprietary information and varies between manufacturers. IC's used in this study were assumed to have a simple layout with feature sizes on the order of 20 nm. The three dimensional IC's were discretized into voxels. Artifacts that arise may due to the sharp voxel edges may be removed by decreasing the pixel size or applying smoothing filters. The materials used in the simulation were silicon, aluminum and copper. 3.2 Material Refractive Indices The refractive indices were determined from those of a 15KeV x-ray source [2]. The effects of different dopants on refractive index of silicon haven't been excessively studied. At lower concentrations, its effect can safely be assumed to be minute. The complex refractive index of the object describes its interaction with x-rays and is written in the form n= 1-6+if, where 5 and f represent the phase delay and absorption respectively. * m I Ea U &5 X,10,6 Figure 2: Shows a typical subsection of a single IC layer. (a) shows the real component, 1-6, of refractive index. b) shows the imaginary component, 8, of refractive index. 3.3 Layer specification Random copper square and rectangular features were populated on each layer. Aluminum wiring connects these features. The layers were assumed to have equal thickness of 20 nm with via layers in between. 8

10 -e.-.. EN.... U. - l U" E m U. * *IU U Figure 3: Images of a model of the layer of an IC and a subsection. IC's will be imaged in subsections and the resulting images will be stitched. 4 Results and Discussion 4.1 Diffraction Images of IC's Small sections of IC will be imaged at a time; this will be achieved with a nano-positioning stage. The resulting images will be stitched together. Figure 4 below shows simulated images from one of these subsections. Figure 4: (a) Diffraction image from a 4-layered IC (b) Diffraction image from an 8-layered IC (c) Diffraction image from a 16-layered IC. Complexity in images increases slightly as the layers are increased. 4.2 Improving Results Diffraction of subsequent layers becomes less prominent as the number layers are increased. This is due to attenuation by absorption of the x-rays and further diffraction as the x-ray propagates. As shown in Figure 4 above, the difference in the images is minute compared to the overall 9

11 contrast. By rotating the IC and causing the incident waves to arrive at an angle, we gain more information about subsequent layers as shown below. Figure 5: Refractive images of a 16-layer IC with (a) X-ray arriving perpendicular to IC (b) Incident waves arrive at an 8' angle. (c) Difference in (a) and (b) Figure 6: Refractive image of the IC in Figure 5. Refractive images taken a) incident wave perpendicular b) at an 1 angle c) the difference between (a) and (b) The figure above shows that even with a small angular rotation of 10, measurable difference can be captured. Limited-angle imaging can then be used to gain more information about subsequent layers. These images make the reconstruction possible and will greatly improve the accuracy of the overall system. The solution of the inverse problem, i.e. obtaining the actual IC layout from images such as these shown in Figures 5 and 6, is beyond the scope of this thesis. Future studies will demonstrate how a neural network can be trained from pairs of actual IC layouts (Figure 3) and diffraction patterns (Figure 4-6) so that, given the diffraction pattern from an unknown IC, the trained neural 10

12 network can then correctly reconstruct the unknown IC layout. Conclusion X-ray phase imaging is a powerful and important tool for non-destructive three-dimensional imaging. This study has shown that the first-order Rytov approximation can be used to create diffraction images for the purpose of training neural networks. This neural network should be able to reconstruct integrated circuit infrastructure from experimentally obtained diffraction images using these images for training and testing. 11

13 References Ardavan F. Oskooi, David Roundy, Mihai Ibanescu, Peter Bermel, J. D. Joannopoulos, and Steven G. Johnson, "MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method," Computer Physics Communications 181, (2010). B.L. Henke, E.M. Gullikson, and J.C. Davis. X-ray interactions: photoabsorption, scattering, transmission, and reflection at E= ev, Z=1-92, Atomic Data and Nuclear Data Tables Vol. 54 (no.2), (July 1993). Yongjin Sung and George Barbastathis, "Rytov approximation for x-ray phase imaging," Opt. Express 21, (2013) Yongjin Sung,. Colin J. R. Sheppard, George Barbastathis, Masami Ando, and Rajiv Gupta, "Full-wave approach for x-ray phase imaging," Opt. Express 21, (2013) 12