EVOLUTION OF ANNEALING TWINS IN SPUTTERED CU THIN FILMS CHANG-KYU YOON. A thesis submitted in partial fulfillment of the requirement for the degree of

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1 EVOLUTION OF ANNEALING TWINS IN SPUTTERED CU THIN FILMS By CHANG-KYU YOON A thesis submitted in partial fulfillment of the requirement for the degree of MASTER OF SCIECE I MATERIALS SCIECE AD EGIEERIG WASHINGTON STATE UNIVERSITY Shool of Mehanial and Materials Engineering AUGUST 2009

2 To the Faulty of Washington State University: The members of the Committee appointed to examine the thesis of CHANG-KYU YOON find it satisfatory and reommend that it be aepted. David Field, Ph. D., Chair David Bahr, Ph. D. Sergey Medyanik, Ph. D.

3 ACKOWLEDGMET Before I begin to aknowledge all amazing people whom I met at Washington State University, I am now thinking of what all I have gone through to be here and I must admit that this thesis would not have been ompleted without a sinere advies and enouragement of my advisor, Professor David Field. I would like to thank Dr. Field not only for his superb guidane and aademi advie but also being a great role model and mentor. He has taught and shown me how to beome a real knowledgeable and philosophial man with humor. He always has endured and enouraged me to pursue a goal onsistently with positive attitude. Dr. Field, It is definite blessing for me to meet you in my whole life. I am forever indebted to you. Thank you so muh for being a trustworthy and warm adviser. My deepest and sinerest gratitude should also go to my family, who have always supported and prayed for me. My wife, Jeong-Huyn, I have taken a spiritual rest and have ompleted this thesis through your prayer and sarifie. My daughter, Esther, you are always be my joy and energy. In addition, my family living in Korea, I know you are always praying for me and I an say I am nothing without your all love. Finally, I give my most sinere appreiation to my God with following onfess: O give thanks unto the LORD, for he is good: for his mery endures forever (Psalms 107:1).

4 EVOLUTION OF ANNEALING TWINS IN SPUTTERED CU THIN FILMS Abstrat by Chang-Kyu Yoon, M.S. Washington State University AUGUST 2009 Chair: David P Field The Monte Carlo Potts model with N-fold method has been used to simulate grain struture evolution in thin Cu films aording to energeti ompetition priniples. The energeti onsiderations of surfae/interfae, grain boundary and strain energy fators were applied to determine grain growth and rystallographi texture evolution as a funtion of film thikness suh as 100 nm, 500 nm and 800 nm. Furthermore, annealing twins in FCC metals with low staking fault energy are typially related to the parent by 60 rotation about the <111> rystal diretion. These twins an hange the rystallographi texture and grain boundary harater distribution in films and line strutures. These fators an be simulated through speifi riteria that arbitrarily insert twin grains into the struture through grain boundary energy onsiderations. Four different types of mirostrutures have been observed experimentally and these have been simulated by the Monte Carlo tehnique. The fous is on the texture and grain struture evolution and how these an be modeled using the Monte Carlo Potts model. Using the overall energy minimization onept to model mirostruture evolution in Cu thin films, the apabilities and diffiulties of using Monte Carlo Potts Model are demonstrated.

5 TABEL OF CONTENTS Page ACKNOWLEDGEMENT ABSTRACT APPENDICES.... LIST OF TABLES LIST OF FIGURES..... CHAPTER 1. INTRODUCTION BACKGROUND Rerystallization Nuleation Grain Growth Annealing Twins Crystallographi Charaterization Tehnique Orientation Mapping Euler Angles MICROSTRUCTURAL SIMULATION Monte Carlo Potts Model Energy Minimization Model Surfae/ Interfae Energies Strain Energy Grain Boundary Energy....24

6 4. SIMULATION RESULTS st ase (100nm film thikness at 300 C annealing temperature) nd ase (500nm film thikness at 300 C annealing temperature) rd ase (800nm film thikness at 300 C annealing temperature) DISCUSSION CONCLUSION FUTURE WORK REFERENCES...44

7 APPEDICES Appendix A- The main subroutine ontrols all other subroutines. Espeially, MCGROWTH1, MCPROBS1 and TWFIND subroutines were modified to simulate mirostruture evolution with annealing twins aording to energeti onsiderations Appendix B- This subroutine onsists of energeti onsiderations suh as surfae/interfae, grain boundary and strain energies to hek the probability a given spin flip for MCGROWTH1 subroutine as a funtion of thikness and temperature. This subroutine shows the ase of 100nm at 300 C annealing onditions Appendix C- This subroutine shows the possibilities to find twins by 60 rotation about the <111> rystal diretion with 24 symmetry operator at ubi system

8 LIST OF TABLES Chapter Three Table 3.1 Listing of 2D lattie types with geometries and anisotropies. 22 Chapter Four Table 4.1 Experimental results of texture aording to different film thikness in Cu films....26

9 LIST OF FIGURES Chapter Two Figure 2.1 Typial rerystallization kinetis during isothermal annealing.. 3 Figure 2.2 The formation of ß nuleus on a flat surfae S Figure 2.3 Shemati representation of grain size distribution during (a) Normal grain growth (ontinuous grain growth) and (b) Abnormal grain growth (disontinuous grain growth)....7 Figure 2.4 Shemati desriptions of various annealing twins morphologies observed in FCC metals Figure 2.5 Figure 2.6 The formation of annealing twins at a three-grain juntion Formation of a twin with a oherent and a non oherent interfae during grain boundary migration. 11 Figure 2.7 Random nuleation and grain growth simulation results (a) - (e) analyzed by using OIM software and (f) orientation olor key Chapter Three Figure 3.1 Diagram of the triangular 2D lattie, showing orientation numbers at eah site and boundary drawn between sites with dissimilar orientations.. 21 Figure 3.2 Shemati of misorientation or the differene in orientation between two grains Figure 3.3 Shemati illustration of grain boundary energy interating with free surfae energie

