NON-EQUILIBRIUM MOLECULAR DYNAMICS OF ELECTROMIGRATION IN ALUMINUM AND ITS ALLOYS

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NON-EQUILIBRIUM MOLECULAR DYNAMICS OF ELECTROMIGRATION IN ALUMINUM AND ITS ALLOYS FAT IH GÜRÇA ¼G ŞEN SEPTEMBER 2006 NON-EQUILIBRIUM MOLECULAR DYNAMICS OF ELECTROMIGRATION IN ALUMINUM AND ITS ALLOYS A
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NON-EQUILIBRIUM MOLECULAR DYNAMICS OF ELECTROMIGRATION IN ALUMINUM AND ITS ALLOYS FAT IH GÜRÇA ¼G ŞEN SEPTEMBER 2006 NON-EQUILIBRIUM MOLECULAR DYNAMICS OF ELECTROMIGRATION IN ALUMINUM AND ITS ALLOYS A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES OF MIDDLE EAST TECHNICAL UNIVERSITY BY FAT IH GÜRÇA ¼G ŞEN IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN METALLURGICAL AND MATERIALS ENGINEERING SEPTEMBER 2006 Approval of the Graduate School of Natural and Applied Sciences Prof. Dr. Canan ÖZGEN Director I certify that this thesis satis es all the requirements as a thesis for the degree of Master of Science. Prof. Dr. Tayfur ÖZTÜRK Head of Department This is to certify that we have read this thesis and that in our opinion it is fully adequate, in scope and quality, as a thesis for the degree of Master of Science. Assoc. Prof. Dr. M. Kadri AYDINOL Supervisor Examining Committee Members Prof. Dr. Şakir BOR (METU, METE) Assoc. Prof. Dr. M. Kadri AYDINOL (METU, METE) Prof. Dr. Ishak KARAKAYA (METU, METE) Assoc. Prof. Dr. Hakan GÜR (METU, METE) Dr. Kaan PEHL IVANO ¼GLU (TÜB ITAK, SAGE) I hearby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required, I have fully cited and referenced all material and results that are not original to this work. Name Lastname : FAT IH GÜRÇA ¼G ŞEN Signature : iii Abstract NON-EQUILIBRIUM MOLECULAR DYNAMICS OF ELECTROMIGRATION IN ALUMINUM AND ITS ALLOYS ŞEN, FAT IH GÜRÇA ¼G M.Sc., Department of Metallurgical & Materials Engineering Supervisor: Assoc. Prof. Dr. Mehmet Kadri AYDINOL September 2006, 99 pages With constant miniaturization of integrated circuits, the current densities experienced in interconnects in electronic circuits has been multiplied. Aluminum, which is widely used as an interconnect material, has fast di usion kinetics under low temperatures. Unfortunately, the combination of high current density and fast di usion at low temperatures causes the circuit to fail by electromigration (EM), which is the mass transport of atoms due to the momentum transfer between conducting electrons and di using atoms. In the present study, the e ect of alloying elements in aluminum on the di usion behaviour is investigated using a non equilibrium molecular dynamics method (NEMD) under the e ect of electromigration wind force. The electromigration force was computed by the use of a pseudopotential method in which the force depends on the imperfections on the lattice at% of various elements, namely Cu, Mg, Mn, Sn and Ti were added into aluminum. The electromigration force was then calculated on the alloying elements and the surrounding aluminum atoms and these forces incorporated into molecular dynamics using the non-equilibrium formalism. The jump frequencies of aluminum in these systems were then computed. Cu, Mn and Sn impurities were found to be very e ective in lowering the kinetics of the iv di usion under electromigration conditions. Cu was known experimentally to have such an e ect on aluminum for several years, but the Mn and Sn elements are shown here for the rst time that they can have a similar e ect. Keywords: Electromigration, Di usion, Non-Equilibrium Molecular Dynamics, Aluminum Interconnect. v Öz ALÜM INYUM VE ALAŞIMLARINDA ELEKTROGÖÇÜN MOLEKÜLER D INAM IK S IMÜLASYONU ŞEN, FAT IH GÜRÇA ¼G Y. Lisans, Metalurji ve Malzeme Mühendisli¼gi Bölümü Tez Yöneticisi: Doç. Dr. M. Kadri AYDINOL Eylül 2006, 99 sayfa Entegre devrelerin sürekli küçülmesiyle elektronik devre ba¼glant elemanlar nda oluşan ak m yo¼gunlu¼gu katlanarak artmaktad r. Bu tür devrelerde geniş ölçüde kullan lan Alüminyum metali, düşük s cakl kta h zl yay n m devinimine sahiptir. Ayn anda hem yüksek ak m yo¼gunlu¼gu, hem de h zl yay n m devinimi, devrelerin elektrogöç nedeniyle h zl bir şekilde bozulmas na neden olmaktad r. Elektrogöç, ak m içinde ilerleyen elektronlar n momentumlar n atomlara aktarmalar yla oluşan kütle hareketidir. Bu çal şmada, alüminyuma eklenen alaş m elementlerinin yay n m davran ş na etkisi, elektrogöç kuvveti alt nda, denge d ş moleküler dinamik yöntemiyle incelenmiştir. Elektrogöç kuvveti, kuvvetin kafes içindeki hatalara ba¼gl oldu¼gu psödopotansiyel metodu ile hesaplanm şt r. Çal şmada, Cu, Mg, Mn, Sn ve Ti elementleri, %1,125 atom oran nda alüminyuma eklenmiştir. Elektrogöç kuvveti, eklenen alaş m elementleri ve bu elementleri çevreleyen alüminyum atomlar üzerinde hesaplanarak denge d ş durumu ile moleküler dinamik yöntemine dahil edilmiştir. Daha sonra alüminyum atomlar n n atlama frekans hesap edilmiştir. Elektrogöç koşullar alt nda yay n m devinimi azaltmada Cu, Mn ve Sn katk lar n n çok etkili oldu¼gu görülmüştür. Bak r n, alüminyum içinde böylesi bir