The Pennsylvania State University. The Graduate School. College of Earth and Mineral Sciences A FIRST-PRINCIPLES METHODOLOGY FOR

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1 The Pennsylvania State University The Graduate School College of Earth and Mineral Sciences A FIRST-PRINCIPLES METHODOLOGY FOR DIFFUSION COEFFICIENTS IN METALS AND DILUTE ALLOYS A Dissertation in Materials Science and Engineering by Manjeera Mantina 2008 Manjeera Mantina Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2008

2 The dissertation of Manjeera Mantina was reviewed and approved* by the following: Zi-Kui Liu Professor of Materials Science and Engineering Dissertation Co-Adviser Co-Chair of Committee Long-Qing Chen Professor of Materials Science and Engineering Dissertation Co-Adviser Co-Chair of Committee Suzanne Mohney Professor of Materials Science and Engineering Jorge O. Sofo Associate Professor of Physics Associate Professor of Materials Science and Engineering Joan M. Redwing Professor of Materials Science and Engineering Chair of Graduate Program of the Department of Materials Science and Engineering *Signatures are on file in the Graduate School. ii

3 Abstract This work is a study exploring the extent of suitability of static first-principles calculations for studying diffusion in metallic systems. Specifically, vacancy-mediated volume diffusion in pure elements and alloys with dilute concentration of impurities is studied. A novel procedure is discovered for predicting diffusion coefficients that overcomes the shortcomings of the well-known transition state theory, by Vineyard. The procedure that evolves from Eyring s reaction rate theory yields accurate diffusivity results that include anharmonic effects within the quasi-harmonic approximation. Alongside, the procedure is straightforward in its application within the conventional harmonic approximation, from the results of static first-principles calculations. To prove the extensibility of the procedure, diffusivities have been computed for a variety of systems. Over a wide temperature range, the calculated self-diffusion and impurity diffusion coefficients using local density approximation (LDA) of density functional theory (DFT) are seen to be in excellent match with experimental data. Selfdiffusion coefficients have been calculated for: (i) fcc Al, Cu, Ni and Ag (ii) bcc W and Mo (v) hcp Mg, Ti and Zn. Impurity diffusion coefficients have been computed for: (i) Mg, Si, Cu, Li, Ag, Mo and 3d transition elements in fcc Al (ii) Mo, Ta in bcc W and Nb, Ta and W in bcc Mo (iii) Sn and Cd in hcp Mg and Al in hcp Ti. It is also an observation from this work, that LDA does not require surface correction for yielding energetics of vacancy-containing system in good comparison with experiments, unlike generalized gradient approximation (GGA). iii

4 It is known that first-principles energy minimization procedures based on electronic interactions are suited for metallic systems wherein the valence electrons are freely moving. In this thesis, research has been extended to study suitability of firstprinciples calculations within LDA / GGA including the localization parameter U, for Al system with transition metal solutes, in which charges are known to localize around the transition metal element. U parameter is determined from matching the diffusivities of 3d transition metal impurity in aluminum with reliable experimental data. The effort yielded activation energies in systematic agreement with experiments and has proved useful in obtaining insights into the complex interactions in these systems. Besides the prediction of diffusion coefficients, this research has been helpful in understanding the physics underlying diffusion. Within the scope of observations from the systems studied, certain diffusion related aspects that have been clarified are: (i) cause for non-arrnenius nature of diffusion plots (ii) definitions of atom migration properties (iii) magnitude and sign of diffusion parameters enthalpy and entropy of formation and migration and characteristic vibrational frequency (iv) trends in diffusivities based on activation energy and diffusion prefactor (vi) cause for anomalous diffusion behavior of 3d transition metals in Al, and their magnetic nature (vii) contributions from electronic contributions to curvature at very high temperatures of bcc refractory elements (viii) temperature dependence of impurity diffusion correlation factors. Finally, the double-well potential of diffusion by vacancy mechanism has been calculated from first-principles. This aided calculation of entropy of migration and thus free energy of migration along with characteristic vibrational frequency. Also for the first iv

5 time, temperature dependence of enthalpy of migration and thus atom jump frequency has been accurately predicted. From the broad perspective of predicting diffusion coefficients from computational methodologies, it can be stated as a result of this work that: static firstprinciples extend an irreplaceable contribution to the future of diffusion modeling. The procedure obviated the use of (i) redundant approximations that limit its accuracy and (ii) support from other computational techniques that restrict its extensibility due to insufficient input or computational resources. v

