3D Simulation of Sputter Etching with the Monte-Carlo Approach

Transcription

1 3D Simulation of Sputter Etching with the Monte-Carlo Approach 3D Simulation des Zerstäubungsätzens mittels der Monte-Carlo Methode Der Technischen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg zur Erlangung des Grades Doktor-Ingenieur vorgelegt von Daniel Kunder Erlangen 2011

2 Als Dissertation genehmigt von der Technischen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg Tag der Einreichung: Tag der Promotion: Dekan: Prof. Dr.-Ing. R. German Berichterstatter: PD Dr. techn. P. Pichler Prof. Dr.-Ing. P. Wellmann

3 Acknowledgment I would like to express my gratitude to all those who gave me the possibility and helped me to write this thesis. First of all, I thank my doctoral adviser Priv.-Doz. Dr. Peter Pichler. His supervision and suggestions gave me new insights and ideas. I am especially grateful for his support to write this thesis in English. I thank Prof. Dr. Peter Wellmann for his readiness to co-examine this thesis. Many thanks go to Dr. Eberhard Bär for his excellent supervision. Especially, the numerous helpful discussions about topography simulations were a large benet to accomplish this thesis. For many useful discussions about sputtering, I would like to thank Dr. Alex Burenkov. The sputtering experiments were carried out by the technology department of the IISB. Special thanks goes to Dr. Mathias Rommel who made these experiments possible and gave me new insights in experiments with focused ion beam milling. I thank Prof. Dr. Heiner Ryssel and Prof. Dr. Lothar Frey for the possibility to work at Fraunhofer IISB. At this place, I would also like to thank Dr. Jürgen Lorenz for his support of this work. I owe special thanks to Dipl.-Phys. Matthias Sekowski for helpful discussions about sputtering and the program MC_SIM. I thank Dr. Alberto Martinez and Dipl.-Phys. Johann Schermer for helpful remarks on this thesis. Besides, I would like to thank all colleagues at Fraunhofer IISB with whom I had the pleasure to work with. In particular, I would like to thank Dr. Peter Evanschitzky, Dr. Thomas Graf, Dipl.-Ing. Christian Kampen, Dipl.-Phys. Oliver Rudolf, Dr. Thomas Schnattinger, and Dr. Bernd Tollkühn. Finally, I want to thank my parents for their support which made it possible to accomplish this thesis. iii

4 iv Acknowledgment

5 Abstract Sputtering as physical process is frequently used for structuring in technology. A typical example is focused ion beam (FIB) milling with which submicron-sized structures can be produced out of an arbitrary material without the requirement of lithography. As a second prominent example, sputtering is used to sharpen the tip of eld emitter of eld emitter arrays. For all these processes, simulation of the topography changes due to sputtering can be very useful to understand and optimize them. A key part of this work is the integration of the Monte-Carlo ion implantation simulator MC_SIM into the 3-D topography simulator ANETCH. This way, ANETCH is now able to simulate topography changes due to sputtering for a wide range of ion/target combinations without a-priori knowledge about the respective yield from experiments. Although the models used in Monte-Carlo programs are usually accurate enough for simulating ion implantation proles, this is not necessarily true for sputtering. In particular, to improve the accuracy of the simulations, a modication of the electronic stopping model in MC_SIM is suggested. Another limitation of Monte-Carlo programs is that they consider only planar surfaces. In the implementation performed in this work, the full surface topography provided by ANETCH was taken into account to calculate sputtering. This is particularly important at corners where sputtering may occur even when the respective surface segment is not exposed to the ion beam. Using the implementation of MC_SIM into ANETCH, characteristic features of sputtering were investigated. The most important ones occurring particularly during FIB processing are the formation of sloped side walls and an increase of the etch rate at the bottom of trenches close to the side walls known as microtrenching. The main inuencing factors determining the slopes of the side walls like redeposition, the spatial distribution of the ion uence, and the angular dependence of the sputtering yield are discussed in detail. Microtrenching was found to depend particularly on the reection of the ions at the side walls. A comparison with the often found assumption of a specular reection of ions at sloped side walls indicates that the ions are in reality reected further away from the side walls which leads to a less pronounced but spatially more extended microtrenching than predicted by the simple assumption of specular reection. Important for the sharpening of tips is also the side-wall propagation at a steep step due to a uniform ion irradiation. Simulations with ANETCH were able to reproduce this phenomenon and an analytical model was developed that explains the slope of the side wall. To validate the implementation of the integration of MC_SIM into ANETCH, the topographies of trenches fabricated by FIB milling with dierent currents were compared to the respective simulations. Good agreement was found and the experiments as well as the simulations showed that microtrenching is reduced and eventually vanishes for higher beam currents because of the more pronounced spreading of the beam. Finally, to validate the dependence of the sputtering yield on the angle of incidence, dedicated FIB experiments were designed rst by simulations. A comparison of the respective experimental results to simulations conrmed the suitability of the model particularly for FIB processing. v

6 vi ABSTRACT

7 Contents Acknowledgment Abstract Symbols and Abbreviations iii v xi 1 Introduction 1 2 Fundamentals of Physical Sputter Etching Geometrical and Electrical Setup Diode Setup (Sputter Etching) Triode Setup (Ion Milling) Setup of Focused Ion Beam Milling Nomenclature of Physical Dry Etching Theoretical Concepts of Physical Sputter Etching Fundamental Concepts Material Characteristics: Amorphous or Crystalline Collision Regime of the Sputtering Process Level of Description of the Sputtering Process Interactions and Binding Forces Nuclear Collisions vii

8 viii CONTENTS Nuclear Stopping Cross Section S n (E 1 ) Collisions with Electrons and Electronic Stopping Cross Section S e (E 1 ) Surface Binding Energy U s Bulk Binding Energy U b Ion Implantation Results of the Sputtering Theory by Sigmund Sputtering Yield for Monoatomic Targets and Normal Incidence Fit of the Yield Formula to Simulation Results Multi-Element Targets Dependence of the Sputtering Yield on the Angle of Incidence θ I Angular Distribution of Sputtered Atoms Energy Distribution of Sputtered Atoms Simulation of Physical Dry Etching Reduction of the Computation Time in ANETCH Constant Atomic Fraction at the Surface of the Substrate Amorphization and Swelling Calculation of the Particle Flux Surface Discretization Monte-Carlo Flux Model (MCFM) for the Calculation of the Particle Flux Calculation of Sputtering Yield Data Table Simulation of Sputtering (DTSS) Monte-Carlo Simulation of Sputtering (MCSS) Surface Evolution Surface Shifting

9 CONTENTS ix Renement and Coarsening Study of Selected Examples with ANETCH Reection of Ions Redeposition of Sputtered Atoms Evolution of Side Walls Propagation of Steep Steps in the Surface Comparison with Experimental Results Calibration of the Electronic Stopping Model Trench Etching with Focused Ion Beams Optimization of the Fluence for an Experiment Summary and Outlook Summary Outlook Bibliography 119 Index 125 Zusammenfassung 129

10 x CONTENTS

11 Frequently Used Symbols and Abbreviations Latin Symbols Symbol Meaning Unit A Area m 2 AR Aspect ratio 1 a Screening length Å a F Screening length by Firsov Å a ZBL Screening length by Ziegler, Biersack, and Littmark Å H S Heat of sublimation ev E 1 Energy of projectile in a nuclear collision ev E I Energy of ion ev E T Energy of sputtered atom ev ER Etch rate m/s F Fluence cm 2 F D (E I ) Energy of an ion deposited in a material per unit length ev/m m Exponent of an interatomic power potential (V (r) 1/r m ) 1 M Atomic Mass u n T Atomic density of target material m 3 p Impact parameter Å r Distance between two atoms during a collision Å R P Projected range Å S Selectivity 1 S n Nuclear stopping cross section ev/m 2 s n Reduced nuclear stopping cross section 1 s KrC n Reduced nuclear stopping cross section (KrC screening function) 1 s W n HB Reduced nuclear stopping cross section (WHB screening function) 1 s ZBL n Reduced nuclear stopping cross section (ZBL screening function) 1 S e Electronic stopping cross section ev/m 2 Se LS Electronic stopping cross section (Lindhard-Schar model) ev/m 2 xi

12 xii Symbols and Abbreviations Symbol Meaning Unit Se OR Electronic stopping cross section (Oen-Robinson model) ev/m 2 Se Z Electronic stopping cross section (Ziegler model) ev/m 2 T Transferred energy during a collision ev U b Bulk binding energy ev U s Surface binding energy ev V (r) Interatomic potential ev x i Atomic fraction of species i 1 Y Sputtering yield 1 Y S Sputtering yield (only atoms from the target) 1 Y R Sputtering yield (only reected ions) 1 Z Atomic number 1 Greek Symbols Symbol Meaning Unit ɛ Reduced energy 1 λ Free path length Å Λ Materials factor in the sputtering theory by Sigmund m/ev φ Screening function (V(r) = Z 1Z 2 q 2 4πε 0 r φ KrC Screening function (KrC parameterization) 1 φ W HB Screening function (WHB parameterization) 1 φ ZBL Screening function (ZBL parameterization) 1 ϕ T Azimuthal emission angle Ω Solid angle 2 θ I Angle of incidence θ T Polar emission angle Fundamental Constants Symbol Meaning Value a 0 Bohr radius ( m) ε 0 Vacuum permittivity ( As/Vm) Reduced Planck constant (h/(2π) = (J kg) 0.5 m) k B Boltzmann constant ( J/K) q Elementary charge ( C)

13 Symbols and Abbreviations xiii Abbreviations Symbol Meaning 2D Two-dimensional 3D Three-dimensional DTSS Data table simulation of sputtering (Section 4.3.1) FBM Flux balancing method FIB Focused ion beam FWHM Full width at half maximum MC Monte-Carlo MCFM Monte-Carlo ux model (Section 4.2.2) MCSS Monte-Carlo simulation of sputtering (Section 4.3.2) Glossary Symbol Meaning ANETCH Topography program for 3D ANETCH2D Topography program for 2D IONSHAPER Topography program [P + 06b] FIBSIM Topography program [BH01] AMADEUS Topography program [K + 07a, K + 07b] MC_Sim Ion implantation and sputtering yield program [U + 05] SRIM Ion implantation and sputtering yield program [Zie04] SDTrimSP Ion implantation and sputtering yield program [Eck91] Sentaurus Technology computer aided design program [Syn09]

14 xiv Symbols and Abbreviations

15 Chapter 1 Introduction Materials have been manipulated since tools have been rst used by humans. By now, the technological challenges have reached the sub-micron range. One technique to fabricate structures in the sub-micron range is sputtering, particularly focused ion beam (FIB) milling. In sputtering, particles impinging on a solid remove material due to momentum transfer. Although sputtering is known since the middle of the 19th century when it was observed in gas discharges, its importance arose in the last decades and thorough investigations have been performed to understand the process [Beh81a, Beh81b, Beh07]. The need to understand and control sputtering is based on the various applications of sputtering for structuring in technology. In the fabrication process of eld emitter arrays, the tips of the eld emitters can be sharpened by sputtering. Focused ion beam milling is often used for circuit repair. Additionally, defective masks for optical lithography can be repaired by focused ion beam milling. It is also possible to fabricate small structures out of arbitrary materials without the requirement of lithography. With this technique, parts of atomic force microscopes can be fabricated. A great advantage of focused ion beam milling compared to other etch processes is the possibility to structure targets which have a pronounced topography. To control these fabrication processes, a better understanding of the physics of sputtering is necessary. Experiments are a good method to gain information about the sputtering process. For example, the angular distribution of sputtered atoms reported by Wehner and Rosenberg [WR60] gave a clue that sputtering is caused by momentum transfer and not evaporation. Due to these experiments, it was possible to develop theories of sputtering for example the one proposed by Sigmund [Sig69]. Another method to study the sputtering process is computer simulations. Programs to calculate the sputtering yield are for example SDTrimSP [BE84], MC_SIM [U + 05], or SRIM [Zie04]. However, these programs do not calculate the topography changes which are associated with sputtering. To calculate the evolution of the surface, programs like AMADEUS [K + 07a] or IONSHAPER [P + 06b] were developed. The main disadvantage of these programs is that they need information about the sputtering yield in the form of look-up tables. The goal of this work was to 1

16 2 CHAPTER 1 INTRODUCTION develop a program that simulates the topography changes due to sputtering for a wide range of ion/target combinations without a-priori knowledge about the respective yield. This was done by integrating the Monte-Carlo ion implantation program MC_SIM into the 3D-topography simulator ANETCH. The structure of this thesis was chosen to rst give an overview of sputtering, then present the integration of MC_SIM into ANETCH, and nally to discuss simulation results obtained with the modied program ANETCH. In the rst part of the overview of sputtering, dierent setups for the sputtering process and how sputtering can be controlled by the setup are summarized in Chapter 2. The dierent setups inuence, for example, the spatial distribution, the angular distribution, and the energy distribution of the ions. This information is important because it is needed as input in the topography program ANETCH. In the second part of the overview, some of the physical processes involved in sputtering are summarized and an introduction of the sputtering theory proposed by Sigmund [Sig69] is given. The results of the theory were studied for gallium ions impinging on a silicon target and compared with results obtained by computer simulations carried out with MC_SIM. This case is particularly important because gallium ions are used usually for FIB milling. The physical processes which are still not well understood are pointed out and for the electronic stopping an alternative model is suggested. In Chapter 4, the integration of the ion implantation program MC_SIM into the topography program ANETCH is presented. A limitation of MC_SIM is that only planar surfaces are considered because at corners sputtering may occur even when the respective surface segment is not exposed to the ion beam. With the integration of MC_SIM into ANETCH, the full surface topography provided by ANETCH is taken into account to calculate sputtering. Although physically more correct than the usage of sputtering yield tables, the drawback of using a Monte-Carlo program to calculate sputtering is a signicantly increased computation time. As possible remedies, the parallelization of the calculations and the automatic generation of sputtering tables via MC_SIM are discussed. In Chapter 5, some characteristic features of sputtering are presented which were investigated with simulations carried out by ANETCH with the integrated MC_SIM. For structuring, two important characteristics which were studied are the formation of sloped side walls and an increase of the etch rate at the bottom of the trenches close to the side wall known as microtrenching. For the sharpening of tips, the side-wall propagation at a steep step is important. This was investigated by the integrated programs of MC_SIM and ANETCH and an analytical model was developed to describe the propagation. In Chapter 6, the integration of MC_SIM into ANETCH was validated by comparing simulations to dedicated experiments. In one experiment, trenches were etched with a focused ion beam machine and two dierent beam currents into silicon and the cross sections are compared with the respective simulations. To validate the dependence of the sputtering yield on the angle of incidence, dedicated FIB experiments were designed. To estimate the uence at which reected ions and redeposition have only a minor inuence on the surface prole, simulations were carried out. With these simulation results, the experiments were carried out with the respective uence.

17 In the last chapter, Chapter 7, a summary of this thesis is rst presented. In the second part of the chapter, the new possibilities to study sputtering are discussed which are now available due to the extended program ANETCH. 3

18 4 CHAPTER 1 INTRODUCTION

19 Chapter 2 Fundamentals of Physical Sputter Etching In the fabrication of microelectronic patterns, dierent etch processes are used to remove material from the wafer. The etch processes can be classied into two main groups: Wet etching and dry etching. In wet etching, liquid-phase etchants are used and in dry etching, typically plasma-phase etchants. For both groups, the substrate can be partially masked to etch structures into the substrate or covering layers. Ideally, the mask is not or only slightly aected by the etchant. Then, the masked parts of the substrate are not etched and the other parts are. The particle removal can, for example, be isotropic or anisotropic as indicated in Figure 2.1. Although an anisotropic structure is often preferred, etched structures are typically between isotropic and anisotropic in a real process. Mask isotropic anisotropic Substrate Figure 2.1: Comparison of an isotropic etched prole and an anisotropic etched prole. In the last decades, the sizes of fabricated patterns have shrinked and, today, it is necessary to etch patterns of sub-micron dimensions and with an anisotropic etch prole. Wet etching is not a reasonable technique to fabricate these patterns because surface tension might preclude a wet etchant from reaching down between photoresist features. For the structuring of patterns with small lateral dimensions, dry etch techniques have become important because anisotropic proles can be achieved. The dierent dry etch techniques 5

20 6 CHAPTER 2 FUNDAMENTALS OF PHYSICAL SPUTTER ETCHING Dry Etching Physical Physical and chemical Chemical Sputter Ion Focused Plasma Reactive Electron Chemical etching milling ion etching ion or photon dry beam (beam) induced etching milling etching dry etching Figure 2.2: Overview of dry etching processes. The focus of this work was on physical dry etch processes (gray shaded). are summarized in Figure 2.2. In chemical dry etching, free radicals are produced in a plasma and react at the surface of the substrate. The free radicals impinge on a planar wafer surface with all possible angles of incidence. For a partially masked substrate, the radicals hit also the side walls of an etched structure causing undercutting of the mask layer. Therefore, anisotropic proles cannot be achieved with chemical dry etching. In physical dry etching, energetic ions are accelerated towards the wafer. Atoms are then removed from the surface due to nuclear collisions with the ions. These ions are almost directional when impinging on the wafer and rarely impinge on a vertical side wall of an etched structure. Therefore, these processes are almost perfectly anisotropic. The third form of dry etch processes are based on a mixture of physical and chemical dry etching. The surface is etched due to chemical reactions with free radicals. However, contrary to chemical dry etching, energy must be supplied to increase the probability for the chemical reactions to occur. Otherwise, the surface is etched signicantly slower. Dierent methods are used to supply the energy at the surface: For example, energetic ions, electrons or photons can be used to supply energy. The etched proles can be anisotropic if ions, electrons, and photons are focused onto the surface. Physical dry etching is used in surface cleaning, unselective thin-lm removal, the fabrication of micro-holes, circuit repair, or the fabrication of microstructures in arbitrary materials. In this thesis, physical dry etching was investigated and, therefore, the setups of physical dry etching processes are further discussed in this chapter. The energy and angular distribution of the ions when impinging on the substrate surface are controlled by the geometrical and electrical setup. The setup also controls the spatial distribution of the uence at the substrate surface. With the geometrical and electrical setup, the etched prole can be inuenced because it depends on the distributions mentioned above.

21 2.1 GEOMETRICAL AND ELECTRICAL SETUP 7 Diode Setup Cathode Substrate Plasma Anode Cathode Substrate Triode Setup Plasma Chamber Plasma Anode Accelerating electrode Plasma Chamber Second Chamber Figure 2.3: Comparison of diode and triode setup Three geometrical and electrical setups have been established: For sputter etching, the substrate is xed to the cathode in the plasma chamber. This setup is called diode setup because only two electrodes are used as shown in Figure 2.3. For ion milling, the substrate is not xed to the cathode but to a substrate holder in a dierent chamber. Ions are accelerated from the plasma chamber into the chamber with the substrate by a grid-like electrode. Because of the third electrode, this setup is named triode setup. A setup similar to ion milling is focused ion beam milling where the ion beam from a liquid metal source is focused down to a sub-micron diameter at the substrate surface. In Section 2.1, the geometrical and electrical setups are discussed and their inuence on the energy and angular distribution of the ions and on the spatial distribution of the uence is studied. In Section 2.2, the nomenclature of physical dry etching used throughout this work is presented. 2.1 Geometrical and Electrical Setup Diode Setup (Sputter Etching) Using the example of a direct current (dc) glow discharge, the diode setup is explained. Afterward, the same setup is used but with an alternating current (ac) at the cathode instead of a direct current. A direct current glow discharge is a plasma formed by the passage of current at 100 V to several kv through a gas at low pressure. A plasma is a mixture of free electrons, ions, and neutral atoms/molecules generated by the partial ionization of a gas. A gas can be transferred from the gas phase into the plasma phase by supplying energy to the

22 8 CHAPTER 2 FUNDAMENTALS OF PHYSICAL SPUTTER ETCHING gas. Energy can be supplied for example by electromagnetic elds, by irradiation, or by heating the reactor wall. By applying electromagnetic elds, free electrons and ions are accelerated. Neutral gas atoms can be ionized by inelastic collisions with other atoms or by collisions with electrons. Collisions also happen when atoms or electrons impinge on the reactor wall. The degree of ionization is the ratio between ionized and neutral gas species. As an example, the degree of ionization is typically between 10 6 to 10 4 for a low-pressure glow discharge plasma and the pressure in the chamber is typically between 0.1 to 1000 Pa. For the setup of a dc glow discharge, two parallel electrodes with the same area are opposed to each other as shown in Figure 2.3. The electrodes are in a tube which is lled with an inert gas, for example argon. One electrode is connected to a current source. In the following, it is rst described how a plasma is established in a glow discharge and, afterwards, the dierent regions in a glow discharge are distinguished. Initially, only a small fraction of the atoms is ionized due to random processes like inelastic collisions between atoms or cosmic radiation. The electrical eld between the two electrodes increases due to the current source and the positively charged ions are accelerated towards the cathode and the negatively charged ions and electrons are accelerated towards the anode. On their way, the ions and electrons will have collisions with other atoms in the gas. During these collisions, further atoms can be ionized. Depending on the density of the gas, the accelerated electrons undergo several collisions with the gas atoms before they reach the anode. The kinetic energies of the electrons depend on the free path length of the electrons and the magnitude of the electrical eld. If it is suciently high, the target gas atoms can be ionized by the collisions. These processes increase the number of free electrons and ions. If enough atoms are ionized, a plasma has been established and can sustain itself. However, the plasma has not yet been established in the whole volume between the anode and cathode but at rst only in a channel connecting anode and cathode as indicated in Figure 2.4. The plasma establishes where most ions and electrons have been generated rst. Due to the current between the electrodes, the initial electrical potential dierence between the two electrodes drops to a xed value which depends on the pressure and the distance between the electrodes [Ben10]. The smaller electrical eld between anode and cathode prevents that a plasma is established elsewhere in the chamber between the electrodes. However, the plasma increases in lateral dimension as shown in Figure 2.4 because ions and electrons leave the channel in lateral dimension due to collisions and cause further collisions and an ionization of the neutral atoms. The current between the two electrodes increases until the plasma is established in the whole volume. The potential between the electrodes in equilibrium is shown in Figure 2.5. Three main regions between the electrodes are distinguished [SN06]: The cathode plasma sheath (CPS), the glow region (GR), and the anode plasma sheath (APS). In the CPS, the electric eld is large and decreases from the cathode towards the interface to the GR where it is almost zero. This electric eld causes electrons to be accelerated from the cathode towards the GR and positive ions to be accelerated from left border of the GR towards the cathode. In the CPS, a small region at the cathode, the Aston dark space, appears dark because the energy and density of the electrons there is too small to excite a larger

23 2.1 GEOMETRICAL AND ELECTRICAL SETUP 9 Anode Anode I 0 < I Cathode Cathode Figure 2.4: The current between the electrodes increases due to the increase of the lateral extension of the plasma (light grey) while the voltage between the two electrodes is constant. V Cathode Emitting photons Anode 0 V e AB C D E F G Cathode Anode Plasma Glow Region Plasma Sheath Sheath Figure 2.5: Schematic voltage distribution in a dc glow discharge in equilibrium. Dierent regions can be distinguished: A = Aston dark space, B = cathodic glow, C = cathode dark space, D = negative glow, E = Faraday dark space, F = positive column, G = anode dark space. number of neutral atoms. In this region, the net space charge is negative and the electric eld is large. Next to the Aston dark space, the electrons accelerated due to the electric eld have sucient energy to excite a larger number of neutral atoms. This part of the CPS is called cathodic glow region. The region between the cathodic glow and the glow region is the cathode dark space. It is relatively dark because the density of electrons is small due to their high energy. The electron energy is high enough to cause ionization

24 10 CHAPTER 2 FUNDAMENTALS OF PHYSICAL SPUTTER ETCHING when colliding with neutral atoms. The net space charge therein is positive. The electrons accelerated through the cathode sheath have high energies when entering the glow region. In the left part of the GR, electrons colliding with neutral atoms will predominantly ionize them in case of high electron energies and excite them in case of low electron energies. In the negative glow region, the densities of ions and electrons are almost the same and the largest in the plasma. The electric eld is negligibly small. The electrons carry almost the entire current due to their higher mobility. Next to the negative glow region, the Faraday dark space establishes where the energy of the electrons is too small to excite the neutral atoms. The right part of the glow region, the positive column, has a small electric eld which is large enough to maintain the current. Due to the electric eld, the region at the anode is almost free of positive space charge. The electrons are accelerated towards the anode and due to their increased kinetic energy their density is small in the region at the anode. For the sputtering process, a substrate can be attached to the cathode. In this case, the energy and angular distribution of the ions impinging upon the substrate are important. Furthermore, the lateral distribution of the uence at the substrate surface is important for the sputtering process. In a rst assumption, the kinetic energy of singly positively charged ions E I is given by E I = qv e, (2.1) where V e is the voltage between the electrodes and q the elementary charge (q = C). The ions impinge on the surface with almost normal incidence if collisions close to the cathode can be neglected and an electrical eld homogeneous in lateral direction is assumed. For typical densities in the chamber, collisions of the ions with neutral atoms close to the cathode cannot be neglected. This reduces the average energy of ions impinging on the substrate and also the direction of the ions is less directional. Computational-uid-dynamic programs like ESI-CFD [EC10] can be used to calculate the energy and angular distribution of the ions when impinging on the substrate. For the setup described so far, the etch rate is low because the degree of ionization in the plasma is small. Another disadvantage is that a dielectric substrate cannot be sputtered in this setup. Ions impinging on an insulator cannot discharge and the insulator will charge in turn. In consequence, the dielectric substrate screens the electrical charge of the cathode. Therefore, the plasma sheath at the cathode would vanish and electrons and ions would not be accelerated anymore. In this way, the plasma could not sustain. To improve the setup of the glow discharge, an ac voltage is applied to one electrode instead of the dc voltage and the other electrode is grounded. To generate the ac voltage, a radio frequency (RF) generator is usually taken. Between the generator and the electrode, a capacitor is inserted. As before, it is assumed that a small fraction of the atoms in the gas is initially ionized due to random processes like inelastic collisions between atoms or cosmic radiation. By applying the ac voltage, these electrons and ions oscillate between the two electrodes. Due to inelastic collisions, more atoms are ionized and a plasma is formed. Some of the energetic electrons impinge on the electrodes, while ions don't reach the electrodes in this stage due to their lower mobility. A positive plasma potential (V P ) establishes. The electrode connected with the rf generator becomes negatively charged because the electrons impinging on the electrode cannot discharge due to the capacitor.