10 Chapter Four Figure 4.1 Orientation maps showing a story board presentation of simulated grain growth for 100 nm and 300C (Figures a-e are in order of inreasing MCS), f) shows the orientation olor key Figure 4.2 Predited texture omponents as a funtion of Monte Carlo time steps for 100 nm and 300C Figure 4.3 The same strutures as shown in Fig. 4.2 with olors that identify {111}, {101} and{001} frations for 100nm and 300 C Figure 4.4 Orientation maps showing a story board presentation of simulated grain growth for 500 nm and 300C (Figures a-e are in order of inreasing MCS), f) shows the orientation olor key...31 Figure 4.5 The same strutures as shown in Fig. 4.4 with olors that identify {111}, {101} {001} and {511} frations for 500nm and 300 C Figure 4.6 The hanging of {511} twins indiated by Σ3 of twins of {111} grains aording to Monte Carlo steps (MCS) Figure 4.7 Orientation maps showing a story board presentation of simulated grain growth for 800 nm and 300C (Figures a-e are in order of inreasing MCS), f) shows the orientation olor key Figure 4.8 Predited texture omponents as a funtion of Monte Carlo time steps for 800 nm and 300C....35

11 Figure 4.9 The same strutures as shown in Fig. 4.7 with olors that identify {111}, {101} and{001} frations for 800nm and 300 C Chapter Five Figure 5.1 Orientation maps showing a story board presentation of 100 nm, 500 nm and 800 nm at 300 C Figure 5.2 Texture map alulated with surfae/interfae and elasti strain energy for Cu-film resulting from grain growth at Tgg in Cu films of thikness h, deposited at Tdep

12 CHAPTER 1 ITRODUCTIO The harateristi mirostruture of a thin film affets its funtion in eletroni appliations. Modern Cu films and line strutures, whih have replaed aluminum due to its high eletrial and thermal ondutivity, often ontain a large fration of annealing twins. Annealing twins in FCC metals with low staking fault energy are related to the parent by 60 rotation about the <111> rystal diretion. These twins, whih are assoiated with rerystallization and grain growth, an hange the rystallographi texture and grain boundary harater distribution in films and line strutures. Beause these annealing twins have properties that affet mirostruture evolution, it is important to inlude these in onsiderations of strutural evolution. Grain growth in thin films generally plays an important role in defining the mirostrutural harateristis of the films [1]. During annealing, normal and abnormal grain growth our in a manner that lowers the total energy aumulated in the film. The total energy is the sum of surfae energy, interfae energy, strain energy and grain boundary energy [2]. These ombined energy variables have a tendeny to be minimized during grain growth and this generally results in texture transformation from the original state. It is well known that texture evolution during grain growth of thin films depends upon a ompetition between priniples of surfae/interfae energy minimization and strain energy minimization [1, 3, 4]. {111} out of plane fiber textures result in thinner films where surfae energy minimization ontrols grain growth, and {100} out of plane fiber textures are

13 observed in thiker films at higher temperatures as a onsequene of strain energy minimization [1]. In the FCC Cu films, the {111} lose-paked surfae is assoiated with the lowest surfae free energy, so strong {111} out of plane texture is present with little dependene on deposition and annealing onditions [5]. The onventional Monte Carlo Potts model has been shown to be apable of making signifiant and preise preditions of grain growth by various researhers sine the 1990s [6-10]. A two dimensional model has been applied with the assumption that thin films are two dimensional in harater. Of ourse, three-dimensional simulations yield more realisti preditions but at the ost of greater omputational effort. In this work it is shown that thikness dependene of struture evolution an be aptured by a two-dimensional model. The fous is on the texture and grain struture evolution and how these an be modeled using the Monte Carlo Potts model aording to the overall energy minimization onept. Experimentally, many researhers have attempted to understand grain struture (grain growth and grain boundary properties) and rystallographi texture as a funtion of various fators suh as stress state [11-13], staking fault energy [14,15], grain size [16,17] and film thikness [2, 5, 18, 19]. The purpose of this thesis is to demonstrate the apabilities and diffiulties assoiated with inluding these important fators of the mirostruture evolution with twins in Cu thin films by using Monte Carlo Potts Model.

14 CHAPTER 2 BACKGROUD 2.1 Rerystallization In order to understand and analyze mirostruture evolution in Cu thin films, it is required to know about rerystallization that ours during deposition of thin films or during post deposition proessing. Primary rerystallization inludes nuleation of new grains, oarsening, and grain growth [20]. During isothermal annealing, the rerystallization proess an be desribed by the volume fration of material rerystallized as a funtion of time as shown in Figure 2.1. This figure indiates that the proess onsists of three parts suh as an inreasing rate of rerystallization at nuleation region, linear region of growth and finally a dereasing rate of rerystallization region as grain growth nears ompletion. Figure 2.1 Typial rerystallization kinetis during isothermal annealing [20].

15 Researh into the rerystallization behavior of materials has been performed over the past 50 years. Stanley and Mehl [21], and Anderson and Mehl [22] initially divided rerystallization into nuleation and growth ontribution to the overall kinetis. Under onsideration that rerystallization is a nuleation and growth phenomenon whih is ontrolled by thermally ativated proesses with stored energy of deformation providing the driving fore, the laws of rerystallization are established [20, 23, 24]. These are desribed as follows: 1. Deformation has to be suffiient to form a nuleus for the rerystallization and to provide a neessary driving fore to maintain growth. 2. When rerystallization temperature dereases, the annealing time inreases. 3. When strain inreases, rerystallization temperature dereases. More rapid nuleation and growth our at a lower temperature in a more highly deformed material. 4. The rerystallized grain size is related to the amount of deformation. The number of nulei or the nuleation rate is affeted by strain more than the growth rate. A higher strain an provide more nulei per unit volume, resulting in a smaller final grain size. 5. When the amount of deformation is fixed, the rerystallization temperature an inrease by a larger initial grain size and higher deformation temperature. A large grain size has relatively few grain boundaries that are preferred nuleation sites so the nuleation rate is lowered and the rerystallization is slower at higher temperature. In addition, more reovery ours at high temperature of deformation and the stored energy is thus lower than for a similar strain at a lower deformation temperature [20].