6 Table of Contents List of Figures... ix List of Tables...xiii Preface... xvii Acknowledgements... xix 1 INTRODUCTION Diffusion Theories describing migration Eyring s reaction rate theory Vineyard s Transition State Theory (TST) Later theories Approximations Harmonic Approximation Local Harmonic Approximation Quasi-harmonic Approximation Other Computational Methodologies Thesis Overview References.16 2 COMPUTATIONAL METHODOLOGY Introduction First-principles Density Functional Theory Ground state energy Exchange-correlation energy Supercell method Nudged elastic band method Implementation Conclusion References.34 3 SELF-DIFFUSION IN FCC METALS Introduction Theory Mono-vacancy mechanism Di-vacancy mechanism Present work Results and discussion FCC Aluminum FCC Copper FCC Nickel FCC Silver Conclusions References.91 4 IMPURITY DIFFUSION IN FCC METALS Introduction vi

7 4.2 Theory Five-frequency model Impurity diffusion equation Literature Study Present work Results and discussions Five jump frequencies (TST-V) Impurity diffusion data (DW) Comparing data from TST-V and DW Comparison with experiments Discussion on trends Conclusions References DIFFUSION OF 3d TRANSITION METALS IN Al Introduction Background study Procedure and System setup Methodology Results and Discussions U parameter Activation energies Magnetic moments Impurity diffusion coefficients Conclusions References DIFFUSION IN BCC AND HCP Introduction Theory Diffusion equations in bcc Diffusion equations in hcp System setup Results and Discussions BCC W BCC Mo HCP Mg HCP Zn HCP Ti Conclusions References CONCLUSION AND FUTURE WORK Conclusion Future work References APPENDIX A.1 VASP relaxation A.1.1 INCAR vii

8 A.1.2 KPOINTS A.1.3 POSCAR (perfect state) A.1.4 POSCAR (initial state) A.1.5 POSCAR (transition state) A.1.6 INCAR (NEB) A.1.7 INCAR (static relaxation) A.1.8 POSCAR (System with impurity) A.1.9 INCAR (CHGCAR) A.1.10 INCAR (+U) A.1.11 INCAR (+U static relaxation) A.1.12 POSCAR (BCC) A.1.13 KPOINTS (HCP) A.1.14 POSCAR (HCP) A.2 ATAT ( fitfc ) phonon calculations A.2.1 vasp.wrap (harmonic approximation) A.2.2 vasp.wrap (quasi-harmonic approximation) A.2.3 fitfc (harmonic approximation) A.2.4 fitfc (quasi-harmonic approximation) A.2.5 Force constant fitting for HA and QHA A.2.6 vasp.wrap (system with non-standard potential) viii

9 List of Figures 1. 1 Figure illustrating a plane of close-packed atoms as solid spheres (a) when the tracer atom (shaded) is in normal lattice position with an adjacent vacant site (b) when tracer atom is in transition to the adjacent vacant site N dimensional energy diagram [9] showing constant potential energy hyper-surfaces (solid lines) about two equilibrium lattice positions A and B. Atom in site A to pass through saddle point P to move to site B. When in transition state, the atom is constrained in its motion to the hyper-surface S that is perpendicular to the direction of motion. M is a representative point on hyper-surface S Flow chart [7] illustrating the iterative scheme for the ground state electron density from density functional theory using Kohn-Sham approximation [6] Schematic illustrating the minimum energy path on the potential energy diagram between two equilibrium positions (solid lines) [17]. Solid dots represent the images positions and dotted line the minimum energy path obtained from NEB relaxation Figure illustrating di-vacancy jumps in pure fcc system [1] Figure illustrating the distribution of the phonon frequencies. The negative frequencies in PDOS of TS are due to the unstable mode. The high peak at low frequencies for the IS are from the weak bonds surrounding the vacancy. The magnitude of low frequencies is high for TS due to weakening of several bonds when the diffusing atom moves to the saddle point. The distribution is smeared from the different neighbors being affected to a different extent. The diffusing atom actually gets closer to some of its neighbors. This is shown by the small peak of high frequencies for this state Equilibrium vacancy concentration from first-principles (both LDA and GGA harmonic and quasi-harmonic calculations) plotted in comparison with experimental data [22-24]. The slightly different slope at higher temperatures of measured data was suggested by experimentalists [24] to be from contributions of di-vacancies. The result from the current work is from mono-vacancies only. The experimental results of Seeger [23] is data for only mono-vacancies Potential representing the motion of the diffusing atom from an equilibrium lattice position to the adjacent vacant position. The potential of pure Al system is a symmetric double-well. The discrete energy states of this potential are illustrated in the figure. The potential between the two equilibrium positions matches well with the NEB potential, as illustrated Figure comparing the results from the transition state theory (TST) to the doublewell (DW) approach for fcc aluminum. The jump frequency results from the two approaches differ only slightly. The difference between HA and QHA results of jump frequency is more evident for TST than that from DW ix