25 2.1 GEOMETRICAL AND ELECTRICAL SETUP 11 A negative net voltage (V DC ) establishes at this electrode. The ion acceleration averages to a mean acceleration towards the electrode connected with the capacitor. Assuming that singly positively charged ions are accelerated without collisions towards the electrode connected with the capacitor, the energy E I of ions impinging on this electrode is E I = q (V P V DC ). (2.2) As before for the setup with the dc voltage, ions typically have collisions with atoms and, therefore, the average ion energy is below E I and the direction of the ions is less directional. For this setup, typical ion energies are between a few ev to several hundred ev. The advantage of the ac setup when compared with the dc setup is the higher density of ions. Furthermore, a dielectric substrate can be sputtered with the ac setup. The diode setup causes the whole substrate surface to be exposed to ions because the substrate is xed to the cathode and the substrate surface is smaller than the cathode surface. To increase the time electrons stay in the plasma chamber, a magnetic eld can be applied. In this case, the electrons do not pass directly from the cathode to the anode but follow helical paths. A review of RF-generated plasmas can be found elsewhere [Mad02] Triode Setup (Ion Milling) In a triode setup, the substrate is in a chamber which is separated from the plasma. The ions are accelerated into the substrate chamber by a third grid-like electrode as shown in Figure 2.3. The plasma can be generated like in the diode setup (Section 2.1.1). An improvement can be made because the substrate is not xed at the cathode. Instead of a planar cathode, a hot lament can be used. Due to the thermal energy, the electrons can overcome the work function to leave the lament, known as thermionic emission. These electrons are accelerated towards the anode. Ions are extracted from the plasma chamber with a grid-like electrode to which a negative charge is applied. This causes positive charged ions to be accelerated from the plasma towards the grid. The substrate is behind the grid in a second chamber and the ions pass the grid and impinge upon the substrate. If the substrate is an insulator, a positive charge is established at the substrate, which slows down the ions. To avoid charging of the substrate, a hot wire can be placed in this chamber. It emits electrons which are accelerated towards the substrate and neutralize the positive charge. The energy of ions impinging on the substrate can range from some hundred ev to several kev. The triode setup allows to use higher ion energies than the diode setup. In the triode setup, the substrate can be tilted so that the ions impinge upon the surface under an angle of incidence dierent from normal incidence. Due to the higher vacuum in the chamber with the substrate, the angular distribution of the ions is more directional than in the diode setup. A problem with the grid is that it is also sputtered by the ions and the sputtered material can contaminate the substrate material. Ion milling might cover an area of the

26 12 CHAPTER 2 FUNDAMENTALS OF PHYSICAL SPUTTER ETCHING order of cm 2 [Mad02]. But it is also possible to focus the ions to a small area (< 10 4 Å 2 ) which increases the etch rate Setup of Focused Ion Beam Milling A special type of ion milling is focused ion beam (FIB) milling. It is a mask-less technique with a very narrow beam which enables direct writing. The beam spots can have diameters down to the order of 10 nm [Mad02]. A focused ion beam machine has typically a liquid ion source. Liquid ion sources are available for dierent elements like Al, As, Au, B, Cs, Cu, Ga, Ge, Ni, Pt and Si. A schematic gure of a liquid metal ion source is shown in Figure 2.6. The metal is inside Gallium reservoir Extractor Electrodes Tungsten needle Figure 2.6: Schematic gure of a liquid gallium ion source the coil and becomes liquid when the coil is heated. The tungsten needle extends into the coil. Liquid metal then ows to the tip of the tungsten needle. An electrical eld is applied between the positively charged tip and the negative extraction electrodes. These electrostatic forces pull on the liquid in outward (away from the tip) direction and surface tension pushes the liquid inward to minimize the surface. As a result of both forces, the liquid forms a cone at the tungsten tip. This characteristic shape is called a Taylor cone [Leh05]. The apex of the cone is not innitesimally small but has a radius of a few nm. At the apex, the electric eld is large enough so that eld evaporation [KW84] occurs. The work function of the atoms at the apex is reduced due to the large electrical

27 2.1 NOMENCLATURE OF PHYSICAL DRY ETCHING 13 elds. Simultaneously, the atomic orbitals are heavily distorted due to the high electrical eld and atoms at the apex are ionized with a high probability. The experiments carried out for this work and which will be described in Section 6 were done with two focused ion beam machines which both have a gallium ion source. In focused ion beam milling, the source is often gallium because of a variety of advantages: First, almost all ions are singly charged [Swa94]. Second, only two stable gallium isotopes exist in nature making it easier to produce mono-isotopic gallium. Third, the melting point is at K which is only slightly above room temperature. Finally, the vapor pressure at the melting point is with approximately Pa [Swa94] so low that only a few atoms are thermally evaporated from the surface. The ions extracted from the source are focused by a system of lenses. Each lens consists of three ring electrodes. The two outer electrodes are grounded and a negative voltage is applied at the middle electrode. This forms an electrical eld which focuses the beam. Other electrodes deect the beam to a certain position on the surface. The beam can also be deected to a Faraday cage where the ion beam current can be measured. To reach a specic kinetic energy, the ions are accelerated by electrodes. With newer gallium-based FIB machines, the energy of ions impinging upon the substrate can typically be selected between 5 to 50 kev. Due to the electrical eld, the ions in the beam are almost directional if collisions of the ions with gas atoms are very unlikely and the repulsion between the ions can be neglected. The rst condition depends on the pressure in the vacuum and the second one depends on the ion current. Ion beam proles extracted by Lugstein et al. [L + 02] from experiments can be approximated by a normal distribution. At the tails of the distribution 1-2 orders of magnitude below its maximum, the measured values exceed the tted normal distribution. Therefore, the beam prole is sometimes approximated by a superposition of two normal distributions. The ion beam can be characterized by the full width at half maximum (FWHM). If areas with diameters larger than the ion beam diameter should be etched, the beam is deected to dierent spots on the surface until the whole area was exposed to ions. A patterning strategy denes to which spots the ion beam is deected and if there is any overlap between the spots. Furthermore, the time, dwell time, must be chosen for which the beam is focused to one spot before moving to another spot. The choice of the time depends on the ion current. Experiments reported by Rommel et al. [R + 10] have shown that the patterning strategy has a large inuence on the etched prole. The following patterning strategy can be used to get an almost uniform etch rate at all spots of a plane silicon surface in the irradiated area: The distance between neighboring spots is the full width half maximum of the beam prole and the time the beam is focused to one spot is 1 µs for a current of around 100 pa. In a cycle, the beam is focused to each of the dened spots. This cycle is repeated until a specic uence is reached.

28 14 CHAPTER 2 FUNDAMENTALS OF PHYSICAL SPUTTER ETCHING 2.2 Nomenclature of Physical Dry Etching The nomenclature used in this work to describe the etch process and especially the interaction of ions with the substrate is presented in this section. In physical dry etching, energetic particles impinge on the surface of a material. In this work, following the nomenclature of ion implantation, these particles are named ions although it is not necessary that they are ionized. All physical quantities of the ions are subscripted by I (for ion). The ions impinge on the surface, as indicated in Figure 2.7 (left), with an ion energy E I. A local coordinate system is dened by the surface normal n as the z-coordinate. The Ion θ I z l Substrate θ T Sputtered Atom Figure 2.7: In the left part, an ion impinges upon a planar substrate surface with an angle of incidence θ I and an energy E I. Sputtered atoms are assumed to leave the solid with a polar emission angle θ T and an energy E T. The distance between the position where an ion has impinged on the surface and the position where an atom is sputtered is denoted l. In the right part, the coordinate system is shown with the polar emission angle θ T and the azimuthal emission angle ϕ T. projection of the ion direction onto the surface denes the x-coordinate. For normal incidence, the distribution of sputtered atoms is axisymmetrical with respect to the z-axis and an arbitrary direction in the xy-plane was chosen for the x-coordinate. The y-coordinate ( z x) is perpendicular to the cross section shown in Figure 2.7 (left). In the local coordinate system, the ion direction is described by spherical coordinates. The polar angle is the angle of incidence θ I and the azimuthal angle ϕ I, the angle between the x-coordinate and the projection of the ion direction onto the surface, is always zero. For a sucient ion energy E I, some substrate atoms are sputtered from the surface with emission energies E T. All physical quantities regarding the sputtered particles are subscripted by T (for target). The spherical coordinate system is also applied to describe the direction of sputtered atoms as shown in Figure 2.7 (right). The polar emission angle θ T is the angle between the z-coordinate and the direction of the sputtered atom. The azimuthal emission angle ϕ T is the angle between the x-coordinate and the projection of the direction of the sputtered atom onto the xy plane. Inside the substrate, the ion has collisions with substrate atoms. If the ion transfers enough energy to a lattice atom during a collision, the atom leaves its lattice site. This x x z ϕ T θ T y

29 2.2 NOMENCLATURE OF PHYSICAL DRY ETCHING 15 atom is then called recoil. In the following, this type of collision is named a hard collision. On the other hand, if the energy transferred during the collision is below the binding energy of the lattice atom, the collision is called a weak collision. Recoils and ions in the substrate are called projectiles. In the substrate, it is necessary to distinguish projectiles from stationary atoms and not ions (I) from sputtered atoms (T). Therefore, the physical properties of projectiles are subscripted with 1 and the properties of stationary atoms are subscripted with 2. The sputtering yield is Y = N A, (2.3) N I where N A is the sum of the number of removed atoms (N S ) and the number of reected ions (N R ) and N I is the number of ions impinging on the surface. In case the number of removed atoms and the number of reected ions must be considered separately, the sputtering yield Y is split into Y S = N S N I, for the removed atoms and (2.4) Y R = N R N I, for the reected ions. (2.5) To consider redeposition and reection, it is necessary to determine the direction and energy of sputtered atoms which can be described by the dierential yield d 3 Y de T d 2 Ω T (2.6) of atoms sputtered with an emission energy E T into the solid angle Ω T (d 2 Ω T = d cos(θ T )dϕ T ). In the following, dω T is used for convenience instead of d 2 Ω T which can both be found in the literature. The radial distance l between the point where the ion impinge on the surface, and the point where the atom leaves the substrate ranges from zero to some hundred Å. The ion uence F is dened as F = N I A, (2.7) where A is the area exposed to the ion beam. The ion ux Φ is dened as Φ = N I A t, (2.8) where t is the time of irradiation. Important for the etch process is the etch rate (ER) which inuences the operational capacity of the process and which is important for the costs. The etch rate is ER = d t, (2.9) where d is the thickness of the removed layer and t is the time needed to sputter the layer. A method to calculate the etch rate without measuring the etched depth is ER = F cos θ I t (Y (θ I ) 1) n T, (2.10)

30 16 CHAPTER 2 FUNDAMENTALS OF PHYSICAL SPUTTER ETCHING where n T is the atomic density of the substrate and θ I is the angle of incidence of the ions. For sputtering of silicon, etch rates around 0.03 µm/min can be achieved with sputter etching and ion milling. Focused ion beam milling can achieve etch rates of 1 µm/min for small areas. For larger areas, the beam must be scanned over a larger area thus reducing the etch rate. Alternatively, more ion beams can be used in parallel. For comparison, in reactive ion beam etching, etch rates of 6 µm/min can be achieved. Beside the etch rate, the selectivity (S) is an important parameter to characterize the etch process and is given by S = ER 1 (2.11) ER 2 When a substrate is masked, a high delity transfer is intended. Therefore, an etch process is preferred where the substrate material has high etch rates (ER 1 ) and the mask material has an etch rate (ER 2 ) close to zero. Otherwise, the mask would be etched, which can lead to undesired etch proles. The selectivity is high if ER 1 ER 2, otherwise the selectivity is low. For FIB, the selectivity is of minor importance because the etching is performed without a mask. The uniformity of the etch rate is another parameter to characterize the etch process. For sputter etching and ion milling, the etch process is uniform if the electrical eld is uniform at the electrodes and if there are only few collisions of the ions with other atoms. The uniformity of a structure fabricated with a FIB machine depends on the patterning strategy described in Section Etched structures, like trenches, are often quantied by the aspect ratio AR. In this work, it is dened as AR = d w, (2.12) where d is the etched depth of the trench and w is the top width of the trench.

31 Chapter 3 Theoretical Concepts of Physical Sputter Etching In this thesis, the topography simulator ANETCH was extended by a modied version of the Monte-Carlo ion implantation program MC_SIM [U + 05] to simulate the sputtering process and the topography changes associated. For a realistic simulation of the sputtering process, it is necessary to understand the theoretical concepts of the process. In this chapter, rst, the sputtering processes are described and the level of complexity of the program is discussed as well (Section 3.1). The theoretical models for the nuclear stopping and the electronic stopping are discussed in the next part (Section 3.2) together with some binding energies which are relevant for the sputtering process. In the last part of this chapter (Section 3.4), a sputtering theory proposed by Sigmund [Sig69] is presented. Some results, as predicted by the theory, are compared with or if necessary tted to simulation results carried out with MC_SIM. 3.1 Fundamental Concepts Material Characteristics: Amorphous or Crystalline Amorphous and crystalline targets have dierent sputtering behaviors and are typically described by dierent models. In this section, the dierences of the targets regarding the sputtering process are explained rst. Afterwards, a reasoning is given why ANETCH was modied to simulate amorphous targets and not crystalline targets. An important reason for the dierent sputtering behavior is an eect called channeling which occurs in crystalline targets. In a crystal, straight channels without any substrate atoms exist. The directions of these channels are dened by the periodicity of the lattice. A projectile in a channel and with almost the same direction as the direction of the channel 17

32 18 CHAPTER 3 THEORETICAL CONCEPTS has only weak collisions with atoms outside the channel. During these weak collisions, only a small amount of energy is transferred and the projectile is reected back into the channel due to the Coulomb repulsion. Such channelled projectiles transfer their energy mainly to substrate electrons. A projectile with a signicantly dierent direction than the channel direction has additionally hard collisions with substrate atoms. During hard collisions, a large amount of energy is transferred and the direction of the projectile is changed in a signicant way which is similar as in amorphous targets. The projectiles following a channel have a larger range until they come to rest than projectiles with other directions. Another important dierence is that projectiles following a channel transfer less energy to substrate atoms at the surface than projectiles with other directions. The sputtering yield depends on the energy transferred to substrate atoms at the surface and, therefore, the sputtering yield is less when ions impinge on the crystal in a direction parallel to one of the channel directions than for ions impinging upon the crystal in dierent directions. For example, the surface of polycrystalline targets consists of regions with dierent crystal orientations and, in sputtering, these regions are not uniformly eroded [BE07] due to the dependence of the sputtering yield on the crystal orientation. For a damage density (density of interstitials and vacancies) beyond a critical value, a crystalline substrate becomes amorphous around the projected range R P. The sputtering behavior becomes almost the same as for an amorphous target when the amorphous region in the substrate extends to the surface. For some materials and depending on the substrate temperature, interstitials and vacancies created during a collision cascade recombine so fast that the critical damage density is not reached. For these materials and conditions, the sputtering behavior can be described with higher accuracy by a model considering a crystalline structure than by a model considering an amorphous structure. Models describing the sputtering process can be classied into two groups: Programs [RT74, Pos94] considering the crystalline structure of the material and programs [BE84] assuming the positions of the atoms to be randomly distributed only according to the density of the substrate. Therefore, programs simulating an amorphous structure are faster than programs simulating a crystalline structure. Hernandez et al. [HM + 02] compared the electron density distribution calculated by Ziegler et al. [Z + 85] for amorphous substrates with an electron density distribution calculated for a crystal structure. Based on this comparison, Hernandez et al. [HM + 02] reported that the electron density distribution is dierent between amorphous and crystalline targets. ANETCH was modied to simulate the topography changes due to sputtering of amorphous targets because of two reasons. First, in most of the simulations carried out with ANETCH, the substrate was silicon due to its importance in the microelectronic industry. At room temperature, for the heavy ions typically used for sputtering, and above a specic uence, silicon becomes amorphous at the surface due to sputtering already for comparatively low uences (> cm 2 ) in comparison to typically used uences (> cm 2 ). Second, the program MC_SIM based on the model proposed by Biersack and Eckstein [BE84] was already implemented and only few changes needed to be done to integrate it into ANETCH. One should keep in mind that MC_SIM considers an amorphous structure of the target. If a crystalline target becomes amorphous at rela-

33 3.1 FUNDAMENTAL CONCEPTS 19 tive low uences (< cm 2 ) during sputtering, simulations carried out with MC_SIM for uences (> cm 2 ) consider the right structure of the target for calculating the sputtering yield for most of the time. Therefore, the error made at the beginning of the sputtering process for low uences can be neglected as studied in Section Collision Regime of the Sputtering Process The physical sputtering process is characterized by the collisions of the incident particle with the substrate atoms and electrons. Those substrate atoms which leave their original position after a collision are called recoils. A collision cascade is generated if recoils cause further atoms to leave their position after a collision. The following collision regimes have been distinguished in literature [Eck07]: The single knock-on regime: The energy transferred to a target atom by the ion is in almost all cases not high enough to generate a collision cascade. This is typically the case when incident particles have low energies which means that they are very slow or very light. The linear cascade regime: In this regime, the energy of the target atom, transferred during the collision, is high enough to establish a collision cascade in most cases. For nuclear collisions, only collisions between a moving particle and a lattice atom are taken into account and nuclear collisions between moving atoms can be neglected. This assumption is valid if the density of the recoil atoms is suciently low. Single ions with energies below some hundred kev typically generate collisions described by the linear cascade regime. The spike regime: Similar to the linear cascade regime, a recoil cascade is generated in the spike regime. The density of recoil atoms, however, is high enough to set the majority of atoms in a certain volume into motion. Very heavy particles impinging on the surface can cause such a high recoil density as described by the spike regime. Electronic sputtering: This is an additional mechanism of sputtering. The moving particles transfer a sucient amount of energy to the substrate electrons. Coupling between electrons and phonons causes large local heating and surface atoms may be liberated by evaporation. Incident particles which are slow and highly charged, or which have energies in the MeV range cause electronic sputtering. This eect is stronger for insulators and semiconductors. In the simulations carried out with MC_SIM, the incident particles considered are singly charged ions with energies below a few hundred kev and above some ev, which is a typical energy range in the microelectronic fabrication. The collisions caused by these ions can be described within the single knock-on regime or the linear cascade regime. Electronic sputtering can be neglected [Sig81] because the ions are assumed to be mainly singly charged.

34 20 CHAPTER 3 THEORETICAL CONCEPTS Level of Description of the Sputtering Process In this work, the collisions of the projectiles in the substrate are described in the linear cascade regime (Section 3.1.2) where only collisions between a projectile and a lattice atom are taken into account. These collisions can be described with dierent levels of accuracy. Models to describe the process are for example Molecular Dynamics (MD), Monte-Carlo method with the Binary Collision Approximation (BCA), or Transport Equation (TE). In Molecular Dynamics, the trajectories of atoms are determined by the calculation of the interactions between the atoms. The interactions are calculated on the basis of many-body potentials. The accuracy of this description depends on the accuracy of the empirical potentials. This description can be very accurate but computer simulations carried out with MD are time consuming. Programs based on MD are applied to study specic cases of ion-solid interaction like implantation proles [BGJ98] or specic cases of sputtering [Urb07] (single knock-on regime or spike regime). In the Monte-Carlo method with the Binary Collision Approximation, interactions are calculated on the basis of pair potentials. As for MD, the accuracy depends on the accuracy of the pair potential. If multiple collisions (small projectile energy) or collisions between projectiles (spike regime) become signicant, the Monte-Carlo method becomes inaccurate. For sputtering and ion implantation, only interactions of the projectiles with lattice atoms are considered. Programs based on BCA are four to ve orders of magnitude faster than programs based on MD [Eck07]. Transport Equations describe transport phenomena like heat transfer, mass transfer, or uid dynamics with partial dierential equations. In ion implantation, Transport Equations can be simplied to approximate the implantation prole. Based on Transport Equation, Sigmund [Sig69] derived a theory of sputtering described in Section 3.4. One result of the theory is an analytical expression to calculate the sputtering yield. Yields calculated with this formula are less accurate than the yields obtained by the other two methods as will be discussed in Section 3.4. ANETCH was extended by MC_SIM, a program based on BCA, because its computation time is faster than programs based on MD while on the other hand it is accurate enough for the sputtering processes studied in this work. Sputtering yields calculated with a program based on BCA can be in fairly good agreement with experiments for a wide range of ion-target combinations and ion energies [Eck07]. The Binary Collision Approximation has the following assumptions: All collisions between atoms are approximated by elastic binary collisions described by an interaction potential. Furthermore, it is assumed that electronic collisions can be decoupled from nuclear collisions. Assuming the potential can be treated as spherically symmetric, the interaction potential between projectile and target depends only on the distance r between the two atoms. To describe the interaction in classical mechanics, the Bohr criterion E 1 M 2 < 100Z 1 2 Z 2 2 kev/u (3.1) must be fullled [Sig04], where E 1 is the energy of the projectile, M 2 is the atomic mass of the target, and Z 1 and Z 2 are the atomic number of projectile and target, respectively. The Bohr criterion gives an upper limit for the projectile energies until which the collision

35 3.2 INTERACTIONS AND BINDING FORCES 21 p M 2 M 2 p r min M 1 θ C M 1 Figure 3.1: Collision in center-of-mass coordinates can be described in classical mechanics. For all simulations carried out with ANETCH for this thesis, the Bohr criterion is fullled. 3.2 Interactions and Binding Forces Nuclear Collisions Most important for programs based on the Binary Collision Approximation (Section 3.1.3) is the description of the collision of two nuclei (nuclear collision) because only these collisions change the direction of a projectile. In this section, the scatter theory is discussed. Important for a realistic description of scattering is a realistic interaction potential. Therefore, dierent statistical models described in literature [Z + 85, W + 77] for the interaction potential are studied which are typically applied in BCA programs. In the frame of the binary collision approximation, the collision between two atoms is discussed in the following. The collisions are described in the center-of-mass reference system (Figure 3.1). In the center-of-mass system, the two atoms approach each other until they are scattered due to the repulsive Coulomb force between the nuclei. The perpendicular distance between the hypothetical unperturbed trajectories of the projectile and the target atom is dened as the impact parameter p. During a collision, the distance between the two atoms r decreases until it reaches a minimum r min and then r increases

36 22 CHAPTER 3 THEORETICAL CONCEPTS again. The angle between the direction of the projectile before the collision and after the collision is the scatter angle θ C. The total energy in the center-of-mass system is given by E C = M 2 M 1 + M 2 E 1, (3.2) where M 1 and M 2 are the atomic masses of the projectile and target, respectively. E 1 is the energy of the projectile before the collision. Considering the conservation of energy and momentum, a time dependence for r can be derived [Eck91]. With the time dependence r(t), Eckstein [Eck91] derived an implicit equation for the minimum of r in the form of V (r min ) E C + p2 r 2 min = 1, (3.3) where V (r) is the spherically symmetric interatomic potential between the two particles. Therefore, the minimum of r depends on E C and p. When the minimum distance between the two atoms has been determined, the scatter angle θ C can be calculated [Z + 85] by θ C (p, E C ) = π 2 r min pdr, (3.4) r 2 1 V (r) E C p2 r 2 where the scatter angle depends on the impact parameter p and on the center-of-mass energy E C. During the collision, the energy T is transferred from the projectile to the target atom. T can be calculated from the conservation of energy and momentum T = 4M 1 E C sin 2 ( θ C ), (3.5) M 1 + M 2 2 as was done for example by Eckstein [Eck91]. For a constant θ C, the transferred energy has a maximum for M 1 = M 2. As described above, the accuracy of the interatomic potential is important for a realistic description of scattering. Dierent approaches to calculate the interatomic potential V (r) have been established. The interatomic potential V (r) between two atoms can be calculated for example by the free-electron method [WB71]. In this method, the interatomic potential V (r) is calculated for the low-energy repulsive range that is most important for nuclear stopping calculations. The interatomic potential is determined by assuming two spherical charge distributions and calculating the interaction energies. As these calculations must be done for all interatomic potentials of interest and thus consume a lot of computation time, a statistical model of the interatomic potential is often used in computer programs [W + 77]. The statistical model ts an analytical expression to calculated interatomic potentials of dierent atom combinations. Often, a screened Coulomb potential is assumed for the analytical expression: V (r) = Z 1Z 2 q 2 φ(r/a) (3.6) 4πɛ 0 r The constant q is the elementary charge, Z 1 and Z 2 are the atomic numbers of the projectile and the target, respectively. The constant ɛ 0 is the vacuum permittivity

37 3.2 INTERACTIONS AND BINDING FORCES 23 (ε 0 = As/Vm), r is the interatomic distance, and a is a screening length which depends on the atomic number of target and projectile. The rst term is the Coulomb repulsion of the two nuclei, and the second term is the screening function φ(r/a). It takes into account that the nuclei are screened by their electrons. In literature [Eck91], the screening function is often approximated by φ(r/a) = 4 i=1 ( A i exp B i r a ), (3.7) where the A i and B i are tting parameters. The A i are restricted such that their sum is one [Eck91]. Other analytical expressions for φ(r/a) were also used, for example for a power potential (V (r) 1/r m ). However, in the program MC_SIM, the parameterization proposed by Wilson et al. [W + 77] and the one suggested by Ziegler et al. [Z + 85] with the analytical expression (3.7) were implemented. In the following, the results of the two approaches are discussed. Wilson et al. calculated the interatomic potentials of 14 atom combinations [W + 77]. For the calculation, they applied the free-electron method [WB71]. For the atom combinations, they took the interaction between light projectiles and heavy target atoms, between atoms of the same element, and between heavy projectiles and light targets into account. Then, the analytical expression (3.7) with the Firsov screening length [W + 77] was tted to each of the 14 atom combinations. The Firsov screening length is given by a F = ( a 0 ) 2/3, (3.8) Z 1/2 1 + Z 1/2 2 where a 0 is the Bohr radius (a 0 = m). One t to all 14 calculated interatomic potentials was evaluated. In the following, this parameterization is named the WHB screening function φ W HB (r/a) after the authors (Wilson, Haggmark and Biersack). It turned out [Eck91] that the parameterization to the atom combination of krypton and carbon is also a good approximation to other atom combinations. This parameterization is known as the KrC screening function φ KrC (r/a). Ziegler et al. [Z + 85] calculated the interatomic potentials of 261 atom combinations. Instead of the Firsov screening length, they proposed a dierent expression for the screening length given by a ZBL = a 0. (3.9) Z Z With this screening length, they made one t of (3.7) to all 261 calculated interatomic potentials. The screening function resulting from this tting procedure is known as the ZBL screening function φ ZBL (r/a) after the authors (Ziegler, Biersack and Littmark). The tting parameters for the dierent approaches are shown in Table 3.1. With V (r), M 1, M 2, E 1 and p all properties of nuclear collisions like the scatter angle and the transferred energy can be calculated. Therefore, (3.3), (3.4), and (3.5) together with one of the screening functions (WHB, KrC, ZBL) are used in MC_SIM to simulate the nuclear collisions.