16 These rerystallization laws still provide a useful guide to the overall behavior of rerystallization. This lassial rerystallization bakground gives insight into materials and proessing parameters in thin films. Thin films have high surfae area to volume ratio, whih leads to greater surfae/interfae energy minimization and the mehanial effets due to the attahment of films to substrates. To understand rerystallization phenomena at a muh finer mirostrutural level, the lassial nuleation and growth phenomena are reviewed in brief following the work of Christian [25] uleation In pratie, homogeneous nuleation ours only in extremely pure solids with very few strutural defets. More usually, impurities or strained regions (phase boundaries) of the lattie ause nulei formation with a muh smaller free energy of ativation than that of homogeneous nuleation [25]. The free energy for heterogeneous nuleation an be explained as a funtion of the ontat angle as follows: G h e t = G h o m f ( θ ) (1) 1 3 f ( θ ) = (2 3osθ + os θ ) 4 Rerystallization and nulei are formed and beome stable when the energy advantage gained by removing stored energy from the bulk is greater than the additional energy required to form the new grain boundary area shown in Figure 2.2. Nuleation depends on a redution in the net surfae energy needed to form a nuleus [25].

17 If the strain energy effets are negleted, the following general free energy of formation an be written s 3 2 S S S G ( η r / v )( g g ) r { ( )} β β β α η αβ σ αβ η α σ β σ α = + + (2) Where, α phase is the deformed or strained state in ontat with a surfae S and ß is the strain free rerystallized grain of developing nuleus that is also in ontat with S. Figure 2.2 The formation of ß nuleus on a flat surfae S. If the surfae energyσ αβ of the α-ß interfae is isotropi, the ß nuleus will be bounded by spherial surfaes of radius r, exept where it is in ontat with S. The volume of ß is β 3 η r and its surfae area of ontat with the α phase isη area of ontat with ß is equal toη α S r 2 αβ 2 r, whereη αβ and β η are the shape fators. The, the area of α-s interfae destroyed when the ß is formed αβ αs β S and σ, σ, σ are the surfae energies per unit area of the respetive surfaes. The free energy S G of formation to reate a nuleus of ritial size, r, when = 0. r

18 This gives the ritial energy differene between strained and un-strained (rerystallized) lattie to be G = S 4 27 αβ αβ α S β S α S β { η σ + η ( σ σ )} ( v ) β 2 α β 2 ( η ) ( g g ) 3 2 (3) Grain Growth Grain growth may be ategorized as normal grain growth or abnormal grain growth [20]. Normal grain growth leads to a uniformly oarsened mirostruture that has a relatively narrow grain size range and shape with a time independent grain size distribution profile. On the other side, during abnormal grain growth, a few grains in the mirostruture grow and onsume the matrix of small grains and a bimodal grain size distribution develops, whih is a disontinuous proess as shown in Figure 2.3 [20, 27]. Figure 2.3 Shemati representation of grain size distribution during (a) Normal grain growth (ontinuous grain growth) and (b) Abnormal grain growth (disontinuous grain growth) [20, 27].

19 Many researhers have performed to verify the ritial harateristis of grain growth and distribution of grain sizes in thin films. It has been shown that surfae/ interfae energy minimization governs the grain growth, for example, (111) in FCC metals at elevated deposition temperature of thin strutures [1, 3, 4, 5] and grain growth in films often stagnates when the average grain diameters is 2 to 3 times the film thikness. Surfae grooves are the apparent ause of stagnant olumnar grain struture [3, 26]. The main fators to ontrol grain growth in thin films are four elements: temperature, texture, preipitates and speimen size [3, 20], whih is desribed as below: 1. Temperature: Grain growth inludes high angle grain boundary migration and the kinetis of grain boundary motion are strongly temperature dependent. Beause the driving fore for grain growth is small, the rate of grain growth is signifiant only at very high temperature. 2. Texture: Strongly textured materials ontain a lot of low angle boundaries of low energy so texture an be a redued driving fore for grain growth. 3. Preipitates: When preipitates our in a thin film, the grain growth is inhibited by the pinning of grain boundaries. This proess leads to a loal energy inrease and thus preipitates exert a drag fore on grain boundary motion. 4. Speimen size: The grain growth rate is dereased when the grain size beomes greater than the sheet speimen thikness. The reason is that the olumnar grains are urved only in one diretion so the driving fore is diminished.

20 Grain growth in thin films generally plays an important role in defining the mirostrutural harateristis of the films [1]. One of the important reasons that grain growth in thin films is different from the grain growth in bulk materials is the interfae of the film with the substrate. The top surfae of the film plays an important role in suppressing normal grain growth and promoting seondary or abnormal grain growth [1, 3]. Beause the energy of the surfaes and/or interfaes is anisotropi, seondary grains generally have restrited rystallographi orientations. These orientations are affeted by the atomi struture of the substrate interfaes as well as the environment of the top surfae of the film. Therefore, seondary grain growth whih is driven by surfae and interfae energy leads to an evolution in the preferred rystallographi orientation of a film as well as the average grain size [3]. 2.2 Annealing Twins Annealing twins are observed in a variety of rerystallized FCC metals and alloys. These twins are exhibited through four prominent morphologies whih are shematially shown in Figure 2.4 [29]. Figure 2.4 Shemati desriptions of various annealing twins morphologies observed in FCC metals [29].