etkiye sebep oldu¼gu deneysel çal şmalar sonuvi cunda uzun zamand r bilinmektedir ancak Mn ve Sn elementlerinin de buna benzer etkileri oldu¼gu ilk defa gösterilmiştir. Anahtar Kelimeler: Elektrogöç, Yay n m, Moleküler Dinamik, Alüminyum ba¼glant eleman vii To my family viii Acknowledgements I would like to express my sincere gratitude to my supervisor Kadri Ayd nol for his valuable guidance and support during the study of this thesis. I also thank to him for his endless patience to me. I deeply thank my friends, Tufan Güngören and Alper K nac for their valuable suggestions and support during the period of this study. I specially thank to my friends Evren Tan, Güher Kotan, Melih Topçuo¼glu, Bar ş Okatan, Hasan Aky ld z, Betül Akköprü, Ziya Esen, Ali Erdem Eken, Doruk Do¼gu, Caner Şimşir, Kemal Davut, Gül Çevik, Selen Gürbüz for their valuable support during the period of writing this thesis. I am grateful to my family, Azize, Şadi and Ayça Şen for their encouragement and constant support during the period of this study. The numerical calculations reported in this study were carried out at the ULAK- BIM High Performance Computing Center at the Turkish Scienti c and Technical Research Council (TUBITAK). I specially thank to the member of this facility Onur Temizsoylu for his constant help during the usage of this super computing facility. ix Table of Contents plagiarism iii Abstract iv Öz vi Acknowledgements ix Table of Contents x CHAPTER 1 Introduction Molecular Dynamics Classical Molecular Dynamics History of Molecular Dynamics Basics of Molecular Dynamics Property Calculations in Molecular Dynamics Non-Equilibrium Molecular Dynamics Potentials Pseudopotential theory Model Potentials Screening of the Pseudopotential x 3.2 Pair Potentials Derived by Pseudopotential Theory Pseudopotential based Electromigration theory Introduction Pseudopotential Based Electromigration Driving Force Methodology Results and Discussion Critical Potential Evaluation for Aluminum Electromigration Force Calculation Results Non-Equilibrium Molecular Dynamics Results Conclusion References APPENDICES A List of Computer Program xi Chapter 1 Introduction Computer simulations play a very important role in materials science today. Computational materials science acts as a bridge between experimental and theoretical approaches. Computer simulations are often used both to solve theoretical models beyond certain approximations and to provide a hint to experimentalists for further investigations. Understanding the properties of materials in terms of their structure and the microscopic interactions between them can be established by them. This serves as a complement to conventional experiments. In addition to that, simulations can be utilized not only to understand and interpret the experiments, but also to study regions which are not accessible experimentally, or which would imply very expensive experiments. Investigation of material properties and materials design by using computers is cost e ective, in which the cost of characterization, synthesis, processing and testing of materials are eliminated with the use of computer experiments. The traditional simulation methods for many-body systems can be divided into two classes of stochastic and deterministic simulations, which are largely covered by the Monte Carlo (MC) method [1] and the molecular dynamics (MD) method [2, 3, 4], respectively. In addition to that there is a whole range of hybrid techniques which combine features from both. Monte Carlo simulations probe the con guration space by trial moves of particles. Within the so-called Metropolis [1] algorithm, the energy change between the following steps is used as a trigger to accept or reject the new con guration. Paths towards lower energy are always accepted, those to higher energy are accepted with a probability governed by Boltzmann statistics. In that way, properties of the system can be calculated by averaging over all Monte Carlo moves (where one move means that every degree of freedom is probed once on average) [5]. 1 By contrast, in MD methods; Newton s equations of motion are integrated to move particles to new positions and to get new velocities at these new positions. The state of a dynamic system is given by its phase space coordinates at a certain time. In MD, a dynamic system moves and changes its phase space coordinates in time, evolving towards an internal equilibrium state of the whole system. The macroscopic observables of the system are then obtained by suitably de ned averages over the phase space trajectory, at the internal equilibrium state of the whole system [6]. Molecular dynamics simulations allow us to calculate static properties as well as dynamic properties of the system including transport properties. Transport properties in materials are important as they describe how a material relaxes back to equilibrium, following application of a mechanical or thermal perturbation [7]. Introducing such perturbations into the equations of motion, produce a time-dependent non-equilibrium