38 24 CHAPTER 3 THEORETICAL CONCEPTS Table 3.1: Fit parameters for the screening function (Equation 3.7) which were found by Wilson, Haggmark and Biersack [W + 77] for WHB and KrC, and by Ziegler, Biersack and Littmark [Z + 85] for ZBL. WHB KrC ZBL A A A A B B B B Nuclear Stopping Cross Section S n (E 1 ) For the transport equations, the average energy E n 1 a projectile loses along its path x due to nuclear collisions is more important than the calculation of each collision. This energy loss per path length x is the stopping power and can be calculated [Z + 85] by E n 1 x = S n(e 1 )n T, (3.10) where n T is the atomic density of the target. The nuclear stopping cross section, S n (E 1 ), is the integral over the transferred energies of the collisions with all possible impact parameters and weighted by the probability of a collision to appear. The probability for a target atom with an impact parameter p in an amorphous material is proportional to p due to geometrical reasons. The nuclear stopping cross section S n (E 1 ) can then be calculated by S n (E 1 ) = 2π 0 T (θ C (p))pdp. (3.11) One result of the sputtering theory proposed by Sigmund [Sig69] is that the sputtering yield is proportional to the nuclear stopping cross section. The accuracy of the nuclear stopping cross section depends signicantly on the accuracy of the interatomic potential. Sigmund [Sig69] originally applied a power potential (V (r) 1/r m ) with an exponent m depending on the ion energy. In the following, the nuclear stopping cross sections with the potential (3.6) and one of the screening functions (WHB, KrC, ZBL) described in Section are studied to compare the results of the Sigmund theory with simulation results carried out with MC_SIM. The transferred energy decreases with an increasing impact parameter and for large impact parameters the transferred energy is negligibly small. In MC_SIM, only collisions are considered with an impact parameter smaller or equal p max to reduce the computation

39 3.2 INTERACTIONS AND BINDING FORCES 25 time. As will be discussed in Section 4.3.2, the maximum impact parameter in MC_SIM is given by 1 p max = (3.12) 1/3 πn T due to geometrical reasons. For a better comparison with MC_SIM, the upper limit of the integration of (3.11) is set to p max. Inserting (3.2) and (3.5) in (3.11) leads to pmax ) pdp. (3.13) S n (E 1 ) = 8πE 1 M 1 M 2 (M 1 + M 2 ) 2 0 sin 2 ( θc (p) 2 The nuclear stopping cross section S n (E 1 ) can be calculated by solving the integrals in (3.4) and (3.13) with one of the screening functions (WHB, KrC, ZBL). Because information about the projectile and target are needed for solving (3.4), the integral must be solved for each projectile-target combination of interest. For p max =, Wilson et al. [W + 77] proposed a parameterization which ts well to the nuclear stopping cross sections of dierent projectile-target combinations: They introduced a reduced nuclear stopping cross section s n (ɛ) which only depends on the reduced energy ɛ. The reduced energy can be calculated by M 2 ae 1 ɛ = 4πε 0 M 1 + M 2 Z 1 Z 2 q, (3.14) 2 where ε 0 is the vacuum permittivity. The nuclear stopping cross section S n (E 1 ) is related to the reduced nuclear stopping cross section s n (ɛ) by S n (E 1 ) = aq2 ε 0 Z 1 Z 2 M 1 M 1 + M 2 s n (ɛ). (3.15) Wilson et al. carried out ts for the WHB screening function and for the KrC screening function. With respect to φ W HB (r/a), they got the best t with an analytical expression given by A ln (Bɛ) s n (ɛ) =, (3.16) C Bɛ (Bɛ) where the tting parameters are A = , B = and C = [W + 77]. The largest deviations from calculations were -6.6% at ɛ = and 8.0% at ɛ = 10 4 [W + 77]. The t was carried out from ɛ = 10 4 to ɛ = 10. The best t to numerical results obtained with φ KrC (r/a) was done with an analytical expression given by s n (ɛ) = 0.5 ln (1 + ɛ) ɛ + Aɛ B, (3.17) where the tting parameters are A = and B = [W + 77]. The largest deviations from calculations were -2.7% at ɛ = 0.05 and 4.1% at ɛ = 10 4 [W + 77]. The t was carried out from ɛ = 10 4 to ɛ = 10. With the same approach described above, Ziegler et al. [Z + 85] tted the analytical expression 0.5 ln (1 + Aɛ) s n (ɛ) =, (3.18) ɛ + Bɛ C + Dɛ0.5

40 26 CHAPTER 3 THEORETICAL CONCEPTS Energy of a silicon atom (ev) sn(ǫ) (1) 10 2 ZBL WHB Kr-C Moliere Reduced Energy ǫ (1) Figure 3.2: Comparison of the reduced nuclear stopping cross section in silicon resulting from dierent screening functions. to numerical results obtained with φ ZBL (r/a). The best t was with the parameters A = , B = , C = and D = The largest deviation from calculation was 4% between ɛ = 10 9 to ɛ = For a reduced energy larger than thirty, they used s n (ɛ) = ln(ɛ) (3.19) 2ɛ instead of (3.18). In Figure 3.2, the analytical expressions of the reduced nuclear stopping cross section for dierent screening functions are plotted versus the reduced energy. For comparison with the reduced nuclear stopping cross sections s n (ɛ) described above, s n (ɛ) determined from the Moliere potential [W + 77] is also shown. For large energies ɛ 1, all s n (ɛ) converge to the reduced nuclear stopping cross section for unscreened Coulomb scattering. For small reduced energies (10 5 < ɛ < 10 3 ), s n (ɛ) with the WHB screening function is larger than s n (ɛ) with the KrC screening function or with the ZBL screening function. The reduced nuclear stopping cross section with the Moliere screening function is larger than the others for reduced energies between 10 4 and 1. The deviations between s n (ɛ) with the KrC screening function and s n (ɛ) with ZBL screening function are small. This inuence on the sputtering yield can be neglected.

41 3.2 INTERACTIONS AND BINDING FORCES 27 A power potential (V (r) 1/r m ) is used in the Sigmund theory [Sig69] to describe the interaction of the projectiles with bonded atoms shortly before the projectiles are sputtered. To determine the exponent m, Sigmund [Sig69] originally tted the power potential to a Born-Mayer potential (V (r) exp( r/a)) for projectile energies in the ev range. Later, he reported that a potential like the KrC potential should be applied to t the power potential [Sig87]. In the following, the nuclear stopping cross section given by (3.15) and s n (ɛ) with the power potential is tted to the nuclear stopping cross section calculated numerically with the ZBL screening function. The screening function of a power potential can be expressed by φ(r/a) = k m m ( a r ) m 1, (3.20) where k m is a constant. For the screening function in (3.20), Lindhard et al. [L + 68] derived the reduced nuclear stopping cross section to be s n (ɛ) = λ m 2 ( 1 1 m )ɛ (1 2 m), (3.21) where λ m is given by λ m = 2 m ( ( 1 k m B 2, m + 1 )) 2/m. (3.22) 2 k m is the same constant as in (3.20) and B(i, j) is the Beta function. The exponent m depends on the ion energy. It is m = 1 for an unscreened Coulomb potential at high projectile energies and increases towards low projectile energies. The exponent m can be estimated for dierent reduced energy ranges by comparing the reduced nuclear stopping cross section of (3.21) with the one shown in Figure 3.2 for the ZBL screening function. For a reduced energy between 0.1 and 1, the exponent m is about 2. For a reduced energy between and 10 3, a parametrization was carried out and the best t was found for m = 5 and λ 5 = 5.9. This result is independent of the target material and can, therefore, be used to describe the interaction of sputtered atoms with lattice atoms for a wide range of target materials as long as the reduced energy of the sputtered atoms is between and However, to compare the Sigmund theory with MC_SIM, the limited maximum impact parameter p max in MC_SIM must be taken into account for the comparison. This was done for the example of a silicon target. For silicon, the maximum impact parameter is p max = 1.53 Å in MC_SIM. The integral in (3.13) was solved numerically with the ZBL screening function and the nuclear stopping cross section based on the power potential was tted to it. The best t was found for m = 10 and λ 10 = 27. To compare simulation results obtained with MC_SIM with results of the Sigmund theory, m = 10 is used for silicon while m = 5 for all other target elements.

42 28 CHAPTER 3 THEORETICAL CONCEPTS Collisions with Electrons and Electronic Stopping Cross Section S e (E 1 ) The projectiles in the substrate transfer energy not only to the nuclei of the atoms but also to free electrons and shell electrons. These are inelastic collisions where the energy transferred is lost by heating the substrate locally. The heat is transferred from the local spot to the whole substrate by free electron diusion and phonon vibration. If the heat transfer is slower than the local heating, local temperatures could be reached at which surface atoms are emitted by evaporation (electronic sputtering). Otherwise, the transferred energy heats the whole substrate. In the experiments carried out for this work, the inuence of the local heating and the inuence of the temperature increase of the substrate on the sputtering yield is assumed to be negligibly small. For the collisions of the projectile with electrons, the description of a single collision is not important due to the following reasons: First, the change of the direction of the projectile due to collisions with electrons is negligibly small. Second, the electron direction before and after the collision is not important for the sputtering process. Also, the energy of the electrons after the collision is not considered because electronic sputtering and substrate heating is not taken into account. Therefore, only the energy loss of the projectile due to collisions with electrons is considered. The average energy E1 e a projectile loses along its path x due to collisions with electrons, the electronic stopping power, can be calculated by E1 e x = S e(e 1 )n T, (3.23) where n T is the atomic density of the target material. S e (E 1 ), the electronic stopping cross section, is the average energy transferred during all possible collisions for a specic projectile energy. For dierent energy ranges of the projectile, dierent theoretical models have been established. For very high energies (E 1 >> Ẽ), the theory of Bethe [Bet30] and Bloch [Blo33] can be applied. This theory was veried with experimental results. The energy limit Ẽ is given by [Eck91] Ẽ = (Z 1 ) 4/3 M 1 (kev/u). (3.24) In the simulations carried out with MC_SIM, the ion with the lowest mass and atomic number was neon. For neon, Ẽ is 430 kev. For ions with larger masses, the energy limit is even above 430 kev. However, in the simulations carried out with MC_SIM, the largest ion energy was 100 kev and, therefore, the theory of Bethe and Bloch cannot be used for the simulations carried out for this thesis. For low energies (E 1 < Ẽ), the uncertainties are particularly large [P+ 06a] and dierent models were proposed. These models can be classied into two groups [BGJ98]: local and non-local models. The local models describe the inelastic energy loss due to close collisions of two atoms. The non-local models describe the loss of kinetic energy due to background electronic stopping. It is necessary to take both kinds of models into account as two distinct mechanisms as was reported by Beardmore and Grønbech-Jensen [BGJ98].

43 3.2 INTERACTIONS AND BINDING FORCES 29 A local model was proposed by Firsov [Eck91]. In this model, energy is transferred to shell electrons during the collision of two atoms. For the interatomic potential, Firsov applied a power potential (V (r) 1/r m ) with m = 4. For the same model, Oen and Robinson [OR76] applied a potential in the form of (3.6) with a screening function in the form of (3.7), taking only the rst term into account. The electronic stopping cross section as the sum of many nuclear collisions with dierent impact parameters is then given by Oen and Robinson [OR76] in the form with Se OR (E 1 ) = 2π Q OR = B2 1 2 pmax 0 K E 1 πa 2 pq OR (p, E 1 )dp, (3.25) ( ) r min (p, E 1 ) exp B 1, (3.26) a where B 1 is typically the rst tting parameter of the screening function (3.7) [Eck91]. For ion implantation proles, Burenkov et al. [B + 01] showed that B 1 should be tted to experimental results. The maximum impact parameter p max is given by (3.12). The symbol a stands for the screening length and K is given by K = 8π (Z 1 ) 7/6 Z 2 1 2a 0, (3.27) ((Z 1 ) 2/3 + (Z 2 ) 2/3 ) 3/2 M1 where is the reduced Planck constant ( = h/(2π) = (J kg) 0.5 m) and a 0 the Bohr radius. In a MD program for ion implantation, Beardmore and Grønbech-Jensen [BGJ98] applied a velocity-dependent pair potential, based on the Firsov model, to describe the inelastic loss during the collision of an ion and a lattice atom. For the pair potential the ZBL potential was used. Additionally, a continuous electronic stopping model is used to describe the interaction of the free electrons and the ions. A similar approach was applied by Peltola et al. [P + 06a] in a MD program and optimized for crystalline silicon. A non-local model was proposed by Lindhard and Schar [LS61] and assumes a free electron gas. Therefore, the moving projectile is assumed to lose energy continuously to the electrons. As was shown by Lindhard [Lin54], the energy loss is proportional to the projectile velocity for a projectile moving through an electron gas of constant density. For the electronic stopping cross section, Lindhard and Schar proposed S LS e (E 1 ) = K E 1, (3.28) where K is given by Equation The non-local Lindhard-Schar electronic stopping cross section is used in the Monte-Carlo sputtering program SDTrimSP together with the local Oen-Robinson electronic stopping cross section. Eckstein [Eck91] suggested that Se LS (E 1 ) should be multiplied with a tting parameter. The reason is that the dependence of the electronic stopping on the atomic number of the target material is not well described by (3.27). The tting parameters included in the constant K take these deviations into account.

44 30 CHAPTER 3 THEORETICAL CONCEPTS The Oen-Robinson model and the Lindhard-Schar model should both independently describe the electronic stopping and, therefore, both should give approximately the same results for high projectile energies E 1. Therefore, the same constant K was taken for the Oen-Robinson model and the Lindhard-Schar model so that Se OR (E 1 ) approximates (E 1 ) for high projectile energies where r min p. This can be seen by inserting r min = p S LS e in (3.26) and solving the integral in (3.25). The dierence between the two models is then given by Se LS (E 1 ) Se OR (E 1 ) = K ( E B ) ( 1p max exp B ) 1p max. (3.29) a a For large maximum impact parameters, the dierence between the two models vanishes. For low energies (in the ev range), the electronic energy loss predicted by the Oen- Robinson model can be lower than the loss predicted by the Lindhard-Schar model by more than a factor of 10 depending on the ion type and the target material [Eck91]. A dierent way to determine the electronic energy loss was proposed by Ziegler et al. [Z + 85]. They tted an analytical expression for the electronic stopping cross section of hydrogen as the projectile to experimental results. The parameterization was done for a wide range of monoatomic targets. For small energies (E 1 < Ẽ), Ziegler et al. [Z+ 85] proposed the analytical expression S ZBL,low e (E 1, Z 1 = 1) = a 1 ( E1 /ev M 1 /u ) a2 ( ) E1 /ev a4 ( + a ) 3 ev m 2, (3.30) M 1 /u for the electron stopping cross section, where a 1, a 2, a 3 and a 4 are tting parameters depending on the target material. Ziegler et al. proposed the analytical expression ( ) Se ZBL,high M 1 /u (E 1, Z 1 = 1) = a 5 log a 7 E 1 /ev + a E 1 /ev / ( ) M1 /u a6 ( ) 8 ev m 2, (3.31) M 1 /u E 1 /ev where a 5, a 6, a 7 and a 8 are tting parameters depending on the target material. For hydrogen energies only slightly above Ẽ, Ziegler et al. proposed to average: 1 e (E 1, Z 1 = 1) = 1 Se ZBL,low (E 1, Z 1 = 1) + 1 Se ZBL,high (E 1, Z 1 = 1). (3.32) S ZBL,int For low energies, the electronic stopping cross section is proportional to the velocity for all target elements except for semiconductors. For high energies, the electronic stopping cross section is proportional to log E 1 as predicted by the theory of Bethe and Bloch. In the following, Se ZBL (E 1, Z = 1) refers to all energy ranges of the above mentioned electronic stopping cross section of hydrogen. For the electronic stopping cross section of He as projectile, Ziegler et al. proposed Se ZBL (E 1, Z 1 = 2) = Se ZBL (E 1, Z 1 = 1)4γHe, 2 (3.33) where γ 2 He is a tting parameter. Ziegler et al. tted this expression to experimental results for a wide range of monoatomic target materials.

45 3.2 INTERACTIONS AND BINDING FORCES Z 1 = 14 and Z 2 = Se(eV A 2 ) Projectile Energy (ev) S ZBL e S LS e S OR e S ZBL n Figure 3.3: Comparison of electronic stopping models for a silicon atom moving in amorphous silicon: The Ziegler, Biersack and Littmark model (Se ZBL ), the Lindhard and Schar model (Se LS ) and the Oen and Robinson model (Se OR ). For comparison, the nuclear stopping cross section is shown as well. For projectiles with Z 1 > 2, Ziegler et al. calculated the electronic stopping cross section by scaling the proton stopping cross section using the Brandt-Kitagawa [BK82] concept of an eective charge. The resulting electronic stopping cross section is Se ZBL (E 1, Z 1 > 2) = Se ZBL (E 1, Z 1 = 1) (γz 1 ) 2, (3.34) where γ [Eck91] is a factor for calculating the eective charge which depends on the ion energy, mass, atomic number, and target material. For projectile energies below a few kev, no experimental results are available [Eck91]. The predictions of the Lindhard-Schar, Oen-Robinson and Ziegler model have not been validated for these energy ranges and the electronic stopping cross section diers signicantly between the models. This can be seen in Figure 3.3 for a silicon projectile moving in a silicon target. For small and large projectile energies, the Ziegler electronic stopping cross section is of the order of the nuclear stopping cross section or larger than the nuclear stopping cross section. The electronic stopping cross section described by the other models is larger than the nuclear stopping cross section only for large projectile energies. The Ziegler electronic stopping model applied in SRIM results in implantation proles for amorphous targets that are in good agreement with experiments [Zie04]. On the other hand, sputtering yields are in most cases too small when calculated with the Ziegler

46 32 CHAPTER 3 THEORETICAL CONCEPTS electronic stopping model. To study the inuence of the dierent stopping models on the sputtering yield, simulations were carried out with MC_SIM for a silicon target exposed to gallium ions with an energy of 30 kev and dierent electronic stopping models. In a rst comparison, the ZBL electronic stopping model was used for projectile energies below 1 kev and dierent models were used for projectile energies above 1 kev. The dierences in the sputtering yield were small. In a second comparison, dierent electronic models were used for projectile energies below 1 kev and the ZBL electronic stopping model was used above 1 kev. For these simulations, the silicon sputtering yield calculated with the Oen-Robinson model was 2.6, with the Lindhard-Schar model it was 2.1 and with the Ziegler model it was 1.4. The choice of the electronic stopping model for small projectile energies (below 1 kev) seems to have a large impact on the sputtering yield. To have a good agreement for implantation proles as well as for sputtering yields between the program MC_SIM and experiments, the electronic stopping model used in MC_SIM was modied. For the implantation prole, the electronic stopping is important at the beginning of the ion-trajectory when the kinetic energy is large. Therefore, the Ziegler model is used as the electronic stopping model when the kinetic energy is larger than E lim,2, where E lim,2 is a tting parameter. It must be chosen so that the implantation proles calculated with MC_SIM still agree with experimental data. On the other hand, for small projectile energies, the electronic stopping has a large inuence on the sputtering yield. As mentioned above, the electronic stopping for small kinetic energies is not validated by experiments and the Oen-Robinson model was chosen in MC_SIM for projectile energies below E lim,1, the second tting parameter (E lim,1 < E lim,2 ). Simulation results show that the sputtering yield is thus in better agreement with experiments because the electronic stopping power of the Oen-Robinson model is smaller than the electronic stopping power of the Ziegler model for small projectile energies. For projectile energies between E lim,1 and E lim,2, an interpolation of the two models is applied. The parameters must be tted to a specic ion/target combination. This was done as described in Section 6.1 for the example of sputtering a silicon target by gallium ions. The electronic energy cross section implemented in MC_SIM is then described by Se OR (E 1 ) E 1 < E lim,1 ev, S e = Se Z (E 1 ) E 1 > E lim,2 ev, Se OR (E 1 )(Elim,2 E 1 )+Se Z(E 1)(E 1 Elim,1) else. Elim,2 Elim,1 (3.35) With this modication, the sputtering yield calculated by MC_SIM agrees well with experiments not only for normal incidence but also for arbitrary angles of incidence as will be discussed in Section In the theory of sputtering proposed by Sigmund [Sig69], the electronic stopping is neglected. The choice of the electronic stopping model has a large inuence on the sputtering yield as described above. By neglecting the electronic stopping, the sputtering yield increases in most cases. An additional simulation with MC_SIM was carried out with the electronic stopping neglected for projectile energies below 1 kev. In the simulation, gallium ions impinged on a silicon substrate with an energy of 30 kev. For the silicon sputtering yield, a value of 3 was calculated which is above the yields calculated with the

47 3.2 INTERACTIONS AND BINDING FORCES 33 electronic loss models described above. For comparison, a value of 2.6 was calculated for the silicon sputtering yield with the Oen-Robinson model. The silicon sputtering yield obtained by experiments is 2.34 for this setup. In Section 3.4.2, the analytical expression to calculate the sputtering yield, as proposed by the Sigmund theory, is tted to simulation results obtained with MC_SIM to consider the electronic stopping in the Sigmund theory for small projectile energies Surface Binding Energy U s The surface binding energy U s was introduced to describe the sputtering process [Eck91]. Atoms sputtered from a target material must overcome a potential barrier of energy U s to leave the target. The energy of a sputtered atom is then reduced by the surface binding energy E T = Ẽ1 U s, (3.36) where Ẽ1 is the energy of the atom after its last collision with a target atom. In the following, rst the form of the potential barrier is discussed and afterwards the value of the potential barrier, the surface binding energy. Assuming a surface which is planar on an atomic scale, the potential barrier in MC_SIM is approximated by a planar potential (the potential depends only on the distance from the surface). The same is done in SDTrimSP [Eck91]. As a consequence of a planar potential, the distribution of the number of sputtered atoms with energy E T is N(E T ) E T, (3.37) (E T + U s ) 3 2/m where m has been determined by the applied power potential (V (r) 1/r m ). This is a result of the theory of sputtering proposed by Sigmund [Sig81]. Two assumptions were made: First, the direction of the projectiles in the substrate is isotropic and second the number of projectiles with energy E 1 is given by N(E 1 ) E 2 2/m 1. The number of sputtered atoms with an energy E T has a maximum for an energy of sputtered atoms at E T = U s /(2 2/m). The form of (3.37) was veried by experiments reported for example by Gnaser [Gna07] for dierent ion/target combinations and normal incidence. Another form of a potential barrier like an isotropic potential (the potential depends on the distance from the spot where an atom has left the surface) would lead to a dierent distribution of sputtered atoms with energy E T. Especially the maximum of the number of sputtered atoms with an energy E T would be at E T = 0 ev. As argued by Sigmund [Sig69], the surface becomes rough by arbitrarily removing atoms from the surface. In consequence, an attractive potential cannot be perfectly planar or isotropic for all removed atoms. In MC_SIM, the same form of the attractive potential is applied for each sputtered atom. To describe the attractive potential of all sputtered atoms by the same form of a potential, it is assumed in MC_SIM that a planar potential gives the best approximation. For a planar potential, sputtered atoms are diracted. This eect is more pronounced for low energies E T and large polar emission angles θ T.

48 34 CHAPTER 3 THEORETICAL CONCEPTS In SDTrimSP [Eck91] and MC_SIM, the heat of sublimation H S is taken as the value for the surface binding energy if the target is monoatomic. The heat of sublimation H S is the heat required to sublime one mol of a substance at a specic temperature and pressure (typically standard temperature 293 K and pressure 101 Pa). The value of the surface binding energy depends on the surface topography on an atomic scale. At a step in the surface, atoms are typically less bonded as in a plane surface. As mentioned above, the surface binding energy is regarded as a mean value in MC_SIM. Typically, the heats of sublimation were measured for arbitrary surfaces and, therefore, the heat of sublimation is assumed to give a good mean value for the surface binding energy. If the substrate composes of more than one element, a surface binding energy for each element is applied in MC_SIM. As an example, a substrate with two elements A and B and an atomic fraction of x A and x B is considered. In MC_SIM, the surface binding energies of sputtered elements A (U A S ) and B (U B S ) are approximated by US A = x A HS A (1 + x B ) + x 2 B HS B (3.38) US B = x B HS B (1 + x A ) + x 2 A HS A, (3.39) where H A S and H B S are the heats of sublimation of the monoatomic substrate A and B, respectively Bulk Binding Energy U b The bulk binding energy U b is the energy with which a target atom is bonded at its lattice side. The kinetic energy of the target atom after the collision is E 2 = T U b, where T is the transferred energy during the collision. As reported by Eckstein [Eck91], the vacancy formation energy is sometimes used for U b. In most calculations carried out with SDTrimSP, U b = 0 ev was chosen [Eck91]. By comparing some sputtering yields from experiments with yields calculated with MC_SIM, good agreement could be reached for U b = 1 ev. Therefore, U b = 1 ev was chosen in MC_SIM. For target atoms bonded at a lattice side at the surface, U b = 0 ev was chosen and only the surface binding energy was applied. The bulk binding energy U b should not be confused with the displacement energy. The displacement energy is the energy a bonded atom must receive to leave its lattice side and form a stable interstitial. This energy is important to calculate the damage density (density of interstitials and vacancies) in a substrate but is not important for the simulation of sputtering as reported by Eckstein [Eck91]. 3.3 Ion Implantation During physical dry etching, ions are implanted into the substrate. The implantation of the ions typically changes the atomic fraction within a specic depth from the surface until a steady state is established. In this section, the change of the atomic fraction due

49 3.3 ION IMPLANTATION 35 to ion implantation is discussed because the atomic fraction at the substrate surface has a large inuence on the sputtering yield. First, an analytical expression is discussed for the temporal and spatial distribution of implanted ions and, afterwards, the case of gallium ions implanted into silicon is studied with the analytical expression and with simulation results obtained with MC_SIM. Assuming a plane surface and normal incidence of the ions, the atomic fraction x i (F, z) of element i depends on the ion uence F (2.7) and the distance from the surface z and it is N i=1 x i(f, z) = 1 where N is the number of dierent elements in the substrate. At the beginning of the process the atomic fractions are assumed to be constant in the whole substrate x i (F = 0, z) = x i (F = 0). For typical ion types and energies in physical dry etching, the atomic fraction is changed within a depth of 10 to 100 nm from the surface due to ion implantation. At the beginning of ion implantation, the fraction of the ions in the substrate is zero if the elements in the substrate are dierent from the ion type. The fraction of the ions in the substrate at the surface increases until one atom with the same elemental type as the ion is sputtered for each ion on average. This indicates a steady state of the spatial distribution of the atomic fraction at the surface. An analytical formula to approximate the distribution of the atomic fraction of implanted ions x I i (F, z) for normal incidence was presented by Ryssel [Rys77] and is x I i (F, z) = 1 2Y (F ) ( erf ( ) z RP + F Y (F ) ( ) ) n T z RP erf, (3.40) 2 RP 2 RP where erf() is the error function (erf(x) = 2/ π x exp 0 ( t2 )dt). R P is the projected range of the implanted ions and R P is the standard deviation. F is the ion uence (2.7), n T is the atomic density in the substrate, z is the distance from the surface, and Y (F ) is the sputtering yield (2.3) depending on the uence. The analytical expression (3.40) describes the implantation prole with its dependence on the ion uence including steady state: The left error function in (3.40) converges to one for uences going to innity (steady state). In this case, the fraction of the elements depends only on the distance from the surface z. In the following, the uence F lim is dened at which steady state can be assumed for the atomic fraction of implanted ions. At the surface, let ɛ be an upper limit for the dierence between the atomic fraction at F = F lim and the atomic fraction at F =. The dierence is normalized by the atomic fraction in steady state: ɛ = xi i (, 0) x I i (F lim, 0) x I i (, 0) = 1 erf ( ) R P +F lim Y (F lim )/n 2 RP T 1 erf ( R P 2 RP ). (3.41) The dierence between Y (F = 0 cm 2 ) and Y (F = cm 2 ) was assumed to be negligible small. In this work, steady state is assumed for a relative dierence ɛ of 5% and below. The error function in the denominator of (3.41) ranges from minus one to zero for any R P and R P. At the surface (z = 0) and F = F lim, the left error function of (3.40) then ranges from 0.9 to The ion uence F lim results from 3.40 as F lim = n T Y (F lim ) ( 2 RP erf 1 ( 1 ɛ ( 1 erf ( RP 2 RP ))) + R P ), (3.42)