21 In 1963, annealing twin nulei whih onsist of staking fault pakets having ompliated morphology were desribed by Dash and oworker [30]. In 1978, Meyers and Murr [31] introdued the pop-out model for formation of twins with staking fault pakets that pop out from ledges present on grain boundaries. Reently, Mahajan [29], Pande [32] and their olleagues have developed a unified model to explain the formation of annealing twins in mehanisti and the twin morphologies shown in Figure 2.4. After Fullman and Fisher (1951), Figure 2.5 shows a standard theory of twin formation at a three-grain juntion where grain G3 is growing at the expense of G1 and G2 [25, 34]. Figure 2.5 The formation of annealing twins at a three-grain juntion [25]. Aording to their model, the onept of twin formation lies in the energy minimization onsideration. During grain growth, when the energy of the boundary between grain G3 and T is low enough to redue the total energy state even though the total grain boundary area is inreasing, the twin will form. The ourrene of T an redue the total interfaial free energy if G3T G3T TG1 TG1 TG2 TG2 TG1 G3G1 TG2 G3G2 o σ + o σ + o σ < o σ + o σ (4) Where o ij are the surfae areas between grain i and j and σ ij are the speifi free energies between grain i and j. When the inequality is sustained, a growth fault may our at some point leading to

22 the marosopi regions for formation of T whih is twinned with respet to grain G3. When the twin interfae is fully oherent, the twin boundary energy will be low and thus the reasonable probability of total interfaial free energy inequality is valid [25]. From the data of Murr (1975), the oherent twin boundary energy is 24 mjm -2, the inoherent twin boundary is 498 mjm -2, and the high angle grain boundary energy is 625 mjm -2 for opper [35]. Therefore, there may be a redution of total boundary energy by formation of a oherent twin boundary beause the energy is signifiantly lower than that of general boundaries. The Fullman-Fisher theory also applies to the situation leading to the formation of a terminated twin shown in Figure 2.6. If a twin T has a lose orientation to that of grain G2, the T and G2 interfae have low energy and the onfiguration with the twin will have a lower energy than the onfiguration without it. The oherent twin interfae beomes tangential to the main migrating grain boundary, after whih another interfae must form. The number of twin lamellae per unit grain boundary area depends only on the number of new grain ontats whih have been made during growth, thus the sequenes of Figure 2.6 is a usual method of introduing a twin [25]. (a) (b) () (d) Figure2.6 Formation of a twin with a oherent and a non oherent interfae during grain boundary migration [25].

23 In general, the formation of annealing twins is assoiated with grain growth that ours during annealing at relatively high temperature. Twins form during reovery, primary rerystallization or during grain growth following rerystallization [20]. There has been interest in improving the properties of alloys by ontrolling the grain boundary harater (grain boundary engineering), and maximizing the number of low Σ boundaries suh as Σ3 twins through thermomehanial proessing [11, 12, 33]. The Σ indiates the reiproal of the ratio of oinidene site lattie (CSL) volume to the primitive lattie volume. The CSL is the lattie formed by superposition of lattie sites from two rystals and used in the ontext of struture analysis to ategorize grain boundary geometry [36]. For ertain orientation relationships having a good fit between the grains, the oherent twin (Σ3) boundary, low angle grain boundaries (Σ1) and high mobility (Σ7) boundaries in FCC metals are good examples of boundaries with speial properties [20]. Twin boundaries are often flat and extend aross an entire grain. Annealing twins in FCC metals with low staking fault energy are typially related to the parent through a speifi misorientation ( g) of 60 rotation about the <111> rystal diretion. For a oherent twin boundary, the twinning plane must be aligned with the boundary plane separating the twin from the parent. This means that the FCC primary rerystallized twin boundary plane must be aligned with one of the {111} planes in eah of the neighboring rystals [37].

24 2.3 Crystallographi Charaterization Tehnique Orientation Mapping Monte Carlo Potts model an inherently mapped onto a two dimensional (2D) mirostruture by a series of FORTRAN programming odes. These odes have been modified to reate speifi files that an be analyzed by using TSL OIM analysis 4.5 version software. Orientation imaging mirosopy or eletron baksattered diffration (EBSD) indiates an automated measurement and storage of orientations to depit the mirostruture aording to a pre-defined pattern of oordinates on the sampling plane of the speimen [36]. Automated EBSD provides spatially speifi measurement of rystallographi orientation information on regular grids. Eah sampling grid is stored with its orientation, spatial oordinates and often a pattern quality index. These data then allow entirely quantitative and aessible map outputs as illustrated in Figure 2.7, for example, whih ontains an orientation map of random nuleation and grain growth of a Monte Carlo simulation. The appliation of orientation mapping an be ategorized depending on the information suh as spatial distribution of texture omponents, misorientation and interfaes, orientation perturbations within grains, true grain size/shape distributions and deformation (i.e. pattern quality) maps [36]. Speifially, misorientation whih means the orientation differene between two individual orientation measurements plays an important role to identify potential twin boundaries. By inspeting a misorientation between neighboring measurements on the grid assigned partiular olors in a map, boundaries an be identified as low angle, high angle, oherent twin, inoherent twin, et. [37].