10 3. 6 Figure illustrating trends followed by the (a) enthalpy and (b) entropy of vacancy formation with changing temperature using within HA and QHA, using LDA and GGA potentials Figures illustrating trends followed by the (a) enthalpy and (b) entropy of atom migration with changing temperature using within HA and QHA, using LDA and GGA potentials Derivatives of enthalpy and entropy of vacancy formation and atom migration with respect to temperature from GGA within QHA First-principles results of self-diffusion coefficients for Al. Results are shown for both the HA and QHA using both LDA and GGA. Calculated results are compared with experimental results [28-32] and theoretical work [21], showing excellent agreement for the GGA (with surface correction included) calculated selfdiffusion coefficient First-principles calculation of self-diffusion coefficients for Cu showing results for HA using both LDA and GGA. Calculated results are compared with experimental data [48-53] showing excellent agreement for the LDA calculated diffusion coefficient without surface correction Entirely first-principles, non-spin polarized calculation results within HA using LDA in comparison with experiments [53-62] Self-diffusion coefficient of Ag from LDA using the HA in comparison with the experimental data [66-69] Equilibrium vacancy concentration from fcc Ag with mono-vacancy in comparison with experiments[66, 70] Figure illustrating silver diffusion coefficient from mono-vacancy and di-vacancy contributions predicted from LDA using HA in comparison with other data in literature [66-69]. The di-vacancy contribution predicted from the present work is seen to have similar slope to experimental measurements by Lam [68] Di-vacancy concentration from the present work in comparison with experiments [71] Five-frequency model illustrated for fcc with dilute impurity concentration. The arrows indicate the direction of the vacancy jump and the numbers n stand for the n th nearest neighboring site to the impurity. In the case of w 1, w2 and w 3 jumps the vacancy position is indicated by solid box and the position of the solvent atom to which the vacancy jumps to is indicated by filled circle. For jumps w0 and w 4 the vacancy position is indicated by dotted box and the solvent position of the jump is indicated by open circle Diffusion coefficient with HA of Mg in fcc Al from simplified approach using LDA and GGA and from TST-V using LDA (without including surface corrections), in comparison to experimental data [26, 28-33] Diffusion coefficient with HA of Si in fcc Al from simplified approach using LDA and GGA and from TST-V using LDA (without including surface corrections), in comparison to experimental data [28, 34-36] x