50 36 CHAPTER 3 THEORETICAL CONCEPTS with ɛ = 0.05 and erf 1 () as the inverse error function. For F > F lim, the spatial distribution of the atomic fraction of implanted ions is approximated by x I i (z) = 1 ( ( )) z RP 1 erf. (3.43) 2Y 2 RP x I i is almost constant within a specic depth from the surface and then decreases. This distance depends on R P and R P. If the specic distance is larger than the depths of the collision cascade which is relevant for the sputtering yield [Sig69], only the constant fraction at the surface inuences the sputtering yield. To give a typical example, the implantation proles and the ion uence F lim are calculated for gallium ions implanted into silicon with 30 kev. For dierent uences, implantation proles are calculated with MC_SIM. The collisions are calculated with the binary collision approximation (Section 3.1.3) and the interatomic potential (3.6) with the ZBL screening function (Section 3.2.1). For the electronic stopping cross section, (3.35) was taken. The analytical expression (3.43) for the spatial distribution was tted to the implantation prole calculated by MC_SIM for a uence of cm 2 assuming steady state is established. Best results for a least square t were obtained with R P = 335 Å and R P = 95 Å. The calculated spatial distributions of the atomic fraction x Ga (z) for dierent uences and the t are shown in Figure 3.4. The dierence between the distribution of the gallium fraction for F = cm 2 and the one for F = m 2 is negligibly small. The variations of the atomic fraction for a uence of F = cm 2 are a result of statistical noise. In MC_SIM, the volume of the target material is divided into layers and the atomic fractions x i are calculated for each layer. In the simulations, the same number of ions were taken and the area was decreased to increase the uence. This also decreased the volume of the layers and, therefore, the statistic noise became larger with an increasing uence. With (3.42) and the values for the projected range and standard deviation of the implanted ions, the uence above which steady state can be assumed is F lim = cm 2. A sputtering yield of Y Si = 2.34 and an atomic density of n Si = Å 3 were taken for the calculation. Until now all estimations are based on the program MC_SIM. For validation, the results of MC_SIM are compared with experiments. The implantation proles carried out with MC_SIM are in good agreement with SIMS measurements reported by Gnaser et al. [G + 08] as shown in Figure 3.5. For a uence of cm 2, the gallium concentration falls of faster in the simulation than in the experiment for implantation depths above 280 Å. A possible explanation is given in the following: In SIMS measurements, the surface is exposed to ions and sputtered charged particles are identied. Due to the ions impinging on the surface, some already implanted ions are pushed further in the substrate and, therefore, increase the concentration of implanted ions deep in the substrate. SIMS measurements reported by Lehrer [Leh05] show a similar implantation prole as the measurements reported by Gnaser. For uences above cm 2, Lehrer found cm 3 for the maximum gallium concentration. This is only about half of what Gnaser reported. An argument in favor of the accuracy of the experiments reported by Gnaser can be given as follows. For amorphous silicon, the concentration is cm 3

51 3.3 ION IMPLANTATION 37 Gallium fraction in target (1) Fit to F = cm 2 F = cm 2 F = cm 2 F = cm 2 F = cm Depth from the surface (Å) Figure 3.4: Prole of gallium implanted in an amorphous silicon target with 30 kev and normal incidence for dierent uences (F). The analytical calculations were done with (3.40). as reported by Custer et al. [C + 94]. The sum of the concentrations of silicon and gallium is assumed to be constant with a value of cm 3 during the sputtering process. Therefore, a gallium concentration of cm 3 (Lehrer) means a fraction of 0.12 and a gallium concentration of cm 3 (Gnaser) means a fraction of Steady state is assumed when the partial sputtering yield of gallium is one (Y Ga = 1). Almost all gallium ions are assumed to be implanted and not reected for normal incidence and, therefore, Y S Ga = 1 (2.4). In a rst assumption, the atoms are sputtered according to their atomic fraction at the surface. From both conditions and the atomic fractions, the partial sputtering yield of silicon can be calculated by Y Si = x Si x Ga Y S Ga. (3.44) From the maximum gallium concentration obtained by Gnaser, the silicon sputtering yield is 2.1 and for the one obtained by Lehrer the yield is 7.3. Yield experiments for these process conditions show that the partial sputtering yield of silicon is 2.34 (Section 6.1 and [Leh05]). Therefore, the experiments reported by Gnaser seem more reasonable and conrm the results of MC_SIM. Until now, ions have been assumed to only impinge upon a planar surface with normal incidence. During the fabrication process the surface topography may change. Then,

52 38 CHAPTER 3 THEORETICAL CONCEPTS Ga concentration (cm 3 ) Exp. Gnaser F = cm 2 MC_Sim F = cm 2 Exp. Gnaser F = cm 2 MC_Sim F = cm Depth from the surface (Å) Figure 3.5: Depth prole for 30 kev Ga implanted in Si with dierent uences (F). The experimental results (Exp.) were obtained with SIMS measurements and reported by Gnaser et al. [G + 08]. The simulation results (MC_SIM) were calculated with the Monte-Carlo program MC_SIM. Table 3.2: Fraction of gallium x Ga in silicon for steady state of sputtering pure silicon with 30 kev gallium ions. A plane surface is assumed and the gallium ions impinge on the surface under dierent angles of incidence θ I θ I x Ga reected ions and sputtered atoms might impinge upon the surface with dierent energies and angles of incidence. In the following, it will be shortly discussed how the spatial distribution of the atomic fraction changes with the angle of incidence. The results were calculated with MC_SIM. The atomic fraction at the surface is shown in Table 3.2. The implantation proles for some angles of incidence are shown in Figure 3.6. The maximum gallium concentration at the surface decreases with an increasing angle of incidence θ I. This can be explained by (3.44) and the dependence of the sputtering yield on the angle of incidence. For gallium ions impinging on a silicon surface, the Si-sputtering yield increases with the angle of incidence until a maximum is reached. Until the maximum is reached, the ratio x Si x Ga must increase as well with the angle of incidence (3.44). For oblique angles of incidence, the share of reected ions in Y Ga = 1 increases and, therefore,

53 3.3 ION IMPLANTATION 39 YGa S decreases signicantly. Although Y Si decreases for large angles of incidence, the ratio Y Si /YGa S increases until all ions are reected and the gallium concentration at the surface becomes zero. The uence F lim at which steady state is assumed for the implantation prole was estimated for normal ion incidence. With an increasing angle of incidence F lim is assumed to decrease due to the following reasons. First, the sputtering yield increases with the angle of incidence and a larger sputtering yield supports the change of the atomic fraction. Finally, more atoms are implanted closer to the surface for increasing angles of incidence. For normal incidence and dierent ion energies, (3.44) can be used to estimate how the concentration of implanted ions changes with the ion energy. The sputtering yield typically increases with the ion energy until a maximum is reached. Due to (3.44), the fraction x Si x Ga must increase as well until the maximum of the sputtering yield is reached and then decreases again. Gallium concentration (cm 3 ) θ I = 0 θ I = 30 θ I = 50 θ I = 60 θ I = 70 θ I = 80 θ I = Depth from the surface (Å) Figure 3.6: Depth prole for 30 kev Ga implanted in Si with dierent angles of incidence θ I. The results were determined with the Monte-Carlo program MC_SIM. For all calculations, the dose was cm 2.

54 40 CHAPTER 3 THEORETICAL CONCEPTS 3.4 Results of the Sputtering Theory by Sigmund In this section, the sputtering theory proposed by Sigmund [Sig69, Sig81, Sig87] is discussed. With this theory, a good approximation of the sputtering process in the linear cascade regime is given. Some results of the theory are compared with simulation results obtained with MC_SIM for the case of gallium ions impinging on a silicon surface. In Section 3.4.1, an analytical expression to calculate the sputtering yield is discussed which was derived by Sigmund for normal incidence and monoatomic targets. In Section 3.4.2, the equation for the sputtering yield is tted to results calculated with MC_SIM. In Section 3.4.3, an extension of the Sigmund theory is studied to describe the sputtering of multi-element targets. Afterwards, in Section 3.4.4, the Yamamura formula and Chen formula, semi-empirical formulas to describe the dependence of the sputtering yield on the angle of incidence, are discussed and compared with simulation results. In Section 3.4.5, an analytical expression for the angular distribution of sputtered atoms proposed by the Sigmund theory is studied and compared with simulation results. The Sigmund theory also presents an analytical expression for the energy distribution of sputtered atoms. This is discussed and compared with simulation results in Section Sputtering Yield for Monoatomic Targets and Normal Incidence A famous result of the sputtering theory proposed by Sigmund [Sig69, Sig81, Sig87] is an analytical expression to calculate the sputtering yield for monoatomic targets and normal incidence. He applied the linearized Boltzmann transport equation to describe the movement of the projectiles in the solid. In his approach, the solid is assumed to be at equilibrium and the projectiles of the collision cascade are treated as deviations from the equilibrium. From this approach, Sigmund derived an analytical expression to calculate the sputtering yield. The analytical expression describes the sputtering yield quite well if the direction of the projectiles in the substrate can be approximated by an isotropic distribution. This is typically a good approximation if the number of projectiles in one collision cascade is large. With this assumption, the analytical expression of the sputtering yield derived by Sigmund is Y = ΛF D (E I ), (3.45) where F D (E I ) is the energy deposited per unit length (ev/m) at the surface. To derive the material constant Λ, the last collisions of projectiles with target atoms before they are sputtered must be described. Sigmund assumed that most of these projectiles have energies of a few ev and applied a power potential (V (r) 1/r m ) to describe the interaction potential. He then derived the material constant to Λ = Γ m 8(1 2/m) 1, (3.46) n T C m Us 1 2/m

55 3.4 RESULTS OF THE SPUTTERING THEORY BY SIGMUND 41 where m is the exponent of the power potential. The unit of the material constant is (m/ev). The unit-free parameter Γ m corresponds to 1/m Γ m = φ(1) φ(1 1/m), (3.47) where φ(x) is the digamma function. C m is a cross section parameter dened by C m = π ( ) Z 2 λ ma 2 2 T q 2 2/m, (3.48) 2πɛ 0 a where λ m was already dened in (3.22). a is the screening length and q the elementary charge. The cross section parameter C m has the unit (m 2 (ev) 2/m ). To consider the bulk binding energy U b (Section 3.2.5) for the sputtering yield, Sigmund [Sig69] suggested to replace the surface binding energy in (3.46) by 1 1 [( 1 + U s U s U b 2U b ) ln ( 1 + 2U ) ] b 1. (3.49) U s Often in literature, only the surface binding energy is considered. If U s U b, the logarithm can be expanded in a Taylor series to the second term and the material constant is again given by (3.46). The density of energy deposited is given by F D (E I ) = α ( MT M I ) n T S n (E I ), (3.50) where S n (E I ) is the nuclear stopping cross section (Section 3.2.2), n T is the atomic density of the substrate and α(m T /M I ) is a function which will be discussed in Section Inserting (3.46) and (3.50) into (3.45) shows that the sputtering yield is independent of the target density and is proportional to the nuclear stopping cross section: Y = Γ 1/m 8(1 2/m) 1 C 1/m U 1 2/m s α ( MT M I ) S n (E I ). (3.51) The nuclear stopping cross section as well as the exponent m of the power potential have a signicant inuence on the sputtering yield. Sigmund [Sig69] originally applied a Born-Mayer potential (V (r) exp( r/a)) to determine the exponent m for projectile energies of a few ev. For the nuclear stopping cross section, also a power potential was applied by Sigmund [Sig69] with a dierent exponent m (Section 3.2.2). With (3.6) and the KrC screening function [W + 77] or the ZBL screening function [Z + 85] (Table 3.1) more realistic statistical models were established to describe the interaction potential. In Section 3.2.2, the power potential was tted to the potential with the ZBL screening function for projectile energies in the ev range. The obtained tting parameters m and λ m together with a nuclear stopping cross section given by (3.15) and (3.18) were used to compare (3.51) with simulations carried out by MC_SIM. The sputtering yield calculated by (3.51) typically overestimates the real sputtering yield because electronic stopping is neglected in the Sigmund theory for low projectile energies (the material constant is assumed to be independent of electronic stopping). In Section 3.4.2, the analytical expression of the sputtering yield (3.51) is tted to sputtering yields calculated with MC_SIM and in this way the electronic stopping is taken into account.

56 42 CHAPTER 3 THEORETICAL CONCEPTS Fit of the Yield Formula to Simulation Results In section 3.4.1, all quantities to calculate the sputtering yield from (3.51) were discussed except for the function α(m T /M I ). Dierent alternatives of an analytical expression were established for α(m T /M I ). Sigmund [Sig69] derived a theoretical form for elastic scattering. Alternative forms were summarized for example by Malherbe [Mal94]. If electronic stopping is considered, the function must be extended. As discussed in Section 3.2.3, the inuence of electronic stopping on the sputtering yield is quite signicant for low projectile energies. Another alternative is presented in this section to achieve agreement between 3.51 and simulations carried out with MC_SIM and thus considering the electronic stopping power for low projectile energies. α (M T /M I ) is determined for a specic energy range by tting sputtering yields calculated with MC_SIM with the analytical expression ( ) c3 MT α (M T /M I ) = c 1 + c 2, (3.52) M I where the c i are tting parameters. Sputtering yields were calculated for dierent ion types and ion energies. Silicon was taken as a target with U s = 4.7 ev, U b = 1 ev. For the electronic energy loss model, the one given by (3.35) was applied. The yield calculated by MC_SIM is denoted by Y MC_SIM. α(m T /M I ) was determined from (3.45) and (3.50) by α (M T /M I ) = Y MC_SIM Λn T S n (E I ), (3.53) where Λ is the material constant (3.46). For the nuclear stopping cross section S n (E I ), (3.15) and for the reduced nuclear stopping cross section with the ZBL screening function, (3.18) were applied. The plot of the factor α (M T /M I ) versus the atomic mass of the ion M I is shown in Figure 3.7. A least square t was carried out for an ion energy of 20 kev with c 1 = 0.191, c 2 = and c 3 = As shown in Figure 3.7, the function α changes with the applied ion energy. One reason for the energy dependence might be the consideration of the electronic stopping in MC_SIM. However, an energy dependence of α was also found when electronic stopping was neglected in MC_SIM. For this case, Sigmund assumed α to be independent of the ion energy E I. There are several dierences between MC_SIM and the Sigmund theory that might explain the energy dependence of α: First, the direction of the projectiles in the substrate is not isotropic and the level of anisotropy depends on the ion energy. Second, the scatter angle is on average larger for light ions than for heavy ions. For normal incidence, some light ions are reected while almost all heavy ions are implanted. Finally, in the Sigmund theory, only a constant nuclear stopping cross section S n (E I ) is considered with the energy of the ion when impinging upon the substrate. In MC_SIM, the sputtering process is inuenced by a dynamic nuclear stopping cross section S n (E 1 ) where the energy of the ion decreases. It is important for a realistic description of the sputtering yield if S n (E 1 ) increases or decreases for lower energies. The Sigmund theory was tted only to simulation results for an ion energy of 20 kev. The energy dependence as predicted by the Sigmund theory is compared with simulation

57 3.4 RESULTS OF THE SPUTTERING THEORY BY SIGMUND 43 α(mt /MI) (1) E I =2 kev E I =10 kev E I =20 kev E I =30 kev E I =100 kev Fit to E I =20 kev Atomic mass of ion M I (u) Figure 3.7: Determination of α (M T /M I ) (3.53) by a t to simulation results. The simulations were carried out with MC_SIM. The target material was chosen to be silicon (Z T = 14, M T = 28) with U s = 4.7eV, U b = 1eV, the ZBL nuclear stopping model and the electronic stopping model given by (3.35). The simulations were carried out for neon, argon, krypton and xenon. results for the example of gallium/silicon ions impinging upon a silicon target with normal incidence. The energy range of the ions is between 0.06 kev and 100 kev. Between 1 kev and 100 kev, the agreement between theory and simulation is quite good, as shown in Figure 3.8, especially for silicon as ion. The simulation results show a cuto for energies below 1 kev while the sputtering yield has a slower decrease in the theory. In the two simulations, the energy dependence is not proportional to the nuclear stopping cross section when the ion enters the substrate for energies below 1 kev. Possible reasons for the dierence between simulation and theory are the assumption of a constant nuclear stopping cross section and the assumption of an isotropic distribution of the projectile directions in the substrate Multi-Element Targets Until now, only monoatomic targets have been considered i.e. for vanishing ion concentration in the target. In the simulations carried out with MC_SIM, ions of element A impinge on a monoatomic target of element B. With an increase of the uence, the atomic fraction of implanted ions within a specic depth from the surface increases as discussed in Section 3.3. For a large ion uence, the partial sputtering yield Y A becomes one. A con-

58 44 CHAPTER 3 THEORETICAL CONCEPTS 10 Si-sputtering yield (atoms/ion) MC_Sim: gallium ion MC_Sim: silicon ion Theory: gallium ion Theory: silicon ion Ion energy (kev) Figure 3.8: The sputtering yield is plotted versus the ion energy for two dierent ion types. Gallium ions or silicon ions impinge on a silicon substrate with normal incidence and with dierent ion energies. The simulations carried out with MC_SIM are compared with the Sigmund theory. stant atomic fraction of element A (x A ) has then been established below the surface. The atomic fraction of element B in this depth range is given by (x B ) with (x A + x B = 1). For elements A and B of similar masses, Sigmund [Sig81] suggested that a good approximation for the partial sputtering yield of element B is Y B = Y A x B x A ( MA M B ) 2/m ( U A S U B S ) 1 2/m, (3.54) where US A and US B are given by (3.38) and (3.39), respectively. The lightest and least bonded element (smallest U s ) is sputtered preferentially. In Section 3.2.2, a value of 5 was determined for the exponent m by tting the reduced nuclear stopping cross section of the power potential to s ZBL n (ɛ) for a reduced energy between and For a better comparison with MC_SIM, a second parameterization was carried out with the same maximum impact parameter, p max = 1.53 Å, as used in MC_SIM for a silicon target. For this parameterization, a value of 10 was obtained for m. The partial sputtering yield for similar masses can be approximated by (3.44) if US A and US B are the same. With (3.54), the partial sputtering yield of B can be calculated if x A or x B is known. Vice versa, the atomic fractions of element A and B can be calculated if Y B is known.

59 3.4 RESULTS OF THE SPUTTERING THEORY BY SIGMUND 45 For a target consisting of two elements A and B with similar masses and the atomic fraction x A and x B in the target, Sigmund [Sig81] suggested that the energy deposited by the ion in the material per unit path length can be calculated by where F AA D and F BB D F AB D (E I ) = x A FD AA (E I ) + x B FD BB (E I ), (3.55) are the energy deposited in the monoatomic targets of element A and B, respectively. The consequences of (3.55) for the partial sputtering yield is discussed at the example of gallium ions A impinging on a silicon target B with normal incidence and 30 kev. Simulations have shown that the partial sputtering yield of silicon is smaller for x Si = 1 than for x Si = Therefore, the energy deposited is smaller for x Si = 1 than for x Si = For a target consisting of two elements with very dierent masses, (3.54) and (3.55) cannot be applied. However, the general predictions of (3.54) and (3.55) are still valid: First, the partial sputtering yield is not stoichiometrical if US A and US B are dierent. Second, a partial sputtering yield of element A can be larger in a multi-element target than the sputtering yield in a monoatomic target of element A Dependence of the Sputtering Yield on the Angle of Incidence θ I In this section, an extension of (3.51) describing the dependence of the sputtering yield on the angle of incidence is introduced. First, an extension suggested by Sigmund [Sig69] is outlined. Afterwards, the semi-empirical formulas proposed by Yamamura [Yam84] and Chen et al. [C + 05] are discussed. Finally, the analytical expressions are tted to simulation results obtained with MC_SIM for silicon sputtered by gallium ions. While the material constant Λ (3.46) is assumed to be independent of the angle of incidence, Sigmund [Sig69] proposed a method to calculate the angular dependence of the density of energy deposition F D (E I, θ I ). To consider the angle of incidence, the function α(m T /M I ) was extended so that it also depends on the angle of incidence α(m T /M I, θ I ). For non-grazing angles of incidence θ I, more energy is deposited at the surface with increasing θ I. For elastic collisions where electronic stopping can be neglected, Sigmund [Sig69] suggested an approximation for the dependence on the angle of incidence by 1 Y (E I, θ I ) = Y (E I ) cos n (θ I ), (3.56) where Y (E I ) is the sputtering yield for normal incidence and monoatomic targets (3.51). With the method of Sigmund, n must be calculated for each ion/target combination. As long as electronic stopping can be neglected, Sigmund [Sig69] calculated that the exponent n is in the range of 1.65 ± 0.25 from M T /M I = 0.1 to M T /M I = 1 and decreases from M T /M I = 1 to M T /M I = 10 where n = 0.9. For ion energies in the range between some hundred ev and below some hundred kev, Sigmund [Sig69] found that the range of values for the exponent n is constrained to 1.82 ± 0.04 from M T /M I = 0.1 to M T /M I = 1 and

60 46 CHAPTER 3 THEORETICAL CONCEPTS decreases from M T /M I = 1 to M T /M I = 10 where n = 0.9. For large ion energies where electronic stopping cannot be neglected, Sigmund [Sig81] suggested to approximate n with 1. For gallium ions impinging on a silicon substrate M T /M I = 0.4 with an energy of 30 kev, Lehrer [Leh05] experimentally found n = 1.9. This result agrees quite well with the calculated value suggested by Sigmund. The method proposed by Sigmund does not consider the reection of ions at grazing angles of incidence and, therefore, the density of energy deposition overestimates the real density of energy deposition for large angles of incidence. Based on the Sigmund extension, Yamamura [Yam84] proposed a semi-empirical formula ( ( 1 Y (E I, θ I ) = Y (E I ) cos n (θ I ) exp C 1 1 )) (3.57) cos θ I for the angle of incidence dependence of light ions. The parameter n is the same as in (3.56) and C needs to be adjusted to experimental results. Starting from normal ion incidence, the sputtering yield increases with increasing angle of incidence until a maximum is reached. Then, the sputtering yield decreases with increasing angle of incidence until only the ion is reected and no substrate atoms are sputtered. In literature [L + 01], expression (3.57) was also applied for heavier ions like gallium. If the angle θ Imax at which the yield has a maximum is known, the parameter C can be calculated by C = n cos (θ Imax ). (3.58) Based on the Sigmund theory and the Yamamura approach, Chen et al. [C + 05] assumed ions to penetrate into the substrate with an average implantation depth R P, with a straggling R P and a lateral straggling σ. Taking the spatial distribution of the implanted ions into account, they proposed the sputtering yield to be calculated by Y (E I, θ I ) = Y (E I ) ( exp 1 cos2 (θ I ) + a 1 sin 2 (θ I ) a a 1 tan 2 (θ I ) + a 2 + C ( 1 1 )), cos(θ I ) (3.59) where a 1 = σ 2 / R 2 P and a 2 = R 2 P /(2 R2 P ). The term exp(c(1 1/ cos(θ I))) is the same as in the Yamamura equation. The Yamamura and Chen equations were tted to simulation results obtained with MC_SIM for the example of gallium ions impinging on a silicon target with an energy of 30 kev. In the simulations, a steady state of the implanted ions was assumed (Section 3.3). With a least square t, n = 2.1 and θ Imax = 80 were found for the Yamamura equation and for the Chen equation the parameters a 1 = 0.52, a 2 = 2.01 and C = were obtained. The simulation results are compared with sputtering yields determined experimentally by Kaito and reported by Vasile et al. [V + 99] as shown in Figure 3.9. The Chen equation is preferred to the Yamamura equation in this work because it describes the angle of incidence dependence more accurately. The simulation results are in good agreement with the experimental results except for an angle of incidence of 80. The uncertainty in the measurement of the sputtering yield for oblique incidence is quite large.