25 In addition, orientation olor key at Figure 2.7 (f) represents that the orientation of the speimen oordinate system an be projeted into the rystal oordination system. In other words, the orientation olor key shows different olors to the rystal orientations aligned with a given diretion. The referene system is the rystal oordinate system and the orientation is defined by the axes of the speimen oordinate system, e.g. rolling diretion (RD), transverse diretion (TD) and normal diretion (ND) Euler Angles Euler angles speify a triplet of rotations about the Z, X and Z diretions. The Euler angles refer to three rotations whih transform the speimen oordinate system onto the rystal oordinate system. The definition of Euler angles adopted here is that aording to Bunge, desribed as follows [36, 38]: 1. φ1 about the normal diretion (ND), transforming the transverse diretion (TD) into (TD ) and the rolling diretion (RD) into (RD ). 2. Φ about the axis RD 3. φ2 about ND The orientation of the rystal is in the form of Bunge Euler angles (φ1, Φ, φ2). The angle φ1 and φ2 range from 0 to 2π and the angle Φ range from 0 to π. These range forms a bounded spae referred to as Euler spae [38]. Analytially, the three rotations are expressed as parameters g = g (φ1, Φ, φ2

26 gϕ 1 = osϕ1 sinϕ ϕ ϕ, gφ = 0 osφ sinφ sinφ osφ sin 1 os 1 0, gϕ 2 = osϕ 2 sinϕ 2 0 ϕ ϕ sin 2 os 2 0 (5) By multipliation of these three matries in order g = gϕ1 gφ gϕ 2, the Euler angles an be obtained as following matrix: o s ϕ 1 o s ϕ 2 sin ϕ 1 o s ϕ 2 sin ϕ 2 sin φ sin ϕ 1sin ϕ 2 os φ + o s ϕ 1sin ϕ 2 o s φ o s ϕ 1 sin ϕ 2 sin ϕ 1 sin ϕ 2 os ϕ 2 sin φ sin ϕ 1 o s ϕ 2 o s φ + os ϕ 1o s ϕ 2 os φ sin ϕ 1sin φ o s ϕ 1sin φ o s φ (6)

27 (a) (b) () (d) (e) (f) Figure 2.7 Random nuleation and grain growth simulation results (a) - (e) analyzed by using OIM software and (f) orientation olor key.

28 CHAPTER 3 MICROSTRUCTURAL SIMULATIO In many irumstanes, omputer simulation has been applied to gain insight into a physial problem for obtaining new theoretial understanding and interpreting the differenes between ideal and pratial situations. In addition, omputer simulation has played an important role in reduing a large expense of pratial experiments in industry. Many tehniques have been developed to simulate mirostruture evolution in materials siene. Beause mirostruture governs the mehanial, eletrial and thermal properties of engineered materials, it is ritial to understand mirostruture. One of the simulation methods applied to mirostruture evolution is the Monte Carlo (MC) methods that Anderson and olleagues developed for simulation of grain growth in two and three dimensions [20]. 3.1 Monte Carlo Potts Model In general, Monte Carlo refers to any method that utilized sequenes of random numbers to perform statistial simulation. The genesis of the Monte Carlo Potts model lies in the development of the Ising model (1925) [6]. The Ising model represents magnetized materials as two states of spin-up and spin-down at every point on a disretized domain struture. Potts (1952) later generalized the Ising model with adding a onept of Q states for eah partile in a system. Considering the Ising model, only two degenerate spin states (Q=2) are equivalent to a Potts model. The Potts model as it is alled a Large-Q model has been extensively used to simulate behavior of materials suh as rerystallization, grain growth and texture evolution. The Monte Carlo Potts model, modified from that developed by Hassold and Holm [8] has been used to

29 simulate two dimensional mirostruture evolution. This simulation method divides materials into a small area or partile whih is assumed to be a nuleus or grain. This nuleus or grain has its own rystallographi orientation (g) expressed by Bunge Euler angles indiated by spin variables ranging from 1 to Q. The Q-states are assigned total orientations (g) whih are randomly seleted from an input file eah having unique orientations and these orientations an be hanged to that of an adjaent grain aording to the effet of this hange on the overall energy state. The unit of simulation time is alled Monte Carlo Step (MCS) and represents an integer time inrement [8]. One iteration represents 1/N time inrements and N iterations are required for eah site in the lattie to have a hane to hange its state. Therefore, 1 unit of system simulation time elapses after N iterations. The inrease in the mean grain size in aordane with time redues the total lengths of grain boundaries and thus the system energy through two algorithms of Conventional Potts Model (CPM) Algorithm and the N-Fold way of the ISING Model. For eah MCS of domain growth in onventional Potts Model Algorithm, the following proedure is performed [8]: CPM 1 Set the energy of eah site in the system to zero at the beginning Ei =0, Ef=0 2 Pik a site i: i{1,, N} and assign an index number (orientation) randomly 3 Pik a spin value Si: Si{1,.,Q} at random 4 For eah site i, if a randomly piked spin value of Si is not equal to the spin of the neighbors spin (Sj), the energy state is Ei.

30 5 Randomly piked spin value of Si is not equal to the Sj, the energy state is Ef. 6 E = Ef Ei 7 Flip (reorientation) site i to spin Si with probability P ( E) 8 Inrement time by 1/N Monte Carlo steps (MCS) Eah MCS requires N iterations (lines 4 and 5) plus N iterations of some onstant time operations (line 1-3 and 6-8) and typial Potts model appliations may require as many as MCS per simulation, so obviously an ineffiient linear dependene of omputing time per MCS exists in the onventional Monte Carlo algorithm. Hene, as grains inrease in size, fewer and fewer iterations of the onventional Monte Carlo loop will result in spin flips atually ourring. As a result, for the purpose of speeding up the simulation time, the extensive simulation algorithm of N-fold way was applied [8]. When the grain growth was nearly omplete, most sites in the lattie were surrounded by sites of similar orientation. Therefore, the probability of hanging orientation is low and omputational time is saved by using the N-fold way as opposed to the onventional Monte Carlo method. Initially, the proessing time of the N-fold method is slower than Monte Carlo Potts algorithm. Beause the N-fold way is set to very fine grained domain struture at the beginning, a high total system ativity is required. When the domains have oarsened and the average domain radius (grain size) reahes a ritial radius aording to individual user assumptions, the N-fold way beomes muh faster than the Monte Carlo Potts algorithm with dereasing the net system ativity [8].