11 4. 4 Diffusion coefficient with HA of Cu in fcc Al from simplified approach using LDA and GGA and from TST-V using LDA (without including surface corrections), in comparison to experimental data [9, 30, 36-38] Diffusion coefficients of Mg, Si and Cu in Al and Al self-diffusion. It is seen that Al self-diffusion is slower than the impurity diffusion in Al Trend of Hubbard U parameter for 3d elements in Al in comparison with values of U parameters calculated for 3d elements from the work of Aryasetiawan et al. [49] Systematic match in the trend of activation energies between current work (LDA+U) and experimental data. Sandberg s [34] results using PAW LDA are also plotted for comparison Partial DOS from the d electrons of the transition element in Al at the Fermi level (E F ) from non-spin polarized, fully relaxed perfect (PS) and initial state (IS). Residual resistivity values are taken from experimental work of Steiner et al. [55] Difference in charge density between un-relaxed Al-V and Al-Cr structures: (a) without vacancy (b) with vacancy adjacent to the solute. The brown colored lines for the charge density difference Al-V to Al-Cr indicate a positive value Figure illustrating the partial density of states of the d-orbital of the transition metals: V, Cr, Mn, Fe, Co and Ni in Al in perfect configuration after full relaxation from non-spin polarized calculation. The d-dos at the fermi energy of the Al-TM system is marked. (Note: The Fermi energy (E f ) of the Al-TM system is deducted from the energy of different states of the d-orbital for the plot, to indicate E f at 0) Figure illustrating the positive charge density (red colored lines indicate positive values as shown in the colorbar) difference between Al-TM systems as captioned in the plots, from charge density of perfect states Figure illustrating the partial density of states of the d-orbital of the transition metals: V, Cr, Mn and Fe in Al in initial configuration after full relaxation from nonspin polarized calculation Figure illustrating the positive charge density difference between adjacent Al-TM systems as captioned in the plots. Charge density of non-spin polarized relaxation of initial states is used d-dos of 3d elements V-Ni in Al without vacancy from their fully relaxed state. Double peaks in the diagrams for Cr, Mn and Fe solutes indicate the magnetic feature of the impurity atoms in Al. Co also exhibits some amount of magnetism, while V and Ni have no magnetic moment Impurity diffusion coefficient from LDA and LDA+U (for systems required) in comparison with experimental data available in the literature Anomalous diffusivities in Al of 3d transition elements Sc, Ti, V, Cr, Mn and Fe, and normal behavior of Co, Ni, Cu and Zn Different jump frequencies in bcc from five-frequency model [2] xi

12 6. 2 Two types of jump in hcp illustrated in figure (a) with vacancy indicated by an open circle, and jump distance components along the three primary axes of the two jumps A and B in hcp with rigid-sphere packing shown in figure (b) Different jump frequencies in hcp system with impurity is illustrated for the case of the jump A (a) and jump B (b). Open box represents the position of vacancy and colored atom represents the position of the solute atom Self-diffusion coefficient in bcc W in comparison with data from experiments [23-25]. Figure illustrates improvement in the diffusion result on inclusion of electronic contributions Impurity diffusion coefficient of Mo in bcc W in comparison with data from experiments [28-30] Self-diffusion coefficient of bcc Mo in comparison with data from experiments [31-35] Impurity diffusion coefficient of W in bcc Mo in comparison with data from experiments [29, 37-39] Impurity diffusion coefficient of Ta in bcc Mo in comparison with data from experiments[35, 37, 38]. Figures (a) and (b) illustrate the effect of including impurity correlation factor Impurity diffusion coefficient of Nb in bcc Mo in comparison with data from experiments [37, 40, 41]. Impurity correlation factor for Nb diffusion in Mo is in not included in this figure Self-diffusion coefficient parallel and normal to the c-axis compared with the data from experiments [49, 50] Predicted Cd diffusivity in hcp Mg in comparison with experimental data [49] Predicted Sn diffusivity in hcp Mg in comparison with experimental data [49] Self-diffusion coefficient predicted in hcp Zn (c/a ratio fixed) in comparison with experiments [51-55] Self-diffusion coefficient from present work after relaxing only the internal coordinates in comparison to experimental data [51-55] Self-diffusion coefficient in hcp Ti along and normal to c-axis in comparison to experiments[59, 61, 62]. In the paper by Brick [62], the direction of the diffusion coefficient is not understood (foreign language article) Al diffusion in hcp Ti from LDA HA (full relaxation) in comparison with the experimental data [63-68] xii