distribution. At this point non-equilibrium molecular dynamics (NEMD) method arises, which is rstly announced by Hoover and Ashurst [8]. NEMD methods are used to calculate transport properties such as viscosity, self- and mutual di usion coe cients, and thermal conductivity more precisely than equilibrium methods [9]. Electromigration is the mass transport of a metal due to the momentum transfer between conducting electrons and di using metal atoms. Electromigration has been known for 100 years, but it is concerned with the development of the integrated circuits. The thin lms used as interconnects in integrated circuits have a thickness in the order of 300 nm. Although a relatively small voltage ( V) passes from these interconnects, because of the small length over the applied voltage, very high electric eld was developed and consequently because of the small cross-section, very high current density (10 6 A/cm 2 ) resulted due to Ohm s Law. Furthermore, the conductors were made of pure aluminum, a material with a low melting temperature, which implies fast di usion at low temperatures. Such very thin lm contains small grains and thus many grain boundaries that are suitable for rapid di usion. This combination of high current density and fast di usion at low temperatures causes the circuit to fail due to vacancy di usion induced void formation and growth especially at grain boundaries. Intensive research has been carried out to overcome this failure, for example, by adding few percentages of alloying elements like copper [10]. The electromigration was investigated theoretically in atomistic and phenomenological point of view in the literature. In the last decade several computer simulations of electromigration in metal lines have been reported from the macroscopic point of 2 view [11, 12, 13]. However, as circuit integration shrinks to smaller dimensions, in- uences of crystal orientation and grain boundary structure on the lifetime must be taken into account. From the physics point of view, Sorbello [14, 15, 16] has made a very comprehensive study about determining the driving force for electromigration for each individual atom in a lattice. Sorbello has studied the atomic con guration-dependent electromigration force including impurities and vacancies. There are very limited studies in the literature involving atomistic simulations. Molecular dynamics study of electromigration has been carried out by Ohkubo et al [17]. In that study H type periodic boundaries for interconnects has been constructed and 2 dimensional molecular dynamics simulations were carried out for aluminum. In that study, the electromigration driving force was calculated using Cloud in Cell (CIC) method and the evolution of the void formation was reported. Shinzawa and Ohta [18] characterized the grain boundary di usion for aluminum interconnects by molecular dynamics. The di usion characteristics with respect to the grain boundary angle was reported. Maroudas and Gungor [19] studied the void evolution and failure in metallic thin lms. In that study, plastic deformations in the vicinity of the voids are examined by the use of molecular dynamics. In the development of an alloy with improved resistance to electromigration, we should somehow eliminate grain boundaries or slow down the di usion process. Because of the stages in the processing of these interconnects, it is inevitable to have grain boundaries. However it is possible to alter the di usion behavior with the addition of alloying elements. Then the answer to the following question should be given: Which alloying element would be e ective in lowering the di usion kinetics. In the present study, therefore the e ect of alloying elements in aluminum to the bulk di usion behavior is investigated using molecular dynamics method, under the e ect of electromigration wind force. Initially the MD method was reviewed in Chapter 2. The construction of a MD simulation is explained in detail, including how a MD simulation can be started, how the atomic interactions are handled, the integration methods used and how the properties of a material can be determined with the given algorithms. In Chapter 3, the derivation of potentials is reported by introducing the pseudopotential method, and how a pair potential can be obtained using this method. In Chapter 4, the pseudopotential approach to the electromigration due to Sorbello was given. In Chapter 5, the methodology followed is reported and the results obtained were given in Chapter 6. 