31 3.2 Energy Minimization Model The onept of interation between sites is based on an energy minimization model. The total system energy an be desribed by interating neighboring spins with eah other. When the spins are the same, no interation energy is ontributed whereas different neighboring spins lead to a ontribution to the system energy [6]. In order to derease the total interating boundary area of a system, boundaries move toward their enters of urvature. High energy state grains have a tendeny to redue their boundary urvature and this urvature-driven diffusive oarsening of grains dominates the grain growth [8]. Randomly seleted sites or grains with adjaent neighboring grains have their unique orientations indentified by Euler angles and these are used to help determine the proposed energy hange of the position ( E) assuming that the orientation was hanged to that of one of its neighbors. In this simulation, the system provided a driving fore to reah a minimum energy state through re-orientation of eah site with probability P ( E) given by p ( E) 1 if E 0 = exp E / k T if E > 0 ( ) B (7) E is the energy hange, T is annealing temperature, and k B is the Boltzmann onstant. If the energy hange is less than or equal to zero, the site would reorient to another orientation with the transition probability given as one. If the energy hange is greater than zero, the reorientation hanges with probability exp (- E/KBT). Aording to the energy minimization ompetition, some nulei are onsumed while others oarsen and grow, resulting in a preferred orientation.

32 Figure 3.1 Diagram of the triangular 2D lattie, showing orientation numbers at eah site and boundary drawn between sites with dissimilar orientations [7]. The Figure 3.1 shows the triangular 2D lattie with orientation numbers at eah site and boundaries between sites with dissimilar orientations [7]. This two dimensional (2D) mirostrutural evolution of grain struture has been traked by a hange of eah initially established orientation aording to energy onsiderations suggested by Park et al [2]. The surfae/interfae, grain boundary and strain energies lead to the preferred final textures of thin films. These energeti fators are desribed in the following Hamiltonian m 1 E = γsurfae ( g) + γinterfae ( g) + γstrain ( g) + γgrain boundary ( g) i= 1 2 j= 1 where, E is the total energy of a system, is the total number of lattie sites, γsurfae ( g) is the surfae energy of a unit lattie as a funtion of orientation (g) aording to normal diretion (ND)(001) out of plane, γinterfae ( g) is the interfae energy as a funtion of orientation (g), γstrain ( g) is the bi-axial strain energy as a funtion of orientation (g), γgrain boundary ( g) (8) is the grain boundary energy as a funtion of misorientation ( g ) between two grains of different index and m is 6, onsidering the number of nearest neighbors in a triangular (1,2) lattie (hexagonal array)

33 indiated in Table 3.1. This governing Hamiltonian is omposed of three individual elements of surfae, interfae and strain energy terms and grain boundary terms. These energeti onsiderations are seen in the software shown in appendies A and B. Table 3.1 Listing of 2D lattie types with geometries and anisotropies [6]. Lattie Type Wulff Shape Coordination Number Grain Growth Square (4.1) Square 4 Inhibited Triangular Hexagonal 6 Normal Square(1,2) Otagon 8 Normal Triangular (1,2) Dodgeagon 18 Normal Surfae/ Interfae Energies For the purpose of omparing surfae energy γ surfae (g) in this model, the broken-bond energy onept given by Sundquist [39] was used. The number of unsatisfied broken bonds will vary from plane to plane so the different atomi paking system has the various rystallographi planes and the surfae energy is dependent on rystallographi orientation (g). The misorientation of an arbitrary grain with the neighboring grain orientations allows one to determine the angular differene of the surfae normal from the referene. This angle is used for omparing surfae energy by following a simple relationship: γ 0{hkl} A B =os θ hkl - h 0k 0l 0 γ 0{h0k0l 0} A B = (9) where γ 0{ hkl} is the surfae energy at 0 K of the{ hkl} plane and θ is the angle between the { hkl} plane and the referene plane. The referene surfae normal unit vetor (hkl) of eah grain is (001) // ND sine this is the high energy plane in an FCC struture. The initial value is J/m 2 for (100) surfae at 200 C with temperature dependene given by dγ dt hkl = 4 2 o J / ( m C) [39, 40].

34 In addition, the interfae energy is assumed to be idential to the surfae energy. Sine there is no hange in barrier layer or staking sequene, the interfae energy for all films is similar. The interfae energy an be expressed by ( ) ( ) γ g = γ g = γ da surfae interfae Strain Energy The strain energy is given by the following equation: ( ) 2 γ strain = hkl A hkl g h e M da hkl M = C + C + K 2 hkl ( C K ) 2 C K ( )( ) hkl hkl (10) A K = C C + C h k + k l + l h (11) T T gg dep ( s f ) e = α α dt α T In the above equations, h is the thikness of the films, e is the elasti strain whih is a funtion of the temperature differene, M hkl is the biaxial modulus for the {hkl} oriented grain, α is the mismath in oeffiients of thermal expansion, and stiffness onstants of C11 = GPa, C12 = GPa, C44 = 75.4 GPa at 25 C with temperature gradient effets inluded as dc dt dc12 dc = GPa/ C = GPa/ C, dt, dt 11 hkl = GPa/ C [41]. 44

35 3.2.3 Grain Boundary Energy The grain boundary energy an be desribed by ( g) h[ L L L ] γ = γ + γ + γ (12) grain boundary LAGB LAGB ICTB ICTB CTB CTB L LAGB, L ICTB and L CTB are the lengths of the large angle grain boundary, inoherent twin boundary and oherent twin boundary and γ LAGB, γ ICTB andγ CTB are the energies of large angle grain boundaries, inoherent twin boundaries and oherent twin boundaries. The large angle grain boundary energy is assumed to be J/m 2 at 925 C, the inoherent twin boundary energy is J/m 2 at 950 C and the oherent twin boundary energy is J/m 2 at 800 C with temperature effets of dγ dt LAGB 4 2 o = 1.0 X10 J / m C dγ dt ICTB 4 2o = 1.0 X10 J / m C, dγ dt CTB 5 2o = 2.0 X10 J / m C [42]. Grain boundaries an be haraterized by the misorientation ( g) between two neighboring sites [43]. The grain boundary energy inreases with inreasing angle of misorientation to 15. With the exeption of twin boundaries, the grain boundaries with larger than 15 misorientation have similar grain boundary energies onsidered to be high angle grain boundaries (HAGB). By analyzing the misorientation ( g) between neighboring sites, twins and twin boundaries an be identified. The next equation shows the misorientation g between two grains 1 and 2, where Ti is the 24 symmetry operators, supersript T denotes transpose.