13 List of Tables 3. 1 First-principles calculated quantities entering calculation of self-diffusion coefficient. QHA results from current work for T from K, using GGA and LDA (with surface corrections as described in the text) are compare to other calculated and experimental data. Data for H in experimental column is deduced from experimental values for Q from [28-32] using H m f =0.67eV. Similarly, using the value of entropy of formation of 1.1k B from experiments, and value of ~ ν of 2.1 THz from the current work, value of entropy of migration is deduced from pre-exponential factor D 0 of experimental data [28-32]. Results of S m from BM referring to ballistic model and cbω model are listed. EAM / MD data in the last column refers to the dynamic simulation results using embedded atom potential fitted to first-principles data (GGA) The frequency factor D0 and activation energy Q of self-diffusion in aluminum from double-well approach in comparison with experimental data. The temperatures represent the ranges over which measured diffusion coefficients were fit to extract D 0 and Q First-principles calculated quantities entering calculation of self-diffusion coefficient for fcc Cu. HA results from current work for T = θ D using GGA and LDA (without surface corrections) are listed, and compared to other calculated and experimental data. From the value of entropy of formation of 2.5k B and entropy of activation of 4.15k B, obtained from analyses of different experimental works, value of entropy of migration listed in the experimental column is deduced Table comparing frequency factor D0 and activation energy Q of self-diffusion in copper within HA from LDA and GGA, with experimental data Table comparing frequency factor D0 and activation energy Q of self-diffusion in nickel within HA from LDA, with experimental data Other parameters obtained in the calculation are compared with data from theoretical works and experiments Tabulated values are for impurity jump w2 from full relaxation using GGA * 4. 2 TST-V results of ν (THz) for the 5 jumps of different impurities. Eq. (3.39) is used for calculating this quantity for each jump from their respective initial and transition states Migration barriers H m (ev) of 5 jumps of different impurities from TST-V Five jump frequencies (Hz) for different impurities in Al. Results for T=400K using ν * from Table 4.2 and H from Table m 4. 5 Enthalpy and entropy of formation of a vacancy adjacent to impurity atom H f, S f and quantities involving the correlation factor (T=400K) for the Arrhenius diffusion parameters xiii

14 4. 6 Five jump frequencies (Hz) for different impurities in Al. Results for T=400K using Eq. (3.46) Quantities describing the impurity diffusion coefficient D 2 (see Eq. (4.4)) calculated from double-well approach are listed. r is the jump distance in angstroms for the impurity atom to reach the vacant site, which is not exactly equal to a / * 4. 8 Comparing the ν (THz) for the impurity jump from double-well approach to Vineyard s TST, of different impurities. For clarity, Eq. (3.37) is used for calculating quantities in column 2 from the gamma point frequencies and Eq. (3.36) is used with the quantities involved calculated within harmonic approximation from double-well Comparing the H m (ev) for the impurity jump from double-well approach to Vineyard s TST, of different impurities. The difference between the two results is that the vibrational contributions to enthalpies of the IS and TS are included in double-well approach and enthalpy of migration from TST is the difference between the ground state energies. Results from TST-V are constant at all temperatures Comparing the impurity jump frequency w 2 (Hz) from double-well approach to Vineyard s TST, of different impurities. Using the results from the respective * approaches from Table 4.8 and Table 4.9 in equation w2 =ν exp( H m k BT ), the jump frequencies obtained are tabulated below Calculated results from LDA and GGA (without surface-correction) to experimental data. The measured data of enthalpy and entropy of solute-vacancy binding listed are the critically assessed data by Balluffi and Ho [24]. The migration o barrier of pure Al is from the difference of assessed Q =29 kcal/mol value of Peterson et al. [9] and the assessed value of enthalpy of vacancy formation (listed below) of Erhart et al. [25]. From the activation energies obtained by the studies referenced in measured column of migration barrier and the value of enthalpy of vacancy formation in impurity system, the migration barrier values listed in measured column are deduced Table listing the diffusion pre-factor and activation energy values from the current work using LDA for the temperature ranging from K, in comparison with experiments and other theoretical calculations Table lists the (i) distances of the first nearest neighboring atoms to the impurity atom obtained from the fully relaxed structures of the Al system with impurity (no vacancy) (ii) energy required for bond breaking, which is defined as the energy of the unrelaxed system with vacancy minus the energy of the relaxed system without vacancy, (iii) total binding energy, which is negative of the difference between energy of relaxed system with vacancy and energy of the unrelaxed system with vacancy. Energies listed are in ev Ground state energies without vibrational contributions and volume after relaxation of perfect state (PS) with no vacancy and initial state (IS) with a vacancy adjacent to the impurity atom Displacements (in Å) of the nearest neighboring atoms towards the vacancy position (0.25, 0, 0.25). The nearest neighboring positions of (0.25, 0, 0.25) includes the impurity at (0, 0, 0). The nearest neighboring atoms to the vacancy (8 of them present in the supercell of the current work) are categorized based on their distances xiv