61 3.4 RESULTS OF THE SPUTTERING THEORY BY SIGMUND 47 Si-sputtering yield (atoms/ion) MC_Sim Exp: Kaito Theory: Yamamura Theory: Chen Angle of incidence θ I ( ) Figure 3.9: Si-sputtering yield for dierent angles of incidence. Gallium ions impinged on a silicon surface with an energy of 30 kev and with dierent angles of incidence. The Yamamura equation (3.57) and the Chen equation (3.59) were tted to the simulation results obtained with MC_SIM. The simulation results are compared with experimental results determined by Kaito and reported by Vasile et al. [V + 99]. In an experiment carried out for this work and described in Section 6.3, a Si-sputtering yield of 19.3 ± 1.6 was determined for an angle of incidence of 80. Therefore, the dierence between experiment and simulation for 80 might be a result of the uncertainty of the experiment. Nevertheless, even if the Si-sputtering yield obtained with the simulation is 9% below the experimental measured yield, the agreement between simulation and experiment is still quite good Angular Distribution of Sputtered Atoms To take redeposition into account, the angular distribution of sputtered atoms and their kinetic energy must be known. An analytical expression of the dierential sputtering yield was derived by Sigmund [Sig81] in the form d 2 Y (θ T, E T ) = Γ 1/m 1 1/m dω T de T 4π C 1/m E T (E T + U s ) 3 2/m cos(θ T )α ( ) MT, θ I S n (E I ), (3.60) M I

62 48 CHAPTER 3 THEORETICAL CONCEPTS where dω T = d cos(θ T )dϕ T is given by the polar emission angle θ T and the azimuthal emission angle ϕ T. Integrating over θ T from 0 to π/2, integrating over ϕ from 0 to 2π, and integrating over E T from 0 to shows that (3.60) is consistent with (3.51). The integration over ϕ is trivial because (3.60) is cylindrical symmetric. Two results of (3.60) need to be discussed: First, the angular distribution of sputtered atoms is given by cos(θ T ) and, second, the number of atoms with energy E T is proportional to E T /(E T + U s ) 3 2/m. The angular distribution is discussed in this section and the energy distribution will be discussed in Section The distribution with the polar emission angle is described by cos(θ T ) in (3.60) because the directions of projectiles in the target are assumed to be isotropic. If the directions of the projectiles cannot be approximated by an isotropic distribution, the distribution with respect to the polar emission angle is expressed by cos α (θ T ) where α depends on the ion/target combination and the ion energy. One of the rst persons to report such a distribution for low projectile energies were Wehner and Rosenberg [WR60] in This gave a rst clue that sputtering is based on momentum transfer and not on evaporation which was a congruous theory of sputtering at that time. The angular distribution was studied for the example of gallium ions impinging on a planar silicon surface with an energy of 30 kev. First, ions impinging on the surface with normal incidence were considered and, afterwards, the dependence of the angular distribution on the angle of incidence was studied. Even for normal incidence, the directions of the projectiles inside the silicon substrate are not isotropic and, therefore, cos α (θ T ) must be taken to describe the distribution of the polar emission angle. The exponent α was tted to simulation results carried out with MC_SIM. In the simulations, a steady state for the implanted ions was assumed (Section 3.3). The simulation results are shown in Figure 3.10 for sputtered silicon and gallium atoms. The dierential sputtering yield is calculated in MC_SIM per solid angle in spherical coordinates. Towards the pole, the area per solid angle decreases. Therefore, the statistical noise increases and the variations increase with a smaller polar angle as can be seen in Figure The values for θ T ± 1 were approximated by the values for θ T ± 2. The best results for a least square t were α = 1.24 for sputtered silicon atoms and α = 1.18 for sputtered gallium atoms. For the angular distribution (cos α (θ T )) of sputtered silicon atoms, Winter and Mulders [WM07] experimentally measured a value of 2.7 for α. The authors also carried out simulations with SRIM [Zie04] from which they got α = 2.7. In the experiment, the distribution of sputtered atoms with the polar emission angle was obtained from measuring the redeposition. Due to a large uence, it is possible that the atoms were not sputtered from a plain surface but from a hole. This could explain that a larger value for α was measured than the one calculated by MC_SIM. The discrepancy between the results obtained with SRIM and MC_SIM is perhaps related to limitations in SRIM reported by Wittmaack [Wit04]. To have another comparison of the angular distribution between experiment and simulation, simulations were carried out with MC_SIM for argon ions impinging on a silicon target with an energy of 10 kev and under normal incidence. Chernysh et al. [C + 04] measured experimentally α = 1.2 for the angular distribution of sputtered silicon atoms. A least square t to the simulation results obtained

63 3.4 RESULTS OF THE SPUTTERING THEORY BY SIGMUND 49 ) atoms ion and degree 1 Y Sim: Silicon Fit: Silicon Sim: Gallium Fit: Gallium c1 cos 1.24 (θ T ) ( dy dθt c2 cos 1.18 (θ T ) Polar emission angle θ T ( ) Figure 3.10: Dierential sputtering yield plotted versus the polar emission angle θ T for gallium ions impinging on a silicon target with an energy of 30 kev under normal incidence. The analytical function cos α (θ T ) was tted to the simulation results with α = 1.18 for gallium and α = 1.24 for silicon. with MC_SIM gave a value of 1.17 for α. For this ion/target combination, simulation and experiment agree quite good. If the value of α measured by Winter and Mulders is realistic and the inuence of redeposition on the surface evolution is signicant, etched topographies calculated with ANETCH can be signicantly dierent from experiments. For the example of a trench etched into silicon by gallium irradiation, two simulations were carried out with an angular distribution for silicon with α = 1.24 and α = 2.7, respectively. The same uence prole was used in both simulations with a uence of cm 2. In Figure 3.11, the dierent side-walls which develop during the etch process are shown. A comparison with experiments discussed in Section 6.2 shows that the side-wall slope with α = 1.24 agrees better with experiments. The lighter silicon atoms have a more pronounced over-cosine (α > 1) emission spectra than the heavier gallium atoms. One possible reason is given in the following. Before leaving the substrate, the last collision of a projectile with a substrate atom reects in most cases the projectile towards the normal direction of the surface due to the distribution of the stationary atoms as shown in Figure The lighter silicon atoms are more deected than the heavier gallium atoms. The exponent α does not only depend on the mass of sputtered atoms but on dierent parameters as reported by Gnaser [Gna07]. For example, the ion energy or the surface roughness have a strong inuence on α. In the following, the inuence of the angle of incidence θ I on the angular distribution of sputtered atoms is discussed. In the extension of the dependence of the sputtering yield

64 50 CHAPTER 3 THEORETICAL CONCEPTS α = 1.24 α = 2.7 Z (µm) X (µm) Figure 3.11: Two cross sections of a side-wall of a trench resulting from two simulations carried out with ANETCH. A trench was etched into silicon by gallium irradiation. The uence prole was in both simulations the same and the only dierence arose from two dierent angular distributions of sputtered silicon atoms. One is described by α = 1.24 as predicted by MC_SIM and the other by α = 2.7 as found by Winter and Mulder [WM07]. Surface normal Surface Projectile Stationary atoms Figure 3.12: Distribution of stationary atoms in an amorphous substrate. In most cases projectiles leaving the substrate are reected towards the surface normal. on the angle of incidence proposed by Sigmund, only the energy deposition depends on the angle of incidence while the material constant is independent because the direction of the projectiles in the substrate is assumed to be isotropic. Due to this assumption, the angular distribution of sputtered atoms in the Sigmund theory is assumed to be independent of the angle of incidence. Simulations were carried out with MC_SIM for gallium ions impinging on a silicon surface with dierent angles of incidence. Again it was assumed that steady state is established for the implanted ions. The angular distribution of sputtered silicon atoms for dierent angles of incidence is shown in Figure Only the sputtered atoms were taken into account which were sputtered with ϕ T = 0 ± 1 and ϕ T = 180 ± 1. For increasing angles of incidence θ I, the direction of the projectiles in the substrate becomes more anisotropic and the angular distribution of sputtered atoms becomes signicantly dierent to the one for normal incidence. At a polar emission angle of θ T = 14, a peak in the distribution forms for larger angles of incidence. The reason for this behavior is

65 3.4 RESULTS OF THE SPUTTERING THEORY BY SIGMUND 51 (1/ ) dy (θi) dθt 1 Y (θi) θ I = 0 θ I = 20 θ I = 40 θ I = 60 θ I = Polar emission angle θ T ( ) Figure 3.13: Angular distribution of sputtered silicon atoms for dierent angles of incidence θ I. In the simulations, carried out with MC_SIM, a silicon target was hit by gallium ions with an energy of 30 kev and under dierent angles of incidence. assumed to be caused by the rst few collisions of an ion with target atoms when entering the substrate. Some of these target atoms are sputtered after the rst collision without having further collisions. These atoms are named primary knocked-on atoms and are sputtered with polar emission angles from 90 θ I to 90. The probability of a primary knocked-on atom increases with an increasing angle of incidence. The larger variations around a polar emission angle of 0 than for larger emission angles are again the result of a small number of sputtered atoms for these angles. For 0 and ±1, not the simulated values are shown but the mean value of ±2. For an angle of incidence of 80, the agreement between Monte-Carlo simulations and experiments is rather poor as was found by Sekowski et al. [S + 08]. In the experiments, the peak at about θ T = 14 was less pronounced. A possible reason is the assumption of a plane surface in most Monte-Carlo sputtering yield programs like MC_SIM, SDTrimSP, and SRIM. A rougher surface would decrease the peak shown in Figure 3.13 for large angles of incidence. Important for the simulation of the etching of structures due to sputtering is also the reection of ions. In Figure 3.14, the angular distribution of the sum of reected gallium ions and sputtered gallium atoms are shown. Only the sputtered atoms were taken into account which were sputtered with ϕ T = 0 ± 1 and ϕ T = 180 ± 1. Like for the sputtered silicon atoms, the sputtered gallium atoms with an angle of 0 and θ T ± 1 are not shown due to the large deviations and instead the mean value of θ T ± 2 is taken.

66 52 CHAPTER 3 THEORETICAL CONCEPTS ) ( atoms ion and degree dy dθt θ I = 0 θ I = 20 θ I = 40 θ I = 60 θ I = Polar emission angle θ T ( ) Figure 3.14: Angular distribution of the sum of sputtered and reected gallium atoms for dierent angles of incidence θ I. In the simulations, a silicon target was hit by gallium ions with an energy of 30 kev and under dierent angles of incidence. For an angle of incidence of 80, the right peak shown in Figure 3.14 is caused by the high reection probability of gallium ions Energy Distribution of Sputtered Atoms For simulating the topography changes due to sputtering, the energy of sputtered atoms is important because these atoms can cause sputtering when impinging on the surface. The energy distribution of sputtered atoms, as predicted by the Sigmund [Sig69] theory, is given by dy E T. (3.61) de T (E T + U s ) 3 2/m In the Sigmund theory, it is assumed that the energy distribution is the same for all emission angles. The energy distribution has a maximum at E T = U s /(2 2/m) where m is the exponent of the power potential V (r) 1/r m. The bulk binding energy is not considered in (3.61) because the results of (3.61) are compared with results simulated with MC_SIM, and in MC_SIM the bulk binding energy at the surface is zero. For E T U s, (3.61) becomes dy 1. (3.62) de T (E T ) 2 2/m

67 3.4 RESULTS OF THE SPUTTERING THEORY BY SIGMUND 53 1 Differential yield dy (1/eV) det dy de T dy E T (E T +U S ) 2.8 de T ET 1.6 MC_SIM Energy of sputtered silicon atoms E T (ev) Figure 3.15: Energy distribution of sputtered silicon atoms. A planar silicon surface was bombarded by gallium ions with 30 kev and under normal incidence. Simulation results carried out with MC_SIM are compared with (3.61) of the Sigmund theory and m = 10. For energies between 100 ev and 1000 ev, (3.62) was tted to simulation results and a value of m = 5 was obtained. The energy distribution was obtained by considering all emission angles. Simulations carried out with MC_SIM are compared with (3.61) and (3.62) in Figure 3.15 for silicon sputtered by gallium ions with an energy of 30 kev and under normal incidence. It was assumed that steady state is established for the gallium fraction x Ga at the surface. The surface binding energy U s for silicon atoms was calculated to 4.6 ev by (3.38) with HS Si = 4.7 ev and HGa S = 2.8 ev. Considering this, the surface binding energy is the same as in the Monte-Carlo simulation. To compare results of the Sigmund theory with simulations carried out with MC_SIM, the exponent m of the power potential was obtained in Section by tting the reduced nuclear stopping cross section of the power potential to s ZBL n (ɛ). For a silicon target and energies below 50 ev, the best t was obtained for m = 10. For increasing projectile energies, m decreases until m = 1 (unscreened Coulomb potential) at energies above approximately 100 kev (Figure 3.2). For energies between 0.1 ev and 100 ev, the angular distribution of sputtered atoms as predicted by the Sigmund theory agrees quite well with the simulation results. For energies between 100 ev and 1000 ev, m decreases to a value of 5 as was found by tting (3.62) to simulation results shown in Figure For increasing energies E T, the exponent m decreases and, therefore, the decrease in the energy distribution becomes less. However, in the simulations, the energy distribution falls o faster for kinetic energies E T above 1 kev. Two assumptions of the Sigmund theory

68 54 CHAPTER 3 THEORETICAL CONCEPTS explain the discrepancy between (3.62) and the results obtained by MC_SIM for kinetic energies above 1 kev: First, in MC_SIM, the maximum energy of a silicon atom in the substrate is determined by (3.5) to 24 kev while in the Sigmund theory there is no upper limit of the kinetic energy of a sputtered atom. Second, the direction of projectiles in the substrate is assumed to be isotropic for all energies in the Sigmund theory while the anisotropy in MC_SIM increases for high projectile energies. The direction of recoils with energies close to the maximum energy (24 kev) is almost the same as the direction of the ion when impinging upon the substrate. For normal incidence, a recoil atom with an initial energy of 24 kev must undergo several collisions to be sputtered. It is very unlikely to loose only a small amount of kinetic energy during these collisions. Agreement and disagreement between the Sigmund theory and MC_SIM has been discussed above for the energy distribution. Gnaser [Gna07] compared the Sigmund theory with experiments and tted (3.61) with m = and U s as a tting parameter to energy distributions obtained by experiments. In the experiments, argon ions impinge on three dierent targets (aluminum, calcium and silver) and the energy distributions of the sputtered target atoms were measured. For U s, the tted values were 1.4 ev (Al), 1.5 ev (Ca), and 2.2 ev (Ag) while the heat of sublimation is given by 3.4 ev (Al), 1.8 ev (Ca), and 3.0 ev (Ag). Therefore, by using the heat of sublimation for U s in (3.61), the energy E T where (3.61) has a maximum was too large and the surface binding energy might be overestimated when using the heat of sublimation. As mentioned in Section 3.4.4, Sigmund assumed the energy distribution to be independent of the angle of incidence. For dierent angles of incidence, simulations were carried out with MC_SIM for gallium ions impinging on a silicon target with an energy of 30 kev. For energies E T below 200 ev, the dierential sputtering yield normalized by the total yield is almost independent of the angle of incidence as shown in Figure For large energies (above about 200 ev), the dierential sputtering yield normalized by the total yield increases with an increasing angle of incidence. As mentioned above, in a collision, the maximum energy transferred from a gallium ion to a silicon atom is 24 kev. In this case, the silicon atom has almost the same direction with which the ions impinge on the substrate. To be sputtered, the silicon atom must change its ight direction. Therefore, it must have collisions with other atoms during which it is slowed down. For increasing angles of incidence, less collisions are necessary to change its ight direction in a way that it can leave the substrate. With a decreasing number of collisions, the silicon atom loses less energy. Therefore, more silicon atoms can be sputtered with higher energies for increasing angles of incidence.

69 3.4 RESULTS OF THE SPUTTERING THEORY BY SIGMUND (1/eV) dy (θi) det 1 Y (θi) θ I = 20 θ I = 40 θ I = 60 θ I = Energy of sputtered silicon atoms E T (ev) Figure 3.16: Energy distribution of sputtered silicon atoms for dierent angles of incidence θ I. In the simulations carried out with MC_SIM, gallium ions impinge on a plane silicon surface with an energy of 30 kev. The dierential sputtering yield dy (θ I )/de T was normalized by the total sputtering yield Y (θ I ). The energy distribution was obtained by considering all emission angles.

70 56 CHAPTER 3 THEORETICAL CONCEPTS

71 Chapter 4 Simulation of Physical Dry Etching In literature, several programs were presented which simulate the prole evolution due to sputtering with dierent levels of accuracy. Here, only the programs AMADEUS [K + 07a, K + 07b], IONSHAPER [P + 06b], and FIBSIM [BH01] are considered for comparison with ANETCH. To illustrate the dierences between these programs, the ow chart (Figure 4.1) describing ANETCH and all programs mentioned above is discussed rst: Starting from an initial surface prole, the surface is decomposed into surface segments. A time interval t is dened. The ion uence F for this time interval is given by F = Φ t, where Φ is the ux from the ion source. In a rst step, the incoming and outgoing uxes are calculated for each surface segment. To calculate the outgoing uxes, the dierential sputtering yield and the dierential reection yield must be determined for each surface segment. In a second step, the etched depth is calculated from the dierence of incoming and outgoing uxes for each surface segment and for the time interval t. The surface segments are then shifted accordingly. The two steps are repeated until a uence F total = F N is reached where N is the number of times the two steps have been repeated. As indicated in Figure 4.1, the dierent steps can be calculated by dierent methods. In Figure 4.1, only the methods are shown which were applied in the programs mentioned above. Two prominent methods to calculate the particle uxes are the Flux Balancing method [Bär98] (FBM) and the Monte-Carlo ux model (MCFM) (Section 4.2.2). In the Flux Balancing method, a discretization is used to calculate the incoming ux of each surface element from the ux from the ion source and from the outgoing ux of all other surface elements. FBM leads to a system of equations which can be solved numerically. Shadowing due to the geometry of the surface can be considered in the Flux Balancing method. In the MCFM, the trajectories of pseudo particles are determined by calculating the intersection points of particles with the surface by a kind of ray tracing method. In this method, the intersection points of the ray (given by ight direction and start position) with the surface elements are calculated and the intersection point with the shortest distance is chosen. The dierences between the uxes calculated by the two methods become negligibly small if the angular discretization is ne in FBM and the number of calculated trajectories is large in the MCFM. Independent of the method it is important that not only the ux 57

72 58 CHAPTER 4 SIMULATION OF PHYSICAL DRY ETCHING Initial Surface Profile Calculation of Particle Fluxes [Section 4.2] Flux Balancing (FBM) Monte Carlo (MCFM) Calculation of Differential Sputtering Yield and Differential Reflection Yield [Section 4.3] Data Table (DTSS) Monte Carlo (MCSS) Surface Shifting for F i[section 4.4] Level Set String Algorithm F i F F i = F (Total Fluence) Final Surface Profile Figure 4.1: Flow chart of ANETCH, AMADEUS, IONSHAPER, and FIBSIM. from the ion source is calculated but also the uxes of the sputtered atoms and reected ions. For example, AMADEUS calculates the ux from the ion source and the ux of sputtered atoms but neglects the ux of reected particles. IONSHAPER calculates all three uxes. However, reected particles are not reected when impinging on the surface again. Both programs calculate the uxes by the Flux Balancing method. The advantage of this method is that the computation time is typically less than the computation time of the Monte-Carlo method. Although the computation takes more time, ANETCH was extended additionally to the FBM by MCFM because it is fast to implement and easy to extend. In ANETCH, all uxes are calculated to ensure a realistic description of the sputtering process. The rst method, named data table simulation of sputtering (DTSS), provides the dierential sputtering yield and the dierential reection yield by data tables. The information can be obtained for example a-priori with a Monte-Carlo sputtering yield program. This method is very fast because the yield is only determined by considering some dependences from the data tables. The disadvantage is that due to the large number of dependences not all can be considered and the dierential sputtering yield and the dierential reection yield become inaccurate for some specic cases. More accurate but slower is the Monte-Carlo simulation of sputtering (MCSS) to calculate the dierential sputtering yield and the dierential reection yield. In this method, the trajectories of ions and recoils in the substrate are calculated using the binary-collision approximation (Sec-

73 4.1 REDUCTION OF THE COMPUTATION TIME IN ANETCH 59 tion 3.1.3). Both methods have their elds of application where they are superior to the other method. Therefore, both methods were implemented in ANETCH so that a specic sputtering process can be simulated with the appropriate method. In AMADEUS and IONSHAPER, the dierential yield is calculated by DTSS and in FIBSIM it is calculated by MCSS. The surface shifting can be calculated for example by the Level Set method or the string algorithm. AMADEUS can calculate the surface shifting by the Level Set method [K + 07b] or the string algorithm [K + 07a]. IONSHAPER and FIBSIM use the string algorithm for the calculation of the surface shifting. In ANETCH, the surface shifting was already implemented with the string algorithm. A discussion of the dierences between the two methods can be found in the literature, for example [Len02]. The structure of this chapter is as follows: First, in Section 4.1, two assumptions are discussed which were made in ANETCH to reduce the computation time. Afterwards, in Section 4.2, the calculation of the particle uxes is described and how it is implemented in ANETCH. The main part of this chapter is the calculation of the sputtering yield in Section 4.3. The two methods, DTSS and MCSS, are presented and their dierences and advantages are studied. Finally, in Section 4.4, the surface evolution and the implementation of the string algorithm are discussed. 4.1 Reduction of the Computation Time in ANETCH ANETCH can be applied to specic cases of sputtering. The elds of application were dened in Section 3.1. Two additional assumptions were made to reduce the computation time and the complexity of the problem. First, a constant atomic fraction at the surface of the material is assumed during sputtering. The second assumption is to neglect the swelling of the substrate due to the amorphization during the sputtering process. These assumptions will be discussed in the following Constant Atomic Fraction at the Surface A constant atomic fraction of the implanted ions within a specic depth from the surface is assumed in ANETCH to reduce the computation time. This atomic fraction must be provided as an input to ANETCH. In the simulations carried out for this thesis, the atomic fraction at the surface was chosen according to Table 3.2 and for the angle of incidence with which most of the ions were assumed to impinge upon the surface. For the example of a trench, the atomic fraction for normal incidence is assumed although some ions impinge upon the side walls with dierent angles of incidence. In this section, the inaccuracy of the sputtering yield is studied for two cases of gallium ions impinging upon a silicon target with an energy of 30 kev and under dierent angles of incidence. As one limiting assumption, the atomic fraction of implanted gallium at the surface is assumed to be the same as for normal incidence and steady state (x Ga = 0.285). As a second one,

74 60 CHAPTER 4 SIMULATION OF PHYSICAL DRY ETCHING Sputtering yield (atoms/ion) Sim: f Ga = 0 Sim: f Ga = Sim: Steady State Angle of incidence θ I ( ) Figure 4.2: For dierent atomic fractions of implanted gallium x Ga at the surface, the sputtering yield is plotted versus the angle of incidence. The sputtering yields were calculated with MC_SIM for gallium ions impinging upon a silicon target with an energy of 30 kev. For Steady State, the gallium fraction at the surface is given by Table 3.2. the atomic fraction of implanted gallium at the surface is assumed to be zero as it is the case for a small uence or grazing angles of incidence. For both atomic fractions, the sputtering yield is calculated for dierent angles of incidence and compared, as shown in Figure 4.2, with the sputtering yield (Steady State) calculated for an atomic fraction given by Table 3.2. For the example of a trench where x Ga = is assumed at the surface, the inaccuracy of the sputtering yield is negligible for small angles of incidence. However, for larger angles of incidence, the gallium atomic fraction at the surface decreases as was discussed in Section 3.3 and the sputtering yield by assuming x Ga = is larger by 10% at most than the yield calculated with a steady state gallium fraction. For the example of a trench, the side wall is etched faster in ANETCH than in other programs where the change of the fraction of implanted ions is considered. The sputtering yield calculated with x Ga = 0 is less than the yield calculated for steady state by 26% at most Amorphization and Swelling Due to ion implantation, the number of vacancies and interstitials close to the surface increases in a crystalline target. Therefore, the degree of long-range atomic order decreases and the crystal structure of the target is transformed into an amorphous one. Defects are located in the target within a specic depth from the surface due to the limited range

75 4.1 REDUCTION OF THE COMPUTATION TIME IN ANETCH 61 Surface shift (nm) Exp. Gnaser -30 Exp. Lehrer -35 Exp. Huey MC_SIM Fluence (cm 2 ) Figure 4.3: Topography changes due to gallium ion implantation in silicon with dierent ion uences. The changes were measured by atomic force microscopy [G + 08], [HL03], and [Leh05] and calculated by MC_SIM. of energetic atoms in a solid. The density of a material with an amorphous structure can be smaller than the density of a crystalline structure. For silicon, it was reported by Custer et al. [C + 94] that amorphous silicon a-si is 1.8 ± 0.1% less dense than crystalline silicon c-si at a temperature of 300 K. The atomic density of c-si is Å 3 at 300 K and according also to Custer et al. [C + 94] the atomic density of a-si is Å 3. During amorphization, the surface is shifted outwards. In this work, swelling is dened as the outward shift of the surface due to the change of the density. The surface does not only shift because of swelling but also because of sputtering and ion implantation. As an illustration, a crystalline silicon target is considered to be implanted by gallium ions with normal incidence. Topography changes measured in experiments for 30 kev gallium ions with atomic force microscopy were reported by Gnaser et al. [G + 08], Huey and Langford [HL03], and Lehrer [Leh05]. Gnaser and Huey exposed a (100) silicon crystal to gallium ions while the crystal orientation was not reported by Lehrer. These results are compared with topography changes simulated with MC_SIM as shown in Figure 4.3. The program calculates only the surface change due to material removal and ion implantation and does not consider amorphization. For ion uences smaller than cm 2, the topography changes due to material removal and implantation can be neglected as indicated by the curve determined with MC_SIM. At uences of cm 2, a swelling of the surface was already measured. The swelling reported by Gnaser et al. was around 0.5 nm and by Huey and Langford it was 1 nm. Lehrer carried out measurements at a uence of cm 2 and above. At

76 62 CHAPTER 4 SIMULATION OF PHYSICAL DRY ETCHING F = cm 2, Lehrer found a swelling of 2.2 nm. Furthermore, Lehrer [Leh05] measured that a surface layer with a depth of 70 nm has become amorphous. It is assumed that the surface shift is due to the amorphization of the crystal structure. This can be veried by calculating the surface shift. With the change of the density reported by Custer et al. [C + 94] and the depth of the amorphous layer measured by Lehrer [Leh05], the shift of the surface is approximately 1.3 nm and, therefore, in the range of the measured surface changes. Signicant material removal begins at a slightly smaller ion uence in MC_SIM than in the experiments. 4.2 Calculation of the Particle Flux The particle ux must be calculated at each point of the surface to determine the etch rate. In ANETCH, the surface is decomposed into segments and the particle ux is calculated for each surface segment. A ner discretization increases the accuracy of the simulation as well as the computation time. First, the way the surface is discretized in ANETCH is described. In the second part, it is described how the particle ux is calculated by the MCFM Surface Discretization In general, sputtering processes have to be simulated in 3D to take into account the complexity of the geometry. However, a large group of sputtering processes can be simulated in 2D as well. For a trench of innite extension, a proof was given by Bär [Bär98]. For the simulation of processes that could be treated in 2D, a modied version of ANETCH was implemented, ANETCH2D, to reduce the computation time. In ANETCH2D, the surface is represented by line segments and the surface shifting is based on the string algorithm. Additionally, the area representing the volume is decomposed into rectangles. These rectangles are separated into three groups. First, rectangles lying completely outside of the material, second, rectangles completely inside of the material, and third, rectangles with a surface segment. The surface of the material must be given as an input. The discretization of the area by rectangles and the renement of the surface segments is done by ANETCH2D. For the discretization of an object in 3D, its surface and/or its volume can be discretized. If processes happen only at the surface, the discretization of the surface is sucient and saves memory. Advantages and disadvantages of the two discretizations concerning the surface evolution were discussed for example by Kistler [Kis06]. The program ANETCH was extended to simulate sputtering and the algorithm to calculate the surface evolution was not changed. This algorithm implemented by Bär [Bär98] is based on a string algorithm and must have a surface discretization. The surface is decomposed into triangles. The triangles do not intersect with each other and the surface described by the triangles has a single continuous boundary. For each simulation carried out for this

77 4.2 CALCULATION OF THE PARTICLE FLUX 63 Figure 4.4: The topography of a masked substrate. The mask has a trench like opening. The surface was triangulated with the program MESH [Syn09]. work with ANETCH, the surface was triangulated with the program MESH which is part of the Sentaurus suite [Syn09]. MESH uses the DF-ISE format. The format is hierarchically ordered with nodes, edges and polygons. Dierent regions of material composition can be dened within this format. Figure 4.4 shows an example of a masked substrate with a triangulated surface Monte-Carlo Flux Model (MCFM) for the Calculation of the Particle Flux The particle ux can be calculated with the Flux Balancing method (FBM) and the Monte-Carlo ux model (MCFM). The FBM, described for example by Bär [Bär98], calculates the incoming and outgoing uxes for each surface element with a system of