36 Figure 3.2 Shemati of misorientation or the differene in orientation between two grains. (13) Geometrially, annealing twins an form and relate to the parent through speifi misorientation ( g) of 60 with <111> axis. For oherent twins, the twinning plane has to be aligned with a grain boundary plane position [37] and the loal relationships between surfae energy and grain boundary energy an be onsidered in the following way [20, 44]: Figure 3.3 Shemati illustration of grain boundary energy interating with free surfae energies os θ 1 = γ γ γ = γ 1 gb, θ 1 = os γ γ 1 gb γ 2 osθ 2 θ 2 = os gb, γ gb In this simulation, the grain boundary grooving phenomenon at Figure 3.3 was onsidered. Thermal grooving forms in order to ahieve a apillary fore balane [44]. Beause grain boundary grooves an develop during annealing, Young s equation (14) was applied to our model for 300 C annealing ondition and grain boundary area inreased through the grooves as grain growth took plae. The software to perform this operation is given in appendix C. (14)

37 CHAPTER 4 SIMULATIO RESULTS For the Monte Carlo ode used in this study, the orientations marked as Bunge Euler angles of grains were ompared and hanged for grain growth as a funtion of the energetis introdued in previous researh [2, 40, 41] summarized by equations (10)-(12). Three different film thiknesses were set for the simulations; 100 nm, 500 nm and 800 nm. The deposition temperature was 25 C and the annealing temperature was 300 C. In this simulation, as Monte Carlo steps (MCS) inreased, the texture evolved as growth onditions favored a lowering of system energy st ase (100nm film thikness at 300 C annealing temperature) Previous experimental results, given in Table 4.1 [2], indiate that the 100nm-thik sputtered film has a {111} out of plane fiber texture. This result was ompared with this simulation result shown in Figure 4.1. Figure 4.1 shows the simulation of mirostruture evolution at 100 nm film thikness and 300 C annealing temperature onditions. Figure 4.1.f ontains the orientation olor key that is used throughout the paper for all orientation images shown. The final stage of this simulation (Fig. 4.1.e) indiates that the texture evolves to a dominant {111} orientation (blue olor) assoiated with <111> // ND (normal diretion) with permissible misorientation range of 15. Table 4.1 Experimental results of texture aording to different film thikness in Cu films [2]. Case Thikness Texture 1 100nm-thik sputtered films {111} fiber 2 200nm-thik sputtered films {111} fiber +twins of {111} 3 500nm-thik sputtered films {001} fiber, very large grain size 4 800nm-thik sputtered films {001}+twins variants of {001}

38 (a) (b) () (d) (e) (f) Figure 4.1 Orientation maps showing a story board presentation of simulated grain growth for 100 nm and 300C (Figures a-e are in order of inreasing MCS), f) shows the orientation olor key.

39 Fig 4.2 and 4.3 show {111}, {101} and {001} texture omponent development respetively as a funtion of Monte Carlo Steps (MCS). These texture omponents are analyzed aording to olor gradients indiated by a degree of misorientation between 0 and 15 from the ideal omponent. Blue is {111}, green is {101} and red is {001}. The final stage shows that the texture is omposed of almost entirely {111} orientations in a thin film of 100 nm thikness. This agrees with well known experimental observation. This phenomenon has been studied by many researhers. For thin films, the {111} orientation is favored beause of the surfae energy minimization effet on the film [3, 4]. In general, thin Cu films have preferred {111} texture as a result of the lose-paked surfae that has the lowest surfae free energy [5, 19]. Figure 4.2 Predited texture omponents as a funtion of Monte Carlo time steps for 100 nm and 300

40 (a) (b) () (d) (e) (f) Figure 4.3 The same strutures as shown in Fig. 4.2 with olors that identify {111}, {101} and{001} frations for 100nm and 300 C

41 4.2 2 nd ase (500nm film thikness at 300 C annealing temperature) The next simulation result, shown in Figure 4.4, was based on the onditions of 500 nm thik films with 300 C annealing temperature. Figure 4.4 indiates that when the {111} grains grow, the {511} twins of {111} grains start to appear as they assist in reduing the total system energy. These twins are about 16 away from {001} grains. These twins appear automatially in the evolving struture through grain boundary energy onsiderations. Even though surfae and/or strain energy are not minimized by a {511} orientation, the total energy onsiderations tend to favor the twin omponent beause of the favorable boundary energy between the {111} and {511} omponents. This result ompares well with observations of struture evolution for 200 nm films (see Table 1). In addition, Figure 4.5 and 4.6 indiate evolution of twin grains having orientation of {511}//ND with 27 perent of the boundaries onsidered to be Σ3 boundaries aording to CSL theory. These twins were traed by the geometrial onept that the misorientation ( g) of twin has the 60 rotation about the <111> rystal diretion with energeti relations between surfae energy and grain boundary energy at equation (14). Figure 4.5 shows the fration hanging of {001}, {101}, {111} and {511} grains for 500nm at 300 C annealing temperature and espeially, Figure 4.6 indiates the hanging of {511} twins marked by Σ3 of twins of {111} grains as a funtion of Monte Carlo steps (MCS).

42 (a) (b) () (d) (e) (f) Figure 4.4 Orientation maps showing a story board presentation of simulated grain growth for 500 nm and 300C (Figures a-e are in order of inreasing MCS), f) shows the orientation olor key.