15 from the impurity atom: 1 st nearest neighbors to the impurity atom i.e. (0.25, 0.25, 0) and (0, 0.25, 0.25), 2 nd nearest neighbors to the impurity atom i.e. (0.5, 0, 0) and (0, 0, 0.5), 3 rd nearest neighbors to impurity atom i.e. (0.5, 0.25, 0.25) and (0.25, 0.25, 0.5) and, 4 th nearest neighbor i.e. (0.5, 0, 0.5) Entropies (k B ) of PS and IS configurations for T=400K. force constant listed is in ev Energies (ev) (no vibrational contributions included), volumes (Å 3 ) of the two configurations are tabulated. b/c is the ratio of the dimensions of the unit cell, indicating the tetragonality of fcc in its relaxed state with a vacancy and the impurity atom in TS and IS Change in distances (in Å) to the impurity atom, from the atoms surrounding the vacancy, when the impurity atom moved from its initial equilibrium position to the saddle point at (0.125, 0, 0.125). (For information on the positions the different nearest neighboring (nn) atoms stand for refer to caption of Table 4. 15) Entropies (k B ) of IS and TS configurations for T=400K. The quantity listed in parenthesis in column 1 is the mass of the diffusing species in amu Distances (in Å) of the first nearest neighboring atoms of the atom at position (0.25, 0, 0.25) (before vacancy is formed) to the vacancy position. 1 and 2 represent 1 st nn to the impurity atom i.e. (0.25, 0.25, 0) and (0, 0.25, 0.25). 3 and 4 represents the 2 nd nn i.e. (0.5, 0, 0) and (0, 0, 0.5). 5 and 6 represent atoms 3 rd nn to impurity atom i.e. (0.5, 0.25, 0.25) and (0.25, 0.25, 0.5). 7 represents its 4 th nn atom i.e. (0.5, 0, 0.5) and 8 represents the impurity atom position (0, 0, 0) Distances of the nearest neighboring atoms of the vacancy position (0.25, 0, 0.25) to the impurity atom position (0, 0, 0) in system with no vacancy. Here 8 represents the atom at position (0.25, 0, 0.25) which is also 1nn to the impurity atom First nearest neighboring distances (in Å) to position (0.25, 0, 0.25) after formation of vacancy. 8 here represents the impurity atom position Distances from the impurity atom of the nearest neighboring atoms to the vacant position (0.25, 0, 0.25) after vacancy is formed. Here 8 represents the position where vacancy is created. The column listed itself is the distance the impurity atom moves after formation of vacancy from its previous position at (0, 0, 0) Distances of the nearest neighbors of vacant site to the vacancy position (0,25, 0, 0,25) after movement of the impurity atom to the saddle position Distances of the nearest neighbors of the vacant site to the impurity atom at the saddle position (0.125, 0, 0.125) Activation energies for 3d elements in Al obtained using LDA and LDA+U, with and without spin-polarization. The U-J values that went into the LDA+U calculations are listed. Activation energy of the Al-3d systems from type of potential and relaxation suitable to it is bolded. Assessed experimental data for activation energy is tabulated for comparison. All the values listed in the table are in ev Magnetic moments (µb) of the fully relaxed perfect, initial and transition states of the impurity jump, obtained from LDA+U relaxation using J=1eV and U-J values from column 2 of Table 5.1, in comparison with results from experimental measurement and predictions from other theories in the literature xv

16 5. 3 Q and D 0 values from the current work in comparison with experimental data Quantities involved in self-diffusion calculation of bcc W for temperatures ranging from K in comparison with other data in literature Arrhenius parameters for self-diffusion in bcc W in comparison with other data in literature Diffusion parameters in bcc Mo from the current work in comparison with experiments. The change in the diffusion parameters over the temperature range in column 4 is listed First-principles calculated atom jump frequencies in hcp Mg as a function of temperature. Partial correlation factors obtained from Mullen s tables [42] based on the ratio of the jump frequencies (listed in column 4) First-principles calculated quantities entering calculation of self-diffusion coefficient for hcp Mg. HA results from current work using LDA (without surface corrections) are listed for temperatures K, and compared to other calculated and experimental data Diffusion prefactor and activation energy for diffusion along c-axis and along a or b-axes in hcp Mg along with data obtained from measurements and temperature range of validity. Anisotropy ratio in hcp Mg with c / a = 1.63 is listed Predicted jumps frequencies and partial correlation factors as a function of temperature for hcp Zn compared with experiments. The data listed here from this work is from the case when shape of the system is not relaxed (constant c/a ratio), only volume and internal co-ordinates are relaxed Diffusion prefactor and activation energy for diffusion along c-axis (D ) and along a or b-axes (D _) in hcp Zn along with data obtained from measurements and temperature range of validity Atom jump frequency in basal plane and between adjacent basal planes in hcp Ti for the temperature rage of experimental measurements First-principles calculated quantities entering calculation of self-diffusion coefficient for hcp Ti Diffusion prefactor and activation energy for diffusion along c-axis (D ) and along a or b-axes (D _) in hcp Ti along with data obtained from measurements and temperature range of validity Diffusion prefactor and activation energy for diffusion along c-axis (D ) and along a or b-axes (D _) for Al diffusion in hcp Ti along with data obtained from measurements and temperature range of validity xvi