78 64 CHAPTER 4 SIMULATION OF PHYSICAL DRY ETCHING coupled dierential equations. The calculation of the particle uxes with this method is very fast. Additionally, for the simulation of the sputtering process, the Monte-Carlo ux model was implemented in ANETCH/ANETCH2D although the calculation is slower than with the Flux Balancing method. The main advantage is that the MCFM is highly exible and, therefore, easy to extend. For the calculation of the particle ux, the spatial distribution of the ion uence from the ion source and the distribution of the initial ight directions must be provided. From these distributions the start positions of the ions and their ight directions are chosen randomly. The trajectories of these ions and the trajectories of reected ions and sputtered atoms are calculated. In the simulation, the trajectories of NI MC ions are calculated while in a real process the number of ions is N I. Therefore, the ions and sputtered atoms in the simulation must be weighted by N I /NI MC. The number of ions NI MC simulated with the MCFMmust be large enough to approach the solution within an acceptable error. On the other hand, the computation time is almost proportional to the number of simulated ions NI MC and, therefore, it is advantageous to reduce NI MC as far as possible. To reduce the computation time further while the solution is kept within an acceptable error, the calculation of the trajectories is distributed among dierent processors. Therefore, instead of calculating NI MC ion trajectories and their sputtered atoms on a single core of a processor, each of the N c cores calculates NI MC /N c ion trajectories and their sputtered atoms. The etch rates resulting from the calculation of each core are averaged and the surface is shifted accordingly. The total time t P total of the parallelized computation is t P total = t P ER + t com + t shift, (4.1) where t P ER is the time the slowest core needs to calculate the etch rate, t com is the time which is needed to average the etch rate and to communicate between the dierent cores, and t shift is the time needed to shift the triangles according to the averaged etch rate. The total time t S total for the serial computation is t S total = t S ER + t shift, (4.2) where t S ER is the time to calculate the etch rate on a single core. One could assume that t P ER = ts ER /N c. However, this is not the case due to two reasons: First, t P ER is not the average time of the cores but the time of the slowest core. Second, the implementation of the random number generator in ANETCH/ANETCH2D is slower for parallel computation. The implementation of the random number generator will be discussed but rst the dierence of the computation time is calculated for one example. As an example, the simulation of one etch step of an initially planar silicon surface is considered. Gallium ions with an energy of 30 kev and normal incidence impinge upon the silicon surface. The surface is decomposed into 64 triangles. The simulations were carried out on a computer cluster which consists of 34 x FSC RX220 (2xDual Core Opterons and 8 GB RAM). For the parallel computation, the simulations were done on 20 cores and the single computation was done on a single core. In both cases, ions were simulated. For the parallel version 10 6 ions were simulated on each core. Therefore, t S ER can be compared with t P ER when t S ER is divided by the number of 20 cores. In addition,

79 4.2 CALCULATION OF THE PARTICLE FLUX 65 Table 4.1: Comparison of computation time between a serial simulation and a parallel one (20 cores). The value in parentheses is t S ER divided by the 20 cores. Time (sec) t S ER (3119) t P ER 3857 t com 39 t shift < 1 t com and t shift are shown in Table 4.1. The time t shift which was used to shift the surface can be neglected as a discretization of 64 triangles is quite small and the etch rate is similar for the triangles. The time t com used for the communication between the cores is independent of the sputtering problem because the values of xed-sized arrays are sent between the cores. t P ER is larger by 24% when compared with t S ER /20. This is mainly due to the implementation of the random number generator which is now discussed. Typically, a large number of random numbers x is computed in one simulation with ANETCH/ANETCH2D. Therefore, a random number generator must have a large period and a uniform probability distribution. Both requirements are met by the Mersenne Twister proposed by Matsumoto and Nishimura [MN98] with a period of and a uniform probability distribution, so that p(x) = { 1 : 0 x 1 0 : otherwise (4.3) For the parallel computation, it must be assured that the dierent cores use dierent random numbers. A pseudo-random number generator like the Mersenne Twister produces a xed sequence of numbers. Even if dierent start values are used for each core, the probability is high that the same part of the sequence is used on more than one. Special start values must be calculated to reduce this probability. A dierent approach was implemented in ANETCH/ANETCH2D. As an example, four cores are considered and a sequence of pseudo random numbers R i is distributed among them as indicated in Figure 4.5. The initial values of the four cores are four directly following random numbers of the sequence. For each core, the next random number R i+4 is determined by the previous random number R i plus four steps in the sequence. This method ensures that the sequence of random numbers doesn't change. However, a special random number generator for parallelization would save quite some computation time. The statistical deviations of the etch rate are larger for triangles which are rarely hit by ions due to geometrical reasons. For example, in the simulation of the etching of a trench, less ions per area impinge upon the side wall than upon the bottom of the trench. In order to decrease the statistical noise at the side wall, the number of ions impinging on the side wall must be increased. This was done in two ways: First, the number of ions impinging on the side wall from the source was increased by a factor w. These ions and their sputtered atoms were weighted by 1/w. This will not change the etch rate at the side wall, but the statistical noise is reduced. Second, the trajectory split method,

80 66 CHAPTER 4 SIMULATION OF PHYSICAL DRY ETCHING Initial Rnd.Nr. Rnd. Nr. Rnd. Nr. Core 1 R R 5 R 9 Core 2 R R R 10 Core 3 R R R 11 Core 4 R R R 12 R i : Sequence of random numbers Figure 4.5: A sequence of random numbers R i is distributed among four cores. proposed by Bohmayr et al. [B + 95], was used when calculating the trajectories in the substrate. If, for example, an ion impinges upon the substrate, the sputtering yield may randomly vary around a mean value. Therefore, the sputtering yield was calculated from w identical ions with dierent random numbers and the simulation was continued with all sputtered atoms. Each sputtered atom was then weighted by 1/w. 4.3 Calculation of Sputtering Yield In ANETCH/ANETCH2D, two methods were implemented to calculate the dierential sputtering yield and the dierential reection yield. The rst method DTSS described is based on data tables where the dierential sputtering yield and the dierential reection yield are determined from look-up tables. The second method MCSS calculates the trajectories of ions or atoms and their recoils in the substrate. From these calculations, the dierential sputtering yield and dierential reection yield are determined Data Table Simulation of Sputtering (DTSS) The rst method to calculate the sputtering yield Y, called data table simulation of sputtering (DTSS), is based on data tables. Its main advantage is the fast computation time. The sputtering yield, without reected ions, Y S (2.4) and the reection yield Y R (2.5)

81 4.3 CALCULATION OF SPUTTERING YIELD 67 are not calculated analytically but instead the user must provide information in form of look-up tables. This data can be determined for example by sputtering experiments. However, sputtering experiments are time consuming and expensive. As an alternative, a modied version of MC_SIM was implemented. With this implementation, the program calculates the sputtering yield data Y S and the reection yield data Y R at the beginning of the simulation and saves it in look-up tables. For both cases, providing data by the user or calculation of data with MC_SIM, the dependences of Y S and Y R must be considered. The sputtering yield Y S depends on dierent parameters Y S (E I, θ I, Z I, M I, Z Ti, M Ti, R). (4.4) These are the ion energy E I, the angle of incidence θ I, atomic number Z I and mass M I of the ion and the atomic numbers Z Ti and masses M Ti of the elements in the target material. Furthermore, Y S depends on the radius of curvature R (R = for planar surface). The reection yield Y R depends on the same parameters as Y S. The sputtering yield Y S and reection yield Y R are sucient for a planar surface where reected ions and sputtered atoms don't cause further sputtering or redeposition. However, for a substrate with a pronounced structure, the angle and energy distribution of the sputtered atoms and reected ions are important to calculate further sputtering and redeposition. Therefore, the number of particles which are sputtered with energy E T, the polar emission angle θ T, and the azimuthal emission angle ϕ T must be considered. Furthermore, the position l at which the particles are sputtered is of interest. Then the sputtering yield Y S has the dependences: Y S (E T, θ T, ϕ T, ) l; E I, θ I, Z I, M I, Z Ti, M Ti, R (4.5) Again, Y R depends on the same parameters as Y S. This vast number of dependences cannot be determined by experiments and it is also not viable to calculate them a-priori with MC_SIM. Therefore, some simplications are needed. For the radius of curvature, a plane surface is assumed and atoms are assumed to be sputtered from the position where the ion has impinged upon the surface. Y S and Y R are treated dierently: Reected ions are described by specular reection. Sputtered atoms described by Y S are assumed to have an energy so small that they cannot cause any sputtering when impinging on the surface. This is a very rough simplication. It is determined by the adhesion probability if these atoms adhere at the surface. The angular distribution of sputtered atoms is approximated by the angular distribution for normal incidence [Sig81] dy cos α (θ T ), (4.6) dθ T where α is a tting parameter which must be provided. The dependence on the azimuthal emission angle ϕ T is approximated by a uniform distribution. With these simplications, the sputtering yield Y S is Y S (E I, θ I, Z I, M I, Z Ti, M Ti ) (4.7) and the reection yield Y R is Y R (E I, θ I, Z I, M I, Z Ti, M Ti ). (4.8)

82 68 CHAPTER 4 SIMULATION OF PHYSICAL DRY ETCHING Si-sputtering yield (atoms/ion) Angle of incidence θ I ( ) Figure 4.6: Sputtering yield of silicon atoms caused by gallium ions impinging on a silicon substrate with an energy of 30 kev and under dierent angles of incidence. The fraction of gallium in the silicon substrate is taken after steady state has been established. For the implementation of this method in ANETCH/ANETCH2D, the ion source is restricted to one ion type (Z I, M I ) and one ion energy (E I ) chosen by the user. The substrate can consist of more than one element. Very rough simplications were made in data table simulation of sputtering described so far. Simulations carried out with this method might become less accurate if the re- ection of ions and sputtered atoms with high energies is important for the etch prole. For normal ion incidence, the angular distribution of sputtered atoms (4.6) was veried experimentally for some metals and semiconductors and the results were summarized by Behrisch [Beh07]. However, for large angles of incidence, the maximum in the yield distribution is shifted to larger polar emission angles as discussed in Section Till now, no analytical expression has been established to describe this behavior. Therefore, the distribution (4.6) is used despite its deciencies. Most of the simulations, which were carried out for this work, calculated the surface shifting due to gallium ions impinging on a silicon target with an energy of 30 kev. In the following, the sputtering yield data and the reection yield data used are presented. The data was calculated by MC_SIM for steady state of the gallium fraction in the substrate (Section 4.1.1) and an amorphous layer at the surface (Section 4.1.2). For the simulations, the electronic stopping model (3.35) was tted in a way that the sputtering yield for normal incidence is in good agreement with experiments (Section 6.1). The dependence of YS Si on the angle of incidence is shown in Figure 4.6 and the reection

83 4.3 CALCULATION OF SPUTTERING YIELD 69 Table 4.2: Probability Y R of a gallium ion with 30 kev to be reected at a plane silicon surface determined by Monte-Carlo simulations. θ I Probability probability is given in Table 4.2. The sum of YS Ga and reected gallium ions Y R is one for all angles of incidence because the fraction of the gallium does not change in steady state. The parameter α for the distribution of the yield with the polar emission angle θ T (4.6) was tted to simulation results. For sputtered silicon atoms, α is 1.24 and for sputtered gallium atoms, α is 1.18 as discussed in Section Sputtered atoms can be redeposited at the surface when impinging on it again. The probability to adhere at the surface is typically described by the sticking coecient. The sticking coecient was assumed to be one for silicon as well as for gallium atoms Monte-Carlo Simulation of Sputtering (MCSS) The second method to calculate the dierential sputtering yield, denoted MCSS to distinguish it from the calculation of the particle uxes (MCFM) in Section 4.2.2, is based on a Monte-Carlo model proposed by Biersack and Eckstein [BE84]. A program based on this model is MC_SIM, an ion implantation program, which was implemented by Ullrich et al. [U + 05]. Other programs based on the Biersack model, for example SDTrimSP and SRIM, can be used to verify results obtained with MC_SIM. To simulate the surface evolution due to sputtering, MC_SIM was integrated into ANETCH. The program MC_SIM was modied to save computation time and to consider the surface topography provided by ANETCH. In the rst part of this section, the model proposed by Biersack and Eckstein is introduced. Second, the modications to reduce the computation time are discussed and in a third part, the implementation is presented that allows to consider the surface topography for the calculation of the sputtering yield. In the MCSS approach, ions and recoils have only collisions with target atoms. After a collision a projectile passes a free path length λ before the next collision. A xed distance (λ = n T 1/3 ) is assumed for the free path length depending on the atomic density n T of the target material. Between the collisions, the only force on the projectile is opposite to the direction of the projectile and is described by the model used for the electronic energy loss. For the determination of the next collision partner, short-range order in the

84 70 CHAPTER 4 SIMULATION OF PHYSICAL DRY ETCHING z Position of projectile d λ Position of target atom y p max ϕ p Figure 4.7: Illustration of the collision cylinder. The volume of the cylinder is the same volume one atom has in the target. x substrate is not taken into account, to save computation time. For each collision, the position of the target atom is determined by constructing a right circular cylinder with the projectile ight direction as the axis and the free path length as the height of the cylinder (Figure 4.7). The radius p max of the circle at the top and bottom is chosen so that the volume of the cylinder V cyl is the same as the volume of one target atom in the substrate (V cyl = n 1 T ). The radius is then p max = n 1/3 T / 1/3 π because λ = n T. The target atom is placed at the bottom face of the cylinder. The exact position of the target atom in the circle is determined by random numbers (ϕ and p). The position of the target atom determines the impact parameter. With the impact parameter, the minimum distance r min can be calculated by solving (3.3). Afterwards, the scatter angle θ C in the center-of-mass reference system can be calculated by numerically solving the integral in (3.4). To save computation time, Biersack and Haggmark [BH80] proposed an analytical expression for the scatter angle, the Magic formula. For the Magic formula, Biersack and Haggmark, as reported by Eckstein [Eck91], found a mean accuracy of the order of 1% by comparing it with calculated data presented by Robinson for the Moliere potential. The direction of the two atoms after the collision can be calculated from θ C and the transferred energy can be calculated from the scatter angle with (3.5). Important for the calculation of the collision process is the choice of the interatomic potential model (3.6) which is used in (3.3) and (3.4). All simulations in this work were carried out with (3.6) and the Ziegler, Biersack and Littmark screening function. The interaction between a projectile and a substrate atom is only considered for distances below (p max = n 1/3 T / π) as was discussed in Section The electronic

85 4.3 CALCULATION OF SPUTTERING YIELD 71 stopping power was calculated by (3.23) and (3.35). The surface binding energy was calculated by (3.38, 3.39) and for the bulk binding energy a value of 1 ev was assumed. In the implementation of MC_SIM in ANETCH/ANETCH2D, the computation time is critical because typically a large number of trajectories must be calculated in ANETCH/ANETCH2D. The rst simplication to reduce the computation time was to neglect the dynamic calculation of the atomic fraction at the surface as discussed in Section A large improvement to save computation time was already implemented in MC_SIM: The calculation of a projectile trajectory is stopped if it is impossible for the projectile to reach the surface. At the beginning of the simulation, a table is calculated with the maximum range each element can pass in the target. The maximum range is determined by the electronic energy loss and the nuclear energy loss for p = p max. After passing the free path length, the shortest distance of the projectile to the surface is compared with the maximum range saved in a lookup table for the current energy of the recoil. If the maximum range is smaller than the shortest distance to the surface, the trajectory is not further calculated. This simplication reduces the computation time signicantly. For example, for gallium ions impinging upon a silicon target with an energy of 30 kev and normal incidence, the mean number of recoils in a collision cascade is about 2500 for the full calculation of the trajectories. With the simplication, the mean number of recoils is reduced to about 60 and the computation time is decreased by a factor of 10. The saving in computation time is less for grazing incidence or pronounced surface topography where recoils are closer to the surface than for normal incidence at a planar surface. Two further simplications were made to save computation time. To determine the scatter angle, (3.3) is typically solved by Newton's method and the integral in (3.4) is approximated by the Magic formula proposed by Biersack [BH80]. This approach was modied in a way that the scattering angle is now determined from a lookup table. At the beginning of the simulation, the lookup table is calculated. Similarly, the energy loss due to electronic stopping is saved in lookup tables at the beginning of the simulation. Then, this energy loss can be determined very rapidly during the calculation of the ion trajectories. No dierences of the sputtering yield were found when comparing simulations carried out with the above mentioned simplications and original simulations with MC_SIM. In Section 4.2.2, it was already discussed that the calculation of the trajectories is distributed among dierent cores which reduces the wall-clock time further. In the program MC_SIM, a planar surface is assumed and, therefore, surface topographies dierent from a planar one can not be considered for the calculation of the sputtering yield. To consider the surface topography, MC_SIM was modied during the implementation in ANETCH/ANETCH2D in the following way: The discretized surface topography in ANETCH/ANETCH2D is taken as boundary for the trajectories in the substrate. Ions or recoils crossing the boundary from the substrate to the gas are assumed to be sputtered if their kinetic energy is larger than the surface binding energy. Dierent methods were used in ANETCH and ANETCH2D to decide if the boundary is crossed. In 3D, not only the position of the projectile is calculated but also the shortest distances to all surface triangles which are within the maximum range of the projectile. From this information it can be decided if the projectile is in the substrate or outside. In 2D, it is rst determined in which rectangle (Section 4.2.1) the projectile is. For rectangles

86 72 CHAPTER 4 SIMULATION OF PHYSICAL DRY ETCHING Z(Å) Ion Sputtered Atoms Silicon X(Å) Figure 4.8: Collision cascade generated by one gallium ion with 30 kev in a silicon substrate. The surface is represented by the drawn line and each collision is represented by a cross.the arrow pointing at the substrate represents the incoming ion and the other arrows the directions of the sputtered atoms. with at least one surface element, it is calculated if the projectile is inside or outside. The method applied in ANETCH2D is faster than the one in ANETCH. For a convex surface topography, a cross section of a collision cascade calculated with MC_SIM is shown in Figure 4.8. In the simulation, a gallium ion is assumed to impinge upon a silicon substrate with an energy of 30 kev. Two surface elements are shown which dene the boundary for the ion and the recoils. If recoils cross the surface, indicated by the crosses outside of the substrate, only such with a kinetic energy exceeding the surface binding energy are assumed to be sputtered. Recoils with less energy are redeposited at the surface. At each surface segment, the number of ions/atoms impinging upon the surface and the number of atoms sputtered from the surface are counted. From the dierence between sputtered and implanted atoms, the etched depth or growth height is calculated. For the example shown in Figure 4.8, the dierence between sputtered and implanted atoms is zero for the left surface element while the dierence is three for the right one. This way to calculate the etched depth of the surface segments does not take into account the ow of projectiles in the solid. In Figure 4.8, it seems that atoms initially closer to the left surface segment have moved to a position closer to the right surface segment. To consider this ow in the calculation of the etched depth, an alternative method to calculate the etched depth was implemented. In this method, the etched depth is calculated by

87 4.4 SURFACE EVOLUTION 73 the dierence of vacancies and interstitials. For the calculation of the etched depth of each surface segment, the vacancies and interstitials are considered which are closest. For this case, trajectories of the projectiles must be calculated until their kinetic energy is zero. Both ways to calculate the etched depth were implemented in ANETCH and the appropriate method can be chosen as an input in ANETCH. 4.4 Surface Evolution In the previous sections, the methods to calculate the etch rate in ANETCH/ANETCH2D for a time-invariant surface were described. In this section, the method to shift the surface according to the calculated etch rates is described rst. Afterwards, renement and coarsening of the surface discretization is discussed Surface Shifting For the 2D case, a string algorithm was implemented in ANETCH2D for this work. In ANETCH, the triangles representing the surface are shifted by a 3D version of the string algorithm. This algorithm was already implemented by Bär [Bär98]. First, the string algorithm is discussed for the 2D case and, then, the extension of the string algorithm to the 3D case is explained. In ANETCH2D, the surface is decomposed into line segments. Each line segment is connected to the ends of other line segments at its ends. The connection points are called vertices and the surface is shifted via the vertices. For each vertex, the direction of the shifting vector and its length are determined by the etched depth of the line segments at the vertex. The shifting vector at the vertex i is s i = (d i n i + d i+1 n i+1 ), (4.9) n i + n i+1 where d i and d i+1 are the etched depths at lines i and i + 1, respectively. The vectors n i and n i+1 with unit length are perpendicular to the lines i and i + 1, respectively. These vectors are dened to point into the substrate. In the following, (4.9) is studied by assuming d i =d. Due to the constant etched depth d, the angles ϕ i between the vectors n i and n i+1 should not change during the surface shifting. However, as shown in Figure 4.9 and reported by Bär [Bär98], this is not the case when the surface is shifted by (4.9). To simulate the evolution of the surface topography in a more realistic way, Bär [Bär98] proposed the correction factor 1 f cor = cos (ϕ/2), (4.10) where ϕ i is the angle between the vectors n i and n i+1. The surface is then shifted by (4.9) times the correction factor. As indicated in Figure 4.9, the angles ϕ i are constant when

88 74 CHAPTER 4 SIMULATION OF PHYSICAL DRY ETCHING Initial surface Vertex Shift vector Shifted surface Shifted surface with f cor Figure 4.9: Schematic view of the shifting of a surface decomposed into line segments. The shifting was carried out with (4.9). For comparison, a second shifting of the surface is shown where the correction factor (4.10) was applied. Initial surface Vertex Shift vector Shifting method 1 Shifting method 2 Figure 4.10: Schematic view of the shifting of a surface decomposed into line segments. Shifting method 1 shifts the surface by (4.9) and the correction factor (4.10). Shifting method 2 shifts the surface by (4.11). the surface is shifted by (4.9) and the correction factor f cor for ϕ = 0 and ϕ = 90. For angles ϕ dierent from 0 or 90, the angles ϕ i can still change as indicated in Figure To have always constant angles ϕ i, an alternative of the surface shifting was implemented in ANETCH2D: The surface can then be shifted by ( ) [ ] 1 ( ) sxi nxi n = yi ERi. (4.11) n yi+1 s yi n xi+1 ER i+1 With this method to shift the surface, the angles ϕ i are almost constant as indicated in Figure The disadvantage of this method is that loops in the surface are more likely

89 4.4 SURFACE EVOLUTION 75 to occur. To reduce this probability, the surface is shifted by (4.9) with the correction factor (4.10) for ( n i n i+1 > 0.9) and by (4.11) for ( n i n i+1 0.9) where simulations have shown that a value of 0.9 is a reasonable limit. In ANETCH (3D), the surface is decomposed into triangles. At each corner of a triangle, the triangle is connected with the corners of other triangles. These points are typically called vertices. For the surface shifting, only the vertices are shifted and the triangles are then constructed from the shifted vertices. For each vertex, the direction of the shifting vector and its length are determined by taking all triangles at the vertex into account. The shifting vector s i is calculated by s i = j γ jd j n j j γ jd j n j j γ jd j j γ j, (4.12) where the sum is over all triangles at the vertex. d j is the etched depth at the j'th triangle, γ j is the internal angle of the j'th triangle at the vertex, and n j is the surface normal of the j'th triangle. The normals of the triangles are directed into the substrate. As for the 2D case, a correction factor is necessary to describe the surface shifting in a more realistic way. The factor depends on the angle between the normals of the triangles at the vertex and the direction of the shifting. The correction factor suggested by Bär [Bär98] is j f cor = γ j j γ j cos(ϕ j ), (4.13) where cos(ϕ j ) is cos(ϕ j ) = sgn(d j ) ( n j n s ). (4.14) sgn() is the sign function and n i n s is the inner product between the normals n j of the triangles at the vertex and the normalized direction n s of the shifting vector. The factor proposed by Bär was originally without the sign function because all triangles were shifted in the same direction. Due to redeposition, some triangles are shifted outwards and some are shifted inwards. Therefore, the sign function became necessary. A simple form of (4.11) for the 3D case could not be found Renement and Coarsening During the shifting of the surface topography, the surface elements (triangles or line segments) can become very small or very large as indicated in Figure 4.11 for the 2D case. For the same uence, line segments with a smaller length are exposed to less ions than line segments with a larger length. Therefore, the statistical variation of the etch rate typically increases with a decreasing length of the line segments. On the other hand, line segments with a large length approximate spatial variations of the ion uence with less accuracy. The same applies to surface triangles. The programs ANETCH/ANETCH2D merge surface elements which are smaller than a predened value with adjacent surface elements and split surface elements which are larger than another predened value.

90 76 CHAPTER 4 SIMULATION OF PHYSICAL DRY ETCHING Figure 4.11: The cross section of the same substrate is compared for two dierent renements. Line segments (between the circles) dene the surface of the substrate. α Figure 4.12: The two cross sections of the same substrate illustrate the renement of two adjacent surface elements with an angle below a lower limit. Line segments (between the circles) dene the surface of the substrate. The angle α between two adjacent surface elements might become too small (the angle is π for a planar surface) as indicated in Figure If the angle is below a lower limit, the programs ANETCH/ANETCH2D delete the vertex with the small angle and connect the vertex before the deleted vertex with the vertex after the deleted vertex.