43 (a) (b) () (d) (e) (f) Figure 4.5 The same strutures as shown in Fig. 4.4 with olors that identify {111}, {101} {001} and {511} frations for 500nm and 300 C

44 MCS Σ3 Frations , , Figure 4.6 The hanging of {511} twins indiated by Σ3 of twins of {111} grains aording to Monte Carlo steps (MCS) rd ase (800nm film thikness at 300 C annealing temperature) Simulation results for 800 nm thik films show a dominant {001} orientation at the final step of the MC simulation (Figure 4.7). The initial textures and grain assignments for the 100 nm and 800 nm simulations were similar but the final textures are distint. The only differene in simulated onditions was film thikness. This 800 nm film thikness ase of simulation indiates the preferred texture is {001}. Figure 4.8 and 4.9 indiate how the frations of {111}, {101} and {001} evolved with MC time steps. Figure 4.8 and 4.9 show that the texture rapidly hanged to {001}. {111} and {101} frations quikly went to zero at the expense of the {001} grains. No twins of {100} grains were observed in the simulations as there is apparently no ondition where the energy would be minimized by reation of suh a struture. The results that preferred texture formation at thinner 100 nm films was {111} and thiker 800 nm films was {001} an be explained by lowering the energy of the system during proessing.

45 (a) () (e) (b) (d) (f) Figure 4.7 Orientation maps showing a story board presentation of simulated grain growth for 800 nm and 300 C (Figures a-e are in order of inreasing MCS), f) shows the orientation olor key.

46 Thompson and o-workers established the onept of energy ompetition between surfae/interfae and strain energy (as a funtion of film thikness and temperature) almost two deades ago [3, 4]. The determination of whih grains will grow is a result of the ompetition between strain energy minimization and surfae/interfae energy minimization. When the hange of the surfae/interfae energy is larger than the hange of strain energy during grain growth, {111} out of plane texture dominates. Otherwise, {100} texture minimizes the strain energy density and dominates growth. Figure 4.8 Predited texture omponents as a funtion of Monte Carlo time steps for 800 nm and 300C.

47 (a) (b) () (d) (e) (f) Figure 4.9 The same strutures as shown in Fig. 4.7 with olors that identify {111}, {101} and{001} frations for 800nm and 300 C

48 CHATER 5 DISCUSSIO This Monte Carlo simulation method with energeti onsiderations based on surfae/interfae, grain boundary, and strain energies showed a struture evolution in sputtered thin films as a funtion of film thikness. Figure 5.1 shows a summary of the final simulation results inluding the resulting frations of {111}, {511} and {001} textures for films of 100 nm, 500 nm and 800 nm thikness. In 100 nm films, the experimental and simulation results show the preferred texture is {111} with surfae/interfae energy minimization dominating struture evolution [3, 4]. The 500 nm film thikness ase of Figure 5.1 shows that {111} grains with twins of {511} tend to form a lower total system energy when all effets (surfae energy, strain energy, and grain boundary energy) are onsidered. In thiker films suh as the result shown for 800 nm, {001} grains grow rapidly during annealing beause of strain energy minimization being the dominant fator in struture evolution [3, 4]. Aording to C. V. Thompson [3], surfae energy dominates growth in the 100 nm films resulting in {111} texture. Strain energy dominates growth in the 800 nm films and {100} texture forms as seen in Figure 5.2 [2].

49 100nm 500nm 800nm Figure 5.1 Orientation maps showing a story board presentation of 100 nm, 500 nm and 800 nm at 300 C

50 Figure 5.2 Texture map alulated with surfae/interfae and elasti strain energy for Cu-film resulting from grain growth at Tgg in Cu films of thikness h, deposited at Tdep [2]. There are no results that show a low energy onfiguration for the thiker films where the twins of {001} grains are present in the simulations. It is hypothesized that during the proess of {111} grains growing, the {511} grains whih are the twin grains of {111} start to be present for the purpose of reduing the total system energy. If a grain forms that is about 15º away from {111} and a twin develops in this struture, it would provide a nuleus for growth of {100} grains. This happens infrequently so the observed grains are very large (nulei spaed far apart). In addition, experimental results show that {211} and other texture omponents that are twin related to either {111} or {001} textures exist in the thiker films (800 nm). The results obtained in this work show no suh struture. In a previous paper [45], twin formation ourred by the brute fore method. Twins grains were arbitrarily inserted into the struture at pre-seleted onditions to ahieve the desired struture evolution. While properly simulating evolved strutures, this approah laks any preditive apability.

51 A realisti model that is apable of aurately prediting struture evolution requires a mehanisti desription of twin formation. Many researhers indiate formation of twins in terms of three fators suh as grain boundary energy, grain boundary mobility and role of the disloation arrangement. Speifially, aording to growth aidents and nuleation of twins by staking faults or fault pakets, many researhers suggest a twin formation mehanism. Gleiter [46] in 1969 proposed a model of annealing twin formation as a result of growth aidents leading to staking faults. Pande et al. [32] arried out a systemati study of annealing twin formation in nikel with the assumption that twin formation is aused by grain boundary migration and their density depends on the driving fore for migration. Mahajan et al. [29] suggested that Shokley partial disloation loops nuleate on onseutive {111} planes by growth aident related to grain boundary migration. In addition, previous researh [13-18, 33] has shown experimental evidene of various twinning frations aused by different kinds of onditions suh as film thikness, stress state, annealing onditions and fabriation proesses. These fators would have to be inluded in the models to predit formation and evolution of twins in thin films. The experimental results presented above are for high purity sputtered Cu films. It is well known that eletro- deposition of Cu films results in a wide variety of strutures subsequent to post-deposition annealing. These an range from randomly oriented, twin-rih strutures to highly oriented {111} fiber textures even for films on the order of 1 µm thikness. The effets of bath hemistry on the energetis of the system and the boundary mobility would have to be inluded in any models of struture evolution for plated films. This will inrease the omplexity of the models dramatially. As mentioned previously, there are several fators that lead to varied