17 Preface Understanding the various phenomena occurring when processing commercial alloys for predicting their properties requires knowledge of diffusion coefficients. To list a few: (i) embrittlement caused in alloys from diffusion of hydrogen from the surface and within the bulk to stress concentrated areas (ii) areas of easy fracture formed from clustered vacancies (iii) transformation of the structure and properties from diffusion flux between species in an alloy (iv) alloy strengthening due to precipitation and coarsening. Knowledge of diffusion coefficients of the pure matrix element and of the alloying elements in the matrix is essential for obtaining inter-diffusion or intrinsic diffusion coefficients in alloys. Since the 1930 s to the present many experimental studies were conducted to study diffusion coefficients in metals and alloys. Purposes of these works have been (i) to measure reliable self and impurity diffusion coefficients, (ii) to know the predominant mechanism of diffusion - vacancy, interstitial, ring etc., (iii) to deduce quantities such as solute-vacancy binding energy, enthalpy and entropy of defect formation and tracer migration, correlation factor etc. for understanding the system behavior (a) in the presence of defects (b) in the transition state, (iv) to explain the non-arrhenius diffusion results from probable anharmonicities of the quantities involved or domination of other mechanisms of diffusion at higher temperatures. In the process of attempting to understand the effect of atomic and electrostatic interactions on diffusion, rigorous models were developed. Even though different approximations were involved, these analytical models could consistently predict the xvii

18 values of individual diffusion parameters, the diffusion mechanism and as a consequence the diffusion coefficients, in qualitative comparison with experiments. In an attempt to minimize the tedious experimental procedures for obtaining volume diffusion data, researchers used different atomistic techniques such as Monte Carlo (MC), Molecular Dynamics (MD), inspite of the fact that they needed high computational resources. Further, reliable electronic structure relaxation procedures also called ab-initio calculations, had also been used for the purpose. With some input from experiments, the computational and analytical techniques were able to predict diffusion coefficients in good agreement with experimental results. Though first-principles calculations were used for computing diffusion coefficients over the past decade, implementation by researchers either involved approximations or inclusion of some empirical techniques that led to inaccurate results or, adopted highly computationally intensive calculations (ab-initio molecular dynamics) that limited the applicability. With a growing need for accurate database of diffusion properties in materials research, this work focuses on proposing a general and straightforward procedure for accurate prediction of volume diffusion coefficients, in various metallic systems with different impurities. xviii

19 Acknowledgements I would like to convey my heart-felt gratitude to my advisor Dr. Zi-Kui Liu for his guidance and encouragement through my four years of research experience in the Phases research lab at Penn state. I would like to convey my sincere thanks and gratitude to my co-advisor Dr. Long-Qing Chen for his advice and support on different occasions. I would also like to thank my committee members Dr. Suzanne Mohney and Dr. Jorge Sofo for serving on my dissertation committee. Finally I would like to thank: All the group members for the useful suggestions and inspiring environment, specifically, Shunli and Dr. Wang for the helpful discussions. Vedic Society members at Penn state whose indirect contribution towards my research has been and will be invaluable by preaching the essence of Bhagavad-Gita - the holy book of the Hindus Now, therefore, inquire into the Absolute Truth. My friends Manju Lata Rao, Priya Kishore, Ravindra Akarapu, Swetha Ganeshan, Komal Tamara, Narasimha Mangadoddy for their direct contribution to my progress at different stages of my Ph.D. Husband Venu Vaithyanathan, sister-in-law Vidhya Sivakumar and mother in-law and other in-laws for their guidance and support that led to the completion of the program requirements after marriage. My parents and my sister for their patient co-operation during the long course. xix