91 Chapter 5 Study of Selected Examples with ANETCH For some examples, interesting for the fabrication of nanostructures with sputtering, the topography changes due to sputtering are investigated with ANETCH. Only the implementation of MC_SIM in ANETCH described in Section makes it possible to study all important eects considered in the examples. A short overview of the examples investigated in this chapter is given in the following: As a rst example, the inuence of reected ions on an eect called microtrenching will be studied in Section 5.1. Close to the side walls at the bottom of a trench, an increase of the etch rate was observed experimentally for example by Lehrer [Leh05]. This eect is typically known as microtrenching. First, for the geometry of a silicon layer masked by a titanium layer with a cylindrical hole, the two methods to calculate the particle ux, the ux balancing method (FBM) and the Monte-Carlo ux model (MCFM), are compared. Afterward, the dierence between the Monte-Carlo simulation of sputtering (MCSS) and the data table simulation of sputtering (DTSS) will be shown for the geometry of a trench. Important for the fabrication process is also redeposition of sputtered atoms as it limits the aspect ratio of deep structures. In Section 5.2, the dependences of redeposition will be identied and their inuence will be studied. Redeposition causes sloped side walls and for deep trenches a v-shape prole. Another parameter inuencing the evolution of side walls is the spatial distribution of the uence and the dependence of the sputtering yield on the angle of incidence θ I. In Section 5.3, it will be investigated how the side-wall slope depends on these two parameters. In this research, it will be taken advantage of the possibility that in computer simulations eects can be studied independent of other eects for example redeposition. For some applications like the fabrication of the tips of eld emitters, the eects of blanket ion bombardment on a steep step in the surface is important. For a realistic description of the topography changes, the dependence of the etch rate on the angle of incidence θ I is important. The evolution process will be studied in Section

92 78 CHAPTER 5 STUDY OF SELECTED EXAMPLES WITH ANETCH Figure 5.1: Cross section of an etched substrate. The silicon substrate is masked with a layer of titanium which has a cylindrical opening with tapered side walls. Due to a simple model of the data table simulation of sputtering, microtrenching is overrated. 5.1 Reection of Ions Reection of ions at sloped side walls can lead to a higher etch rate close to the side walls at the bottom of a structure, called microtrenching. The eect of reected ions on the etch rate is studied in this section for dierent mask geometries. In the rst part of this section, two dierent methods to calculate the particle uxes are compared with an analytical model for a structure with a cylindrical mask opening (Figure 5.1). In the second part, two methods to calculate the ion reection are compared for a trench with sloped side walls. Large parts of this section have been published in [KB08]. The two models to calculate the particle uxes, Monte-Carlo ux model MCFM (Section 4.2.2) and Flux Balancing method FBM [Bär98], are compared in the following. To study the dierences between the two methods, it is important how accurate the uxes are calculated for each triangle of a surface discretization. For this purpose, a very simple model for the data table simulation of sputtering DTSS (Section 4.3.1) is advantageous because it allows to overrate microtrenching and thus allows to have larger dierences between the etch rates of adjacent triangles. The sputtering yield is assumed to be independent of the angle of incidence θ I and ions are assumed to be reected specularly without loosing energy if their angle of incidence θ I is above a given limit θi lim. Below this limit, the ions are assumed to be implanted. To compare the particle uxes, a geometry was chosen which causes a large variation in the etch rate at the substrate. A silicon substrate is masked with a layer of titanium with a cylindrical opening. First, the etch

93 5.1 REFLECTION OF IONS 79 rates are compared for a test case assuming an isotropic angular distribution of incoming gallium ions and, afterwards, the etch rates are compared for a perfectly directional angular distribution of incoming gallium ions. For both cases, the uence was assumed to be uniform. The cross section shown in Figure 5.1 was calculated by assuming an isotropic angular distribution of incoming ions. An isotropic angular distribution has the advantage that some ions are reected more than once and, therefore, the accuracy of the calculation of the particle uxes can be tested. The etch rate of the titanium mask was assumed to be negligibly small compared with the etch rate of the silicon substrate and, therefore, the mask layer was not etched at all in the simulation. The simple model for DTSS has also the advantage that the etch rates for this setup can be calculated additionally by a third method which is described below. The ux for a location at the substrate is calculated by integration. The calculation takes advantage of the axisymmetry of the setup and the etch rate is only calculated from the center to the mask edge. Part of the integration had to be carried out numerically. The advantage of this method is that the accuracy of the calculated uxes only depends on numerical errors and the surface must not be decomposed into surface segments. For the geometry shown in Figure 5.1 but with a side-wall angle of 90, the particle uxes were calculated with the three approaches (i.e. MCFM, FBM, and analytical calculations). The geometry and the incoming ions are cylindrical symmetric and, therefore, the etch rate at the silicon surface is also cylindrical symmetric. From the center of the substrate exposed to ions to the mask edge the etch rate varies signicantly giving the opportunity to compare the three approaches. For the etched depth along the radius, good agreement was found between the three fully independent approaches as indicated in Figure 5.2. At the right hand side in Figure 5.2, a dierence of the topography was found between the FBM and the MCFM on the one side and the analytical calculation on the other side. This is caused by the triangles representing the bottom of the trench and which are in contact with the side wall of the mask. For the isotropic angular distribution of the ions, the agreement between the two methods, FBM and MCFM, to calculate the particle uxes can be seen in Figure 5.2. For the same geometry considered above but this time with a side-wall slope of 86 (90 for perpendicular side walls), the etch rates caused by a perfectly directional ion beam were calculated. For the data table simulation of sputtering, the model described above was taken (i.e. specular ion reection and a sputtering yield independent of the angle of incidence θ I ). Due to the simple model, the etch rates can be calculated by an analytical model described in the following. The calculation of the etch rates for a directional ion angular distribution is performed by summing up the particles directly from the ion source and the particles arriving due to reection from the mask. To this end, as shown in Figure 5.3, the surface is decomposed into rings. For example, the etch rate ER for the substrate ring 1 results from ions arriving directly from the ion source and ions reected by the mask ring 1. More generally and for innitesimally thin rings i, the etch rate ER for ring r i is given as ( ER(r i ) = const n T 1 + A ) m,i Φ, (5.1) A s,i

94 80 CHAPTER 5 STUDY OF SELECTED EXAMPLES WITH ANETCH Vertical direction (µm) Analytical calculation Flux-Balancing method Monte-Carlo method Distance from the center (µm) Figure 5.2: Cross sections of a cylindrical hole are shown which were obtained by three methods to calculate the particle uxes (i.e. Monte-Carlo ux model (MCFM), Flux- Balancing method (FBM), and analytical calculations). A silicon substrate was masked with a layer of titanium which had a cylindrical mask opening. The aspect ration is one and the diameter of the opening is 0.5 µm. The angular distribution of incoming ions was assumed to be isotropic. (a) Side view (b) Top view Figure 5.3: Schematic gure of the analytical approach to calculate the etch rate for a directional ion angular distribution and sloped side walls. where A m,i and A s,i are the projections of the areas of the substrate ring i and of the mask ring i onto a plane perpendicular to the direction of the ions from the ion source. Φ is the ux (2.8) from the ion source to the substrate.

95 5.1 REFLECTION OF IONS 81 Etch ratemax / Etch ratecenter Analytical: AR = 1.0 Analytical: AR = 1.5 Analytical: AR = 2.0 MCFM: AR = 1.0 MCFM: AR = 1.5 MCFM: AR = Angle of inclination ( ) Figure 5.4: Ratio between the maximum etch rate and the etch rate at the center of the substrate exposed to ions for dierent angles of incidence θ I and aspect ratios AR. The etch rates were obtained by calculating the particle uxes with the Monte-Carlo ux model (MCFM) and with an analytical approach. The substrate was masked by a layer of titanium with a cylindrical opening. A perfectly directional angular distribution has been assumed. As can be seen from (5.1), the etch rate is more than twice as high as the etch rate without reection because A m,i > A s,i. The ratio A m,i A s,i increases with decreasing i for A m,1 being the area of the outermost ring of the mask. Therefore, the etch rate increases from the mask edge towards the center of the substrate until a maximum is reached and then falls o to the etch rate given only by the direct ion ux from the source. This maximum of the etch rate depends on the aspect ratio AR and the side-wall slope. The ratios between the maximum etch rate to the etch rate without reection were obtained by calculating the uxes with the MCFM and by the analytical method. The results are shown in Figure 5.4 for dierent aspect ratios and side-wall slopes. The ratios shown in Figure 5.4 correspond to the values at the very beginning of the etching process. The dierence observed between the Monte-Carlo approach and the analytical one is assumed to be due to the nite triangle size which leads to an averaging of the etching rates over a lateral range corresponding to the spatial extension of the triangles. An increase of the maximum etch rate with a decreasing triangle size was observed in the simulations. After dierent approaches to calculate the particle uxes have been studied, two different methods to calculate the sputtering yield and the reection yield are compared in the following. The rst method (DTSS) determines the sputtering yield and the reection yield from data tables (Section 4.3.1) and the second method (MCSS) calculates the ion and recoil trajectories in the substrate (Section 4.3.2) and determines the yields from these calculations. The particle uxes were calculated by the Monte-Carlo ux model. For these simulations, redeposition was taken into account. The simulations were carried

96 82 CHAPTER 5 STUDY OF SELECTED EXAMPLES WITH ANETCH Data table simulation of sputtering Monte-Carlo simulation of sputtering 0.02 Etched depth (µm) Distance from side wall (µm) Figure 5.5: Etched depth at the bottom of a trench versus the distance from the sloped side wall (angle of inclination 84 ). The results were obtained by a simulation with the data table simulation of sputtering (DTSS) and by a simulation with the Monte-Carlo simulation of sputtering (MCSS). The particle uxes were calculated by the Monte-Carlo ux model (MCFM). Only the side wall was exposed to gallium ions under an angle of incidence of 84. out for a trench with sloped side walls. The height of the side wall was 3 µm and the angle of inclination of the side wall was 84 (90 for perpendicular side walls). The diameter of the trench was assumed to be large enough to neglect the inuence of the other side wall. To investigate the eects of reection, gallium ions were assumed to impinge only on the side wall of the trench under an angle of incidence of 84 and with a directional ion distribution. The energy of the ions from the source was 30 kev, and the substrate material was silicon. A signicant dierence of the etch rate at the bottom of the trench was found between the two simulations as shown in Figure 5.5. A discontinuity in the etch rate can be seen for the simulation with the DTSS at a distance of µm from the side wall. In the DTSS, the ions are reected specularly and, therefore, all ions reected from the side wall impinge on the bottom of the trench within a distance of the thickness of the side wall at the base which is µm. For distances exceeding µm, only atoms sputtered from the side wall impinge on the bottom and are redeposited in the data table simulation of sputtering with an adhesion probability of one. In the other model (MCSS) where the trajectories of ions and recoils are calculated in the substrate, the ions are not reected specularly (Figure 3.14). Reected ions also impinge on the bottom of the trench at distances exceeding µm from the side wall. Furthermore, some sputtered atoms have enough energy to cause sputtering when hitting the substrate

97 5.2 REDEPOSITION OF SPUTTERED ATOMS 83 Etched depth (µm) Angle of inclination: 72 Angle of inclination: 76 Angle of inclination: 80 Angle of inclination: 84 Angle of inclination: Distance from side wall (µm) Figure 5.6: Etched depth at the bottom of a trench versus the distance from a sloped side wall. The etched depth is shown for dierent side-wall slopes (72, 76, 80, 84, and 88 ). The simulations were carried out with the Monte-Carlo simulation of sputtering. again. Both eects cause a positive etch rate even for larger distances from the side wall. The etch rate calculated with the MCSS is smaller within a distance of µm than the etch rate calculated with the DTSS because ions are reected also to larger distances in the MCSS while in the DTSS all ions are reected to positions within µm. For the MCSS, simulations were carried out for the same setup described above but this time the side-wall slope was varied and the inuence on the sputtering yield was studied. With a decreasing angle of inclination of the side wall, the etch rate at the bottom of the trench becomes more uniform, as shown in Figure 5.6. For small angles of inclination, the etch rate even decreases towards the side wall. Thus, microtrenching only occurs for large angles of inclination when a large increase in the etch rate can be seen. 5.2 Redeposition of Sputtered Atoms For sputter etching, redeposition limits the aspect ratio of deep structures. In this section, redeposition is studied at the geometry of a trench because an analytical model can be applied to calculate the redeposition at the side walls of the trench. With the analytical model, the methods to calculate the sputtering yield in ANETCH, the data table simulation of sputtering (DTSS) and the Monte-Carlo simulation of sputtering (MCSS), are veried. The DTSS and the MCSS were described in Section and 4.3.2, respectively.

98 84 CHAPTER 5 STUDY OF SELECTED EXAMPLES WITH ANETCH Ion Sputtered Atom θ T z 0 ϕ T x 0 y 0 Figure 5.7: Schematic representation of the surface topography to study redeposition: Innitely long trench (y 0 = ) in a silicon substrate. Gallium ions were assumed to impinge on the bottom of the trench with an energy of 30 kev and under normal incidence. A part of the sputtered atoms with polar emission angle θ T and azimuth angle ϕ T was redeposited at the side walls. z 0 = 10µm and x 0 = 1µm After the comparison of the models, the inuence of the angular distribution of sputtered atoms on the redeposition at the side walls is studied. In the last part, the inuence of the side-wall slope on redeposition is investigated. For the validation of the models, a typical substrate geometry, a perpendicular trench, as shown in Figure 5.7, has been chosen. The width of the trench is x 0 = 1µm and the depth is z 0 = 10µm while in y-direction periodic boundaries were taken to approximate an innitely long trench. Gallium ions were assumed to impinge on the bottom of the trench with normal incidence and an energy of 30 kev. Steady state was assumed to be established for the implanted ions and the silicon substrate was assumed to be already amorphous at the surface. The side wall, not exposed to the ion beam, was hit only by sputtered atoms because the probability of an ion to be reected at the bottom can be assumed to be zero for this setup. A time-invariant surface topography was assumed in the simulations thus giving a snap-shot of the etch process. The analytical calculation to determine the redeposition for the surface geometry described above is presented in the following: For normal ion incidence, the angular distribution of sputtered atoms can be described by cos α (θ T ), where θ T is the polar emission angle and α is a tting parameter as was discussed in Section The tting parameter considers the anisotropic directions of the projectiles in the substrate. For an isotropic distribution, α becomes one. Due to the rotational symmetry for normal ion incidence and a plane surface, the angular distribution of sputtered atoms is uniform regarding the azimuthal emission angle ϕ T. For sputtered silicon atoms, α is 1.24 and for sputtered gallium atoms, α is 1.18 as was determined with MC_SIM in Section The thick-

99 5.2 REDEPOSITION OF SPUTTERED ATOMS 85 Thickness of deposited film (Å) Analytical 2D-Simulation 3D-Simulation Vertical position z at side wall (µm) Figure 5.8: Thickness of the lm (silicon and gallium) deposited at the side wall of a trench versus the vertical position at the side wall. The simulations were carried out with ANETCH and the data table simulation of sputtering (DTSS). Simulations in 2D and 3D are veried by (5.2). The uence was cm 2. ness D(z) of the redeposited lm at the vertical position z at the side wall can then be calculated by ( ( ) D(z) = F Y (θ α ) P = 0) 1 1, (5.2) αnc 1 + x02 /z 2 where F is the uence (2.7), Y (θ P = 0) is the sputtering yield for normal incidence, n is the atomic density of the redeposited atoms, and C is a normalization constant. The equation for the thickness D(z) of the redeposited lm was obtained by integrating the uxes from the bottom of the trench. Sputtered atoms impinging on the side wall adhere to the side wall with a specic probability, the sticking coecient, which was assumed to be one. In the analytical calculation and in the simulations, redeposited atoms are assumed to occupy the same volume as they would in an amorphous solid of their elemental type. This allows to calculate the growth rate. Simulations carried out with ANETCH/ANETCH2D and the data table simulation of sputtering (DTSS) are veried by (5.2). In the simulation, the same angular distribution of sputtered atoms is used as in the analytical model and the sticking coecient is also one. Two simulations were carried out: One simulating the process in 2D and one simulating the process in 3D to verify that this sputtering process can be simulated in 2D as well as in 3D. The spatial distribution of the redeposited lm at the side wall is in good agreement between the simulations and the analytical formula as shown in Figure 5.8. A simulation for this setup was also carried out with ANETCH and the Monte-Carlo simulation of sputtering (MCSS). ANETCH with the MCSS was modied to have a sticking coecient of one for all sputtered atoms impinging on the side wall. Then,

100 86 CHAPTER 5 STUDY OF SELECTED EXAMPLES WITH ANETCH Thickness of deposited film (Å) Modified MCSS DTSS Vertical position z at side wall (µm) Figure 5.9: Thickness of the lm (silicon and gallium) deposited at the side wall of a trench versus the vertical position at the side wall. The Monte-Carlo simulation of sputtering (MCSS) was veried with results calculated with the data table simulation of sputtering (DTSS). To obtain the same results, the Monte-Carlo model was modied so that the sticking coecient (the probability for sputtered atoms to adhere at the surface) was one. The simulations were carried out with ANETCH and a uence of cm 2. the results are the same as for the simulation carried out with ANETCH and the DTSS as shown in Figure 5.9. However, in the real process, some sputtered atoms have large kinetic energies and by impinging on the substrate they cause sputtering. This behavior is taken into account in ANETCH with the unmodied MCSS. The absolute and relative dierence between the redeposition calculated by the two models in ANETCH is shown in Figure Below a vertical position of 1.5 µm at the side wall, less particles are redeposited when calculated with the MCSS than when calculated with the DTSS. In the MCSS, sputtered atoms with large energies cause sputtering when impinging on the side wall and, therefore, the side wall growth is less than when a sticking coecient of one is assumed. For a vertical position above 1.5 µm at the side wall, substrate atoms are still sputtered by energetic atoms. One would expect that the sputtering yield at such heights is even larger due to the larger angle of incidence (Section 3.4.4). However, as can be seen in Figure 5.10, the number of redeposited atoms is larger when calculated with the MCSS than with the DTSS. The reason for this are the atoms which were sputtered from the opposite side wall. Most of them have low kinetic energy and, therefore, adhere at the side wall. This changes the spatial distribution of redeposition at the side wall and depends on the aspect ratio AR (2.12). Programs like IONSHAPER [B + 07] or AMADEUS [K + 07a] consider redeposition in the same way as ANETCH with the DTSS. However, as shown above, redeposition is then overestimated at the bottom of the side walls when compared to more realistic simulations calculated by ANETCH and the MCSS.

101 5.2 REDEPOSITION OF SPUTTERED ATOMS R1 - R2 (Å) R 1 : Redepositon (MCSS) R 2 : Redepositon (DTSS) R 1 - R (R 1 - R 2 )/(R 1 + R 2 ) Vertical position z at side wall (µm) Figure 5.10: Absolute and relative dierence between the redeposition (silicon and gallium) calculated by ANETCH with the Monte-Carlo simulation of sputtering (MCSS) R 1 and with the data table simulation of sputtering (DTSS) R 2 versus the vertical position at the side wall. 0 2(R1 - R2)/(R1 + R2) The redeposition at the side wall depends on the angular distribution of the sputtered atoms at the bottom of the trench. For the example described above, silicon and gallium were sputtered from the bottom of the trench with dierent angular distributions. Both can be described by cos α (θ I ), where θ I is the polar emission angle and α is 1.24 for silicon and 1.18 for gallium. The absolute and relative dierence between redeposited silicon and redeposited gallium is shown in Figure 5.11 for dierent vertical positions at the side wall. For comparison, redeposited silicon and gallium were weighted by the inverse of their sputtering yield for normal incidence (Y Si = 2.34 and Y Ga = 1): The redeposition of silicon and gallium would then be the same if their α were the same. The simulation was calculated with ANETCH and the MCSS model to calculate the sputtering yield. Less silicon than gallium is redeposited at the side wall, as shown in Figure This is due to the dierent angular distribution of sputtered atoms (α = 1.24 for silicon and α = 1.18 for gallium). As a last example, the dependence of redeposition on the side-wall slope is studied. The simulations were carried out with ANETCH and the MCSS model for the setup described above. In the simulations, the height of the trench was 10 µm and the width of the trench at the bottom was 1 µm. To vary the side-wall slope, the width at the top of the trench was changed. Simulations were carried out for four dierent slopes, 90, 80, 70, and 60. The thickness of the redeposited lm is plotted versus the vertical position at the side wall as shown in Figure The number of atoms redeposited at the side

102 88 CHAPTER 5 STUDY OF SELECTED EXAMPLES WITH ANETCH RSi - RGa (Å) R Si : Redeposition (silicon) R Ga : Redeposition (gallium) R Si -R Ga (R Si -R Ga )/(R Si +R Ga ) Vertical position z at the side wall (µm) 0 2(RSi - RGa)/(RSi + RGa) Figure 5.11: Absolute and relative dierence between redeposited silicon R Si and redeposited gallium R Ga versus the vertical position at the side wall. The dierence results from the dierent angular distributions of the sputtered elements. R Si was weighted by the inverse of the sputtering yield of silicon (Y Si = 2.34) for normal incidence while R Ga was weighted by the inverse of the sputtering yield of gallium (Y Ga = 1) wall decreases with a decreasing slope of the side wall. The variations in the distribution arise from stochastic noise because the triangles at the side wall had dierent sizes and the variation of the number of ions per triangle is large for small triangles. In summary, redeposition is taken into account by the DTSS and the MCSS with dierent degrees of accuracy. The MCSS considers redeposition in a more realistic way. Especially in cases where redeposition has a large impact on the prole evolution, the MCSS is, therefore, preferable. During the sputtering process of deep trenches, the side walls become sloped because more atoms are redeposited at the bottom of the trench than at the top as shown in Figure Redeposition depends on the angular distribution of sputtered atoms as shown in Figure A possibility to decrease the inuence of redeposition within certain limits is to chose the ion type and energy in a way to maximize the exponent α of the angular distribution. Redeposition is one eect inuencing the sidewall slope during the sputtering process. The other eects will be studied in Section 5.3.

103 5.3 EVOLUTION OF SIDE WALLS 89 Thickness of deposited film (Å) Side-wall slope = 90 Side-wall slope = 80 Side-wall slope = 70 Side-wall slope = Vertical position z at side wall (µm) Figure 5.12: Thickness of the lm (silicon and gallium) deposited at the side wall of a trench versus the vertical position at the side wall. Simulations were carried out with ANETCH and the Monte-Carlo model for dierent side wall slopes. 5.3 Evolution of Side Walls Structures on the nanometer scale can be fabricated by sputter etching. A problem arising with sputter etching are the sloped side walls of etched structures. Due to the stochastic nature of MCFM and of MCSS, the results calculated by ANETCH include statistical variations of the etch rate. As discussed in Section 4.2.2, the variations are large at parts of the surface where the number of impinging particles per triangle is small. To study the inuence of this statistical noise on the side-wall evolution, four simulations were carried out with dierent levels of renement. The setup of these simulations is as follows: A plane silicon surface was exposed to gallium ions with normal incidence and an energy of 30 kev. The area exposed to ion bombardment was 1 µm in x-direction and innite in y-direction. The innite y-direction was described by exposing 0.1 µm in y-direction and applying periodic boundaries. One simulation with a coarse discretization was carried out until a trench with a depth of 0.47 µm (z-direction) had been etched. The prole of the trench was taken as the initial prole for four subsequent simulations. These simulations were carried out with dierent levels of renement until the trench had reached a depth of 0.5 µm. Due to the dierent levels of renement, the variation of the etch rates can be studied if the number of particles are the same in each simulation. The dierent nal proles and dierent discretizations are shown in Figure The side wall is smoother when the statistical variation of the etch rate is smaller (Discretization 4). For ne discretizations (Discretization 1), an initially smooth side wall becomes rough as indicated in Figure 5.13 because the variation in the etch rate is large. The variation of the etch rates is larger for small triangles

104 90 CHAPTER 5 STUDY OF SELECTED EXAMPLES WITH ANETCH Discretization 1 Discretization 2 Z Z X Y X Y Discretization 3 Discretization 4 Z Z X Y X Y Figure 5.13: For dierent levels of discretization, a section of the trench prole is compared. The simulations were carried out with ANETCH (MCSS and MCFM) for gallium ions impinging upon a silicon surface. The level of renement decreases from Discretization 1 to 4. Only the surface of the substrate is visible. (ne discretization) than for large triangles (coarse discretization) because the number of ions impinging on a triangle decreases. In the simulations, it was also observed that a small bump in an otherwise smooth side wall becomes large and moves to the bottom of the side wall if the discretization is small enough. In ANETCH, the level of renement depends on the angle α between two adjacent surface elements (α = 180 for a plane surface). If α is below a specic limit, a ner discretization can be applied. This allows to minimize errors in the surface prole arising from the MCFM and the MCSS. The side-wall slope depends on the spatial distribution of the ion uence. In the fabrication of structures with a FIB, the spatial distribution is given at the edges by the prole of the ion beam which is often approximated by a normal distribution or a superposition of two normal distributions as described for example by Lugstein et al. [L + 02]. Simulations were carried out with the four dierent spatial ion distributions shown in Figure For the spatial distribution, a prole linearly decreasing at its edges and a box prole were considered. Additionally, two proles were calculated by assuming a normal distribution of the ion beam for the two currents 93 pa and 460 pa. As mentioned in Section 2.1.3, the normal distribution can be described by the Full Width at Half Maximum (FWHM). For the two currents 93 pa and 460 pa, a FWHM of 32 nm and 105 nm was assumed. The cross section of the spatial distribution (Figure 5.14) is

105 5.3 EVOLUTION OF SIDE WALLS 91 Fluence (10 18 cm 2 ) Distribution 1 Distribution 2 Distribution 3 Distribution Distance from the center (µm) Figure 5.14: Dierent spatial distributions of the uence at one of their edges: A linear prole (1), a box prole (2), a FIB prole for a current of 93 pa (3), and a FIB prole for a current of 460 pa (4). Depth (µm) Fluence profile 1 Fluence profile 2 Fluence profile 3 Fluence profile Distance from the middle of the trench (µm) Figure 5.15: Etched proles resulting from dierent spatial distributions of the ion uence (Figure 5.14) of gallium ions impinging on a plane silicon surface with an energy of 30 kev and normal incidence. innitely extended in y-direction (perpendicular to the cross section). In the simulations carried out with ANETCH, gallium ions are assumed to impinge on a silicon surface with an energy of 30 kev. Reected ions and redeposited atoms were neglected to focus on the inuence of the uence prole on the side-wall slope. The calculated cross sections of the proles etched with the dierent distributions of the uence are shown in Figure The dierent spatial distributions of the ion uence lead to dierent side-wall proles.

106 92 CHAPTER 5 STUDY OF SELECTED EXAMPLES WITH ANETCH Silicon Molybdenum Yield (1) Angle of incidence θ I ( ) Figure 5.16: Dependence of the sputtering yield on the angle of incidence. Gallium ions impinge on a plane substrate surface with 30 kev and dierent angles of incidence. In one simulation the substrate was silicon and in the other molybdenum. Due to this behavior, the side-wall slope can be controlled within limits by the current of the ion beam which inuences the spatial distribution of the ion uence as indicated by the uence prole 3 and 4. In the following, the etching of a trench is simulated. In the simulations, an area of an initially planar surface was exposed to gallium ions with an energy of 30 kev and under normal incidence. The number of atoms redeposited at the side wall depends on the sputtering yield for normal incidence while the number of atoms sputtered from the side wall depends on the sputtering yield for grazing incidence. To study the inuence of the ratio of sputtering yield for normal incidence to sputtering yield for oblique incidence on the side-wall slope, the trench is etched in one case into silicon and in the other into molybdenum. These materials were chosen because their fraction of the sputtering yield for normal incidence to the sputtering yield for oblique incidence is dierent as indicated in Figure The uences in both simulations were chosen so that the same depth was etched and the uence prole is the one for 420 pa (Figure 5.14). As can be seen in Figure 5.17, two dierent side-wall slopes have evolved. By comparing the etched prole in Figure 5.15 (Fluence prole 4) with the one in Figure 5.17 (Silicon) the inuence of redeposition can be seen. For the simulation results shown in Figure 5.15, redeposition was neglected while for the simulation results shown in Figure 5.17 redeposition was considered. The evolution of a side wall during a sputtering process is a complex process and depends on dierent parameters. At the beginning of the sputtering process where the aspect ratio is small, the main inuence on the angle of inclination of the side walls is determined by the spatial distribution of the ion uence. In this regime, the inuence of redeposition and the dependence of the sputtering yield on the angle of incidence can be neglected. For larger aspect ratios, the inuence of the spatial distribution of the uence

107 5.4 PROPAGATION OF STEEP STEPS IN THE SURFACE 93 Depth (µm) Silicon Molybdenum Distance from the middle of the trench (µm) Figure 5.17: Cross section of a part of a prole etched in silicon/molybdenum by exposing a plane surface to gallium ions with 30 kev and normal incidence. The uence in each simulation was chosen so that the same depth was etched and the uence prole is the one for 93 pa. on the side-wall slope decreases. The evolution of the side wall is then mainly inuenced by the dierence between etch rate and growth rate. This dierence is inuenced by the ratio of the sputtering yield at the bottom of the structure and the sputtering yield at the side wall. In ANETCH with the MCSS model, the above mentioned inuences on the side-wall slope are considered implicitly. Statistical variations of the etch rate at the side wall which arise from the Monte-Carlo method are reduced by applying a coarse discretization for plane surfaces and a ner discretization for curved surfaces. 5.4 Propagation of Steep Steps in the Surface A steep step in an otherwise plane surface propagates laterally when the surface is exposed to a uniform ion ux Φ (2.8). This behavior is used in micro-machining to sharpen the tips of eld emitter arrays [N + 04]. An understanding of this behavior can also help to describe the etching of the side wall of a mask when exposed to a uniform ion uence. In the following, the propagation of a steep step is studied in 2D, as indicated in Figure For an ion ux Φ, the etch rate ER in x-direction at an arbitrary point at the sloped side (except for the two edges) is given by ER x = Y (θ 2)Φ cos(θ 2 ), (5.3) n T sin(θ 2 ) where Y (θ 2 ) is the sputtering yield of the target material, Φ cos(θ 2 ) is the number of ions impinging on the side wall per area and time, θ 2 is the slope of the side wall, and n T is

108 94 CHAPTER 5 STUDY OF SELECTED EXAMPLES WITH ANETCH 2 θ2 ER 2 3 2x Ions 3 Surface ER ER 1 ER 2 Figure 5.18: Schematic cross section of a steep side wall represented by the three line segments 1, 2, and 3. Due to uniform ion bombardment, the surface is etched with the etch rates ER 1, ER 2, and ER 3 = ER 1. The shifted surface is given by the dashed lines. At the side wall, the angle of incidence is given by θ 2. the atomic density of the target material. The term sin(θ 2 ) describes the projection of the etch rate onto the x-direction. However, more important than ER x at an arbitrary point at the step is the etch rate at the two edges (top and bottom). This is studied in the following. The surface is assumed to be decomposed into small line segments and the slope of each line segment is assumed to be constant during the evolution process while the length of each line segment can change. For this condition, Townsend et al. [T + 76] have proposed a method to calculate ER x12 at an intersection point O of two line segments 1 and 2, as indicated in Figure The line segments 1 and 2 are exposed to ions with angles of incidence of θ 1 and θ 2, respectively. For a uniform ion ux Φ, as indicated in Figure 5.19, the etch rate in x-direction at the intersection point O is z x ER x12 (θ 1, θ 2 ) = Φ (Y (θ 2) Y (θ 1 )) n T (tan(θ 2 ) tan(θ 1 )), (5.4) where Y (θ 1 ) and Y (θ 2 ) are the sputtering yields at line segment 1 and 2, respectively. The dierence between (5.3) and (5.4) is shown in Figure Originally, Townsend et al. [T + 76] only considered the evolution of two adjacent line segments with dierent angles of inclination, as shown in Figure They distinguished three cases for θ 1 > θ 2 : First, the sputtering yield at the two segments are the same Y (θ 1 ) = Y (θ 2 ) so that the etch rate in x direction is zero. Second, for Y (θ 1 ) > Y (θ 2 ), the etch rate in x direction is positive and third, for Y (θ 1 ) < Y (θ 2 ), the etch rate in x-direction is negative. From this consideration, they assumed a side wall should evolve to a slope where the sputtering yield has a maximum. To get a more realistic description of the evolution process, more than two segments must be considered and, therefore, the Townsend model is in the following extended to more realistic geometries.