3 Chapter 2 Molecular Dynamics 2.1 Classical Molecular Dynamics History of Molecular Dynamics The foundations of the Molecular Dynamics (MD) method has been laid by the development of Newtonian dynamics by Isaac Newton. He proposed that the same mechanical laws can explain the motions of all bodies, large and small, with suitable de nitions of the forces operative. Thus the Newtonian laws applies to systems of any size, even to atoms. This concept has been modi ed by quantum mechanics. Then in classical statistical mechanics, Ludwig Boltzmann investigated the problem of correlating the detailed dynamic behavior of a system of atoms and molecules with the macroscopic experimentally measurable properties of the same system. These two scienti c approaches of classical dynamics and classical statistical mechanics constitute the basis of the MD method. The trajectories and velocities of the particles corresponding to atoms (or molecules, or ions) are generated using Newtonian equations of motion. Classical statistical mechanical concepts are then used to obtain the correspondence of this system to a thermodynamic system. The problem of studying the interaction of many atoms or molecules has only become possible, using the methods of MD and MC, with the development of powerful electronic computers after 1950 s. The rst papers reporting a molecular dynamics simulation were written by Alder and Wainwright in 1957 [2, 3]. The purpose of the paper was to investigate the phase diagram of a hard sphere system, and in particular the solid and liquid regions. In a hard sphere system, particles interact via instantaneous collisions, and travel as free 4 particles between collisions. Probably the rst example of a molecular dynamics calculation with a continuous potential based on a nite di erence time integration method was done by Gibson et al [20]. In that study, the calculation for a 500-atoms system was performed on an IBM 704, and took about a minute per time step. A successful attempt to solve the equations of motion for a set of Lennard-Jones particles was made by Rahman [4]. Since that time the properties of the Lennard-Jones model have been thoroughly investigated by Verlet [21, 22] in which Verlet time integration algorithm was used.[23] After initial foundations on atomic systems, molecular dynamics simulation developed rapidly. Today molecular dynamics simulations are used in various research areas, such as liquids, defects, fracture, surfaces, friction, clusters, biomolecules, electronic properties and dynamics. [23] Basics of Molecular Dynamics In molecular dynamics, the laws of classical mechanics are followed, and most particularly Newton s law of equation of motion: F i = m i a i (2.1.1) for each atom i in a system constituted by N atoms. Here m i is the atomic mass, a i = d2 r i its acceleration, r dt 2 i its position and F i the force acting upon it, due to the interactions with other atoms. Therefore in MD, given an initial set of positions and velocities, the subsequent time evolution of the system is determined. The macroscopic or thermodynamical properties are then calculated with the help of statistical mechanics. MD simulation consists of three steps; initialization, equilibration and thermalization. In initialization part, initial positions and velocities are given to the atoms and initial system parameters of the simulation are set. Then, equations of motion are solved until the system is reached to thermal equilibrium in the equilibration part. After the thermal equilibrium, the thermodynamic properties are calculated in thermalization step. In Figure the algorithm of a basic molecular dynamics simulation has been shown, where in the following sections the parts of the molecular dynamics method will be covered in detail. 5 Give initial positions and velocities to atoms Calculate the forces on the atoms Apply the calculated forces to atoms and obtain new position and velocities No Calculate thermodynamical quantities Enough time? Yes Store the thermodynamical properties and final configuration Figure 2.1.1: Basic molecular dynamics simulation algorithm. Initialization To start a simulation, the MD box should be de ned and the set of positions and velocities should be assigned initially to the particles. There are two common ways of doing this: i) Starting from scratch If the simulation is starting from scratch, a set of initial positions and velocities should be created. Positions are usually de ned on a lattice, assuming a certain crystal structure. This structure is typically the most stable one at 0 K with the given potential. Assigning positions to atoms should be carefully handled so that there should not be any overlap of atoms. The initial velocities may be taken to be zero or can be assigned by taking them from a Maxwell distribution at a certain temperature T set. When doing this, the system will have a small total linear momentum, corresponding to a translational motion of the whole system. Since this is somewhat inconvenient to have, it is common practice to subtract this component from the velocity of each particle in 6 order to operate in a zero total momentum condition. Initial temperature of the system is also given in the initialization step by adjusting the kinetic energy. The instantaneous temperature, T (t) of the system is given by the relation N f k b T (t) = NX m i vi 2 (2.1.2) i=1 where N f is the degrees of freedom (N f = 3N 3 for a system of N particles with zero total linear momentum), k b is the Boltzmann constant and v i is
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