109 5.4 PROPAGATION OF STEEP STEPS IN THE SURFACE 95 Ion Beam 2 Surface O θ1 ER1 2 ER θ 2 2 ER x12 O 1 z 1 Figure 5.19: Schematic cross section of a surface represented by two lines 1 and 2. After a specic ion uence, the surface has shifted and is described by the lines 1' and 2'. The intersection of the planes are laterally (x-direction) shifted by d 1, d 2 and d 3. x ER x ER x12 Figure 5.20: Dierence between E x (5.3) and E x12 (5.4). The Townsend approach is applied to the original geometry of the steep step shown in Figure The cross section of the surface is then represented by three lines and two intersections. With (5.4), the etch rate at the two intersection points can be calculated with θ 1 = 0 and θ 3 = 0. The etch rate ER x calculated by ER x (θ 2, 0 ) or ER x (0, θ 2 ) will be dened as the propagation velocity V P (θ 2 ) of the steep step. Then, the etch rates (ER x12 and ER x23 ) of each intersection are given by ER x12 =ER x23 = V P (θ 2 ). Therefore,

110 96 CHAPTER 5 STUDY OF SELECTED EXAMPLES WITH ANETCH ER x12 ER x34 ER x z Ions 4 Surface Figure 5.21: Schematic cross section of four planes 1, 2, 3 and 4 representing the surface of a substrate. After a specic ion uence was applied the surface has shifted and is represented by the planes 1', 2', 3' and 4'. x Initial Surface 1. Variation 2. Variation θ 3 θ 4 θ 4 + θ 4 θ 4 3 θ 2 + θ θ 2 θ 2 θ Figure 5.22: Two possible variations at the top of an initial step. the slope of the step does not change. It is worth to notice the symmetry at the bottom and top. In a realistic process, small changes in the slope occur at the top or the bottom of a step. First, only changes at the top are considered and, afterwards, changes at the bottom are studied. A geometry is considered with a dierent slope at the top of the side wall as shown in Figure This geometry has three intersections and, therefore, three etch rates (ER x12 = V P (θ 2 ), ER x23 and ER x34 = V P (θ 3 )) must be calculated. The dierences (V P (θ 2 ) ER x23 ) and (ER x23 V P (θ 3 )) determine if the perturbation in the side-wall slope given by line 3 decreases or increases to nally determine the side-wall slope. For example, if (V P (θ 2 ) ER x23 ) > 0, line 2 vanishes during the surface evolution while the slope of the step is then given by the slope of line 3. At the top of a step, two small variations from the initial geometry can occur as shown in Figure For the rst variation shown in Figure 5.22, the Townsend formula

111 5.4 PROPAGATION OF STEEP STEPS IN THE SURFACE 97 1 Variation 2. Variation 3. Variation 3 θ 4 4 θ 2 θ 2 θ θ 3 θ θ 2 θ 2 θ 2 + θ θ 1 + θ Figure 5.23: Three possible variations at the bottom of the initial step shown in Figure becomes ER x23 (θ 2, θ 2 + θ 2 ) = Φ n T Y θ cos 2 (θ 2 ). (5.5) θ2 This will be dened as the advection velocity V A (θ 2 ). If the advection velocity V A (θ 2 ) is larger than the propagation velocity V P (θ 2 ), small changes of the slope θ 2 at the top of the steep step do not change the slope θ 2 of the step. Therefore, as long as (V A (θ 2 ) > V P (θ 2 )) and neglecting small changes of the slope at the bottom of the step, the slope of the side wall is constant during the propagation process. In the following, the second variation with a dierent slope of line segment 3 is assumed given by θ 3 = θ 4. As long as V A ( θ 4 ) < ER x23, the slope of the side wall is constant during the propagation process. The etch rate ER x23 is almost V P (θ 2 ). In general, the slope of a steep step θ SW is insensitive to small changes at the top of the step as long as V A (0 ) < V P (θ SW ) and V A (θ SW ) > V P (θ SW ). However, if the two conditions are fullled, the slope of the step can change due to small changes of the slope at the bottom of the step. At the bottom of the step, the process is more complex due to redeposition and the reection of ions at the step. Similar to the top of the step (Figure 5.22) small changes can occur at the lower part of the step. The rst variation, shown in Figure 5.23, typically does not persist because ions reected at the step increase the etch rate at the bottom where the small variation has occurred. The second variation, shown in Figure 5.23, does also typically not persist because the number of atoms redeposited at the small variation is larger than for the original step. Only the third variation snails foot, shown in Figure 5.23, was observed to evolve during the sputtering process. It seems that the inuence of the increase of the etch rate due to ions reected at the step is less than for the rst variation discussed. The reason of the snails foot might be atoms which were sputtered from the side wall and are redeposited at the bottom. The number of redeposited atoms at the bottom is largest close to the step and decreases with the distance from the step. With the Townsend model and the extension presented above, it is possible to describe some of the characteristics of the propagation of a steep step under uniform ion bombardment. In the following, the condition (V A (θ SW ) > V P (θ SW )) is called the Lax condition to show the relation of the evolution of a steep step with shock dynamics. For the example of gallium ions impinging on a silicon target with an energy of 30 kev, the advection velocity and the propagation velocity are calculated for dierent angles of incidence and

112 98 CHAPTER 5 STUDY OF SELECTED EXAMPLES WITH ANETCH Velocity (arbitrary units) advection velocity Gallium ions bombarding silicon Lax condition fulfilled propagation velocity Angle of incidence θ P ( ) 66.5 Lax condition not fulfilled Figure 5.24: Propagation velocity and advection velocity versus the angle of incidence. The Lax condition is fullled if the advection velocity is above the propagation velocity. The critical angle θ Cr = 66.5 where propagation and advection velocity have the same value is indicated. the results are shown in Figure The range of the angles of incidence where the advection velocity is larger than the propagation velocity is shown as well as the critical angle θ Cr where both velocities have the same value. For steep side walls where the Lax condition is violated, the slope of a step decreases until the condition is fullled. Initial slopes below θ Cr = 66.5 are constant during the evolution process if the snails foot can be neglected. The results described above are now shown for the example of the evolution of cuboids of dierent materials under ion bombarded. The evolution was simulated with ANETCH. For the calculation of the ux the MCFM was used and for the calculation of the sputtering yield the MCSS was taken. For the example of molybdenum as target material, the propagation shown in Figure 5.25 is discussed. Initially, the block had the geometrical shape of a cuboid and partially masked a silicon substrate. The top of the molybdenum mask and the unmasked silicon were exposed to gallium ions with normal incidence and an energy of 30 kev. The uence was uniform over the mask and the substrate. The growth at the two sides between the rst and second prole, as shown in Figure 5.25, is due to the redeposition of sputtered silicon atoms. After a specic uence, the top corners of the mask became rounded as indicated in Figure After the top corners have been rounded, the side walls became sloped starting from the round corners, as

113 5.4 PROPAGATION OF STEEP STEPS IN THE SURFACE Z (µm) Molybdenum -0.2 Silicon X (µm) Figure 5.25: Cross section of the etching of a molybdenum block sitting on a silicon substrate. Cross sections for dierent ion doses are shown. shown in Figure 5.25, until the whole side wall had the same slope. Due to the larger etch rate at the top of the slope, the side-wall slope seems to evolve until the advection velocity is the same as the propagation velocity (at this point the Lax condition is fullled). The side-wall slope in the silicon substrate is dierent from the slope of the molybdenum mask because both materials have dierent propagation and advection velocities. At the bottom, the snails foot can be seen. The reason is variations at the bottom of the step. As stated above, as long as the Lax condition is fullled, the advection velocity is larger than the propagation velocity. At the bottom this leads to the snails foot. Redeposition and reection of ions have an impact on the evolution of the snails foot. Similar simulations were carried out for dierent target materials. The slopes given by the angle of inclination θ SW are compared in Table 5.1 with the angle θ Cr where the propagation and advection velocity have the same value. For the materials carbon, silicon, titanium, and molybdenum, fairly good agreement between the side-wall slope and θ Cr was found. In experiments, steeper side walls were observed. This can not be explained by the Townsend model with the presented extensions or reproduced by ANETCH. The reason for the steeper side walls is assumed to be surface diusion. Chen et al. [C + 05, Che05] presented a model with surface diusion and found a constant slope which is larger than θ Cr. Even if the Lax condition is violated, constant side-wall slopes can exist, as was found by Chen. For this solution found by Chen, the surface diusion term in the Chen model is

114 100 CHAPTER 5 STUDY OF SELECTED EXAMPLES WITH ANETCH Table 5.1: Comparison of the steady side wall slope θ SW of a mask block under ion bombardment with the critical angle θ Cr at which propagation and advection velocity have the same value. Material θ SW θ Cr Carbon Silicon Titanium Molybdenum assumed to be important. However, surface diusion is not considered in ANETCH and, therefore, only constant side-wall slopes can be found if the Lax condition is fullled.

115 Chapter 6 Comparison with Experimental Results As a nal test for the models available in ANETCH, specic test examples were manufactured with the focused ion beam (FIB) systems, a FEI Helios Nanolab and a FEI 800. In both FIB systems, gallium ions can be focused onto a substrate with an energy of 30 kev. As target material, silicon was chosen due to its importance in the microelectronic industry. To simulate the fabrication of silicon in a FEI Helios Nanolab, the electronic stopping model (Section 3.2.3) implemented in MC_SIM was calibrated. This calibration will be described in Section 6.1. For the calibration, the sputtering yield of silicon when processed with the gallium beam had to be determined in an experiment rst. After the calibration, sputtering yields calculated by MC_SIM are compared with sputtering yields reported by Wittmaack and Poker [WP90] and Zalm [Zal83] for dierent ion types and kinetic energies to estimate how good the calibration was. As a rst test example, the topographies of two trenches in silicon after FIB preparation will be compared to simulations in Section 6.2. A trench was chosen because the accuracy of reection and redeposition in the simulation can be tested for this geometry. In a second test example, simulations were carried out to optimize a process parameter, the uence, for sputtering experiments. In these experiments, to determine the dependence of the sputtering yield on the angle of incidence θ I, trenches were etched in silicon by gallium ion bombardment under dierent θ I. The topographies of the fabricated trenches will be compared to the ones predicted by the simulations. 6.1 Calibration of the Electronic Stopping Model In this section, the calibration of the electronic stopping model presented in Section and implemented in MC_SIM is described for the sputtering yield of silicon when exposed to gallium ions with an energy of 30 kev and under normal incidence. To calibrate the model, the sputtering yield must be determined by experiments for this setup. 101

116 102 CHAPTER 6 COMPARISON WITH EXPERIMENTAL RESULTS A typical method to obtain the sputtering yield from experiments is to sputter a trench into silicon with, for example, a focused ion beam and to calculate then the yield from the removed volume. The sputtering yield depends on the atomic fraction of implanted gallium and, therefore, changes until a constant atomic fraction of implanted gallium has been reached. To determine the constant sputtering yield, a steady state of implanted gallium must be reached rst. Therefore, a large region of the silicon substrate was exposed with a FIB to a gallium uence of cm 2 (Section 3.3) before the fabrication process of the trench. Then the trench is etched and the sputtering yield is calculated as the number of atoms in the removed volume of the trench divided by the number of ions impinging on the surface. For this calculation, the sum of reected gallium ions and sputtered gallium ions divided by the number of ions impinging on the surface is one because, as assumed, steady state has been established for the implanted gallium ions. With these considerations the yield is calculated by Y (θ I ) = Vol n T + 1, (6.1) F cos(θ I )A where F is the uence, n T is the atomic density of silicon, Vol is the sputtered volume, and A is the surface area which was exposed to the ions. The rst term on the right hand side of (6.1) is the partial sputtering yield of silicon and the second term is the partial sputtering yield of gallium. A problem arising with this method is to determine the volume. The side walls of the trench are sloped and, therefore, were hit by ions under oblique angles of incidence. Furthermore, ions are reected at the side walls and increase the etch rate at the bottom of the trench and close to the side walls. Both eects make it dicult to determine the volume which was etched with the uence F and under normal incidence. This inaccuracy can be reduced by calculating the sputtering yield from the depth d in the middle of the etched trench by Y (θ I ) = d n T + 1. (6.2) F cos(θ I ) In an experiment which will be described in Section 6.2, a trench was etched with a uence of F = cm 2 resulting in a depth of z = 0.94 µm. With n Si = Å 3 and θ I = 0, the sputtering yield is Y (0 ) = With this result, the partial sputtering yields of gallium and silicon are Y Ga (0 ) = 1 and Y Si (0 ) = 2.34, respectively. Having determined the sputtering yield, the electronic stopping model in MC_SIM was adjusted. In MC_SIM, the electronic stopping is calculated by the Oen-Robinson model for projectile energies below E lim,1 and by the Ziegler model for projectile energies above E lim,2. Between E lim,1 and E lim,2, an interpolation between the Oen-Robinson model and the Ziegler model is used. This was described in more detail in Section Good agreement between the sputtering yield simulated with MC_SIM and the one obtained from experiments was found for E lim,1 = 700 ev and E lim,2 = 850 ev. The electronic stopping model with these energies, E lim,1 and E lim,2, is denoted by Se. I The calibration of the electronic stopping model was published in [K + 10]. In the following, sputtering yields calculated with MC_SIM and Se I are compared with some sputtering yields reported by Wittmaack and Poker [WP90] and Zalm [Zal83]

117 6.1 CALIBRATION OF THE ELECTRONIC STOPPING MODEL 103 Sputtering Yield Exp S I e S LSOR e S ZOR e S Z e Ion Energy (kev) Figure 6.1: Sputtering yields for neon ions impinging onto a silicon target at normal incidence with dierent ion energies. The experimental results (Exp) were reported by Wittmaack and Poker [WP90]. The simulations were carried out with MC_SIM and dierent electronic stopping models Se, I Se Z, Se ZOR, and Se LSOR. for dierent ion types and ion energies. The target material was always silicon. The uncertainties of the sputtering yields are 10 % in the experiments reported by Wittmaack and Poker [WP90] and in the experiments reported by Zalm [Zal83], the uncertainties are 5 % for energies of 2.5 kev and below and 10 % otherwise. For comparison with other electronic stopping models, the sputtering yields were calculated with MC_SIM and one of the following models: Se Z, Se ZOR = 0.5Se Z + 0.5Se OR, or Se LSOR = 0.5Se LS + 0.5Se OR. The electronic stopping models were described in Section As a rst example, neon is considered and the respective sputtering yields are shown in Figure 6.1 for dierent ion energies (2.5, 5, 8, 10, and 20 kev). The best agreements between the experiment of Wittmaack and Poker [WP90] and simulations were obtained with Se LSOR and Se ZOR. The sputtering yield calculated with Se I is too large compared with experiments. The energy limits of Se I are not sucient to describe the electronic stopping of neon ions in silicon in a realistic way. The sputtering yields calculated with Se Z are signicantly smaller than the experimental results. Next, sputtering yields are compared for argon ions impinging on a silicon target with dierent ion energies (2, 4, 7, 10, 20 kev). Good agreement was found between the experiments of Wittmaack and Poker [WP90] and the values calculated with Se I as shown in Figure 6.2. The sputtering yield with Se LSOR is also in good agreement with the experiments. The sputtering yields calculated with Se Z and Se ZOR are both signicantly smaller than the experimental results. The sputtering yield of krypton was calculated for dierent ion energies (1, 2.5, 10, 20 kev). Again, good agreement between the experiments of Zalm [Zal83] and the simulation with Se I or Se LSOR was found as shown in Figure 6.3. The sputtering yields calculated

118 104 CHAPTER 6 COMPARISON WITH EXPERIMENTAL RESULTS Sputtering Yield Exp S I e S LSOR e S ZOR e S Z e Ion Energy (kev) Figure 6.2: Sputtering yields for argon ions bombarding a silicon target at normal incidence with dierent ion energies. The experimental results (Exp) were reported by Wittmaack and Poker [WP90]. The simulations were carried out with MC_SIM and dierent electronic stopping models Se, I Se Z, Se ZOR, and Se LSOR. Sputtering Yield Exp S I e S LSOR e S ZOR e S Z e Ion Energy (kev) Figure 6.3: Sputtering yields for krypton ions bombarding a silicon target at normal incidence with dierent ion energies. The experimental results (Exp) were reported by Zalm [Zal83]. The simulations were carried out with MC_SIM and dierent electronic stopping models Se, I Se Z, Se ZOR, and Se LSOR.

119 6.1 CALIBRATION OF THE ELECTRONIC STOPPING MODEL 105 Sputtering Yield Exp S I e S LSOR e S ZOR e S Z e Ion Energy (kev) Figure 6.4: Sputtering yields for xenon ions bombarding a silicon target at normal incidence with dierent ion energies. The experimental results (Exp) were reported by Wittmaack and Poker [WP90]. The simulations were carried out with MC_SIM and dierent electronic stopping models Se, I Se Z, Se ZOR, and Se LSOR. with Se Z and Se ZOR are both signicantly smaller than the experimental results. Finally, the sputtering yield of xenon was calculated for dierent ion energies (2, 3, 4, 5, 8, 10, 20 kev). The sputtering yields of the experiments of Wittmaack and Poker [WP90] and the calculated ones are shown in Figure 6.4 for dierent ion energies. There is good agreement between experiment and the calculation with Se I and Se LSOR. Again, the sputtering yields calculated with Se Z and Se ZOR are signicantly smaller. The sputtering yields calculated with Se I are in reasonable agreement with experiments using dierent ion types and ion energies, except for neon. With the Ziegler electronic stopping, realistic implantation proles can be calculated as reported by Ziegler [Zie04]. The method of considering local electronic stopping (Oen-Robinson) for low projectile energies (<700 ev) and continuous electronic stopping (Ziegler) for large projectile energies (>850 ev) results in realistic sputtering yields as shown for the four noble gases above and realistic implantation proles as for example the one shown in Figure 3.5. For better results, the energy limits must be adjusted for each ion/target combination. A mixture of the Lindhard-Schar and the Oen-Robinson electronic stopping model also results in sputtering yields which are in good agreement with experiments. For high projectile energies, the electronic stopping calculated with the Lindhard-Schar and Oen-Robinson models is always proportional to the velocity v of the projectile. However, as reported by Eckstein [Eck91], for light ions Z 1 < 19 and semiconductor targets, a better agreement was found for S e v 0.75 due to the band gap. This is considered in the Ziegler electronic stopping model.

120 106 CHAPTER 6 COMPARISON WITH EXPERIMENTAL RESULTS 6.2 Trench Etching with Focused Ion Beams For the comparison of ANETCH with experimental results, two trenches were fabricated with focused ion beam (FIB) milling into a silicon substrate to estimate the accuracy of ion reection and redeposition calculated by ANETCH. Additionally, the dependence of the topogography on the ion beam current is investigated and, therefore, the one trench was fabricated with an ion current of 96 pa and the other with an ion current of 460 pa. The trenches were fabricated in a FEI Helios Nanolab where gallium ions were focused onto a silicon substrate with an energy of 30 kev and normal incidence. In focused ion beam milling, the beam must be deected to dierent spots on the surface to expose a specic surface area to ions. To assure a uniform etch rate, the patterning strategy described in Section was taken in the experiments. As already mentioned, before the fabrication of the trenches, a large area was exposed to ions with a uence of cm 2 to ensure a constant implantation prole during the fabrication process. Afterwards, the two trenches were etched in the prepared region. The width of each trench is approximately 1 µm and the length is 6 µm to reduce 3D eects. The only dierence between the two trenches was the current of the ion beam. The dierent current causes a dierent beam prole and, therefore, a dierent spatial distribution of the uence as shown in Figure To obtain these spatial distributions, the beam prole was approximated by a normal distribution where the full width at half maximum (FWHM) to describe the normal distribution was provided by the FEI company. To simulate the fabrication process of the trenches with ANETCH, the calibration of the electronic stopping model described in Section 6.1 was taken. For a current of 96 pa, the etched surface prole, especially the side-wall slope, agrees quite well between experiment and simulation, as shown in Figure 6.5. The simulation reproduces the typical w-shape of the experiment, called microtrenching. However, microtrenching is a little bit underestimated by the simulation and the prole at the top of the side wall is less curved in the simulation. A reason for less pronounced microtrenching is a smoother side wall in the experiment than in the simulation. A rougher surface causes ions to be reected into a larger area and thus the local increase of the etch rate at the trench edges at the bottom is smaller. The surface roughness depends on the discretization and statistical noise at the side wall, as was discussed in Section 5.3. Another uncertainty of the simulation arises from the spatial distribution of the uence. To approximate the ion beam by a normal distribution leads to the discrepancy between experiment and simulation for the topography at the top of the side walls. Lugstein et al. [L + 02] suggested to approximate the ion beam by two normal distributions to take the tails of the beam prole into account. As discussed in Section 5.3, the decrease of the uence at its edges has an inuence on the side-wall slope of the trench. Due to the good agreement of the side-wall slope between simulation and experiment, the value of 32 nm for the FWHM at a current of 96 pa seems realistic. With an increase of the current to a value of 460 pa, the ion beam becomes wider and a value of 105 nm for the FWHM was assumed. For this FWHM, the comparison between simulation and experiment can be seen in Figure 6.6. To get agreement between

121 6.2 TRENCH ETCHING WITH FOCUSED ION BEAMS 107 Figure 6.5: Comparison between simulation (white line) and experiment for etching a trench with a focused ion beam of 30 kev gallium into silicon. The cross section of the trench is shown and the dark grey region is silicon. Before cross sectioning, the structure was lled with Pt (light grey) for protection. The uence was cm 2 and the ion current 96 pa. simulation and experiment for the depth of the trench, a uence of cm 2 was assumed. In the experiment, a uence of cm 2 was intended. However, in the experiments only the ion current is controlled and due to a wider ion beam prole the current density in the center of the beam decreases slightly. Therefore, the uence in the center of the trench might be slightly less than expected. The side-wall slope is a little bit larger in the simulation than in the experiment and the top of the side wall is less curved in the simulation. However, most striking in the experiment is the rounding at the two bottom edges of the trench. This rounding is not reproduced in the simulation and gives an indication that the approximation of the uence is inaccurate. A larger value for the FWHM than 105 nm would lead to a lesser decrease of the uence which is necessary for the rounding. This should also cause a less sloped side wall. An increase of the ion beam current has lead to a dierent topography of the etched trench. The width at the top of the trench has increased from 1.06 µm to 1.13 µm and the slope of the side wall has decreased from 84.3 to Most interesting, however, is that no microtrenching was observed for a current of 460 pa.

122 108 CHAPTER 6 COMPARISON WITH EXPERIMENTAL RESULTS Figure 6.6: Comparison between simulation (white line) and experiment for etching a trench with a focused ion beam of 30 kev gallium into silicon. The cross section of the trench is shown and the dark grey region is silicon. Before cross sectioning, the structure was lled with Pt (light grey) for protection. The uence was cm 2 and the ion current 460 pa. 6.3 Optimization of the Fluence for a Sputtering Experiment The focus of this section is not on the investigation of specic characteristics of sputtering but to show how the program ANETCH can be of advantage to optimize process parameters for sputtering experiments. As a meaningful example, the ion uence was optimized for experiments to determine the sputtering yield for oblique angles of incidence. Parts of this sections have been published in [K + 10]. All simulations were performed with ANETCH and the MCSS model (Section 4.3.2) to calculate the sputtering yield. For the electronic stopping model, (3.35) was applied with the parameterization explained in Section 6.1. Therefore, no further parameterization was necessary for the simulations carried out for this section. Trenches were fabricated with focused ion beam (FIB) milling into a planar silicon substrate under dierent angles of incidence θ I. This is a typical experimental setup to determine the dependence of the sputtering yield on the angle of incidence. The trenches were fabricated in a FEI 800 where gallium ions are focused to a surface with an energy

123 6.3 OPTIMIZATION OF THE FLUENCE FOR AN EXPERIMENT Figure 6.7: Comparison of a cross section determined experimentally by Beuer [B 07] (silicon = dark grey) and by simulation (white lines). The angle of incidence was 60. For the simulation di erent ion uences (white lines) are shown to visualize that there is a part of the bottom of the trench that is always etched under the same angle of incidence. Before cross sectioning, the structure was lled with Pt (light grey) for protection. of 30 kev. To establish steady state for the implantation pro le, a large area had been 17 2 exposed with the FEI 800 to gallium ions with a uence of 10 cm and under the same angle under which the trench was afterwards fabricated in this region. The sputtering yield can be calculated from the removed volume (6.1). However, as shown in Figure 6.7, an asymmetric trench forms during sputtering so that the angle of incidence changes along the cross section. To determine the yield for one speci c angle of incidence, it is advantageous to calculate the yield from the depth d of the trench (6.2). The surface at which the depth is measured must have been etched by the ions under the same angle of incidence during the whole sputtering process. The in uence of re ected ions and redeposited atoms on the depth of the trench cannot be estimated from the experiment. However, with ANETCH, the in uence of re ected ions and redeposition on the surface topography can be estimated by comparing a full simulation with a simulation that neglects re ected ions and redeposition. By comparing a full simulation with one that neglects re ection and redeposition, an upper limit for the uence Fmax was estimated until which re ection and redeposition can be neglected for the depth of the trench. The maximum uence Fmax depends on the angle of incidence θi and the beam height h which