20 1 INTRODUCTION 1.1 Diffusion Diffusion is a non-equilibrium phenomenon that occurs commonly in solids. There are two main approaches to the theory of diffusion 1 : (i) continuum approach (ii) atomistic approach. In the continuum approach, the diffusion process is treated as a continuum from the concentration profile across the system. From the atomistic approach, a more complete picture of diffusion phenomena is obtained; microscopic quantities such as atom jump frequencies that link directly to the diffusion coefficients are obtained from considering the details of the atomic motions. The theory discussed below is of the atomistic approach. Diffusion occurs (i) in the bulk, (ii) along dislocations or grain boundaries or (iii) at the surface. In the bulk of the crystal, where there is regular lattice structure, volume diffusion takes place. In the case of dislocation or grain boundary, or surface, lattice structure almost breaks down. Hence elementary jumps in these cases are not welldefined. In the case of volume diffusion, there are eight possible ways elementary jumps can occur [1]: (i) Ring mechanism (ii) Exchange mechanism (iii) Interstitial mechanism (iv) Interstitialcy mechanism (v) Crowdion mechanism (vi) Vacancy mechanism (vi) Divacancy mechanism (vii) Relaxion mechanism. Of these mechanisms, vacancy, interstitial and di-vacancy mechanism are most frequently encountered. Diffusion by vacancy mechanism is usually preferred for self-

21 diffusion or for diffusion of substitutional impurities. Interstitial mechanism is preferred in the case of loosely packed crystal structures and cases when impurities are considerably small in size compared to host atoms. Di-vacancy mechanism is common at high temperatures when there is a high concentration of vacancies and the di-vacancy binding energy causes the system to be energetically more favorable. Substitutional tracer atoms, either an isotope of the host or an impurity atom, diffuse predominantly by vacancy mechanism in close packed materials. In such materials, jumps occur mainly to the first nearest neighboring sites. Hence for diffusion of a substitutional tracer, a vacant site must be present in its adjacent sites. This causes a lattice imperfection with a specific local symmetry about the tracer atom. When the tracer atom makes the jump, there is a change in the direction of the lattice imperfection compared to its previous position. This difference in local symmetry about the diffusing atom causes the movement of the tracer to be correlated i.e. after the first jump the tracer has more than random probability to make the next jump in the reverse direction of its previous jump. Interstitial mechanism of diffusion on the other hand is mostly uncorrelated. If jumps are uncorrelated or random with no preferred direction, each atom follows a random walk. For such a random walk, Einstein [2] described diffusion in a specific direction as 1 X 2t 2 D =, (1. 1) where X is the net displacement in the direction of interest over a time t. The mean square value of the net displacement over several observations is represented by 2 X. 2

22 In the cubic structure [3] all jumps have the same length and have identical 2 2 correlation with later jumps. Hence X = Nx, where N is the number of jumps in time t and x is the distance in a specific direction from any jump. Defining N/t as Γ, we have D = Γ x. (1. 2) The total tracer jump frequency Γ [4] in a system that depends on the concentration of vacancies adjacent to the tracer C, the tracer atom jump frequency w and the increased probability of tracer jump due to the possibility of existence of vacancy at any one of the nearest neighboring sites to the tracer z: Γ = zcw, (1. 3) Cz is the probability that the tracer or diffusing atom will have a vacant site in its nearest neighboring site. This multiplied by the probability with which an atom successfully jumps w gives the total frequency Γ with which vacancy-mediated jumps can occur in a system. Also in cubic crystals the jump distance along a specific direction x can be 2 2 expressed in terms of the jump length r as x = r 3. Diffusion equation (1.2) for a cubic crystal can then be written as D = zcw r. (1. 4) When a successive jump depends on the direction of the previous jump(s), the motion is said to be correlated. When the motion is correlated, the tracer atom has more than random probability of tracing its path back. This implies that correlation causes decrease in the net displacement of the tracer. The extent to which jumps are correlated in a pure system depends on the mechanism of diffusion and symmetry of the crystal structure, 3

23 indicated by correlation factor f. From this, the actual displacement of Eq. (1.1) in the case of correlated motion is given by X 2 2 = f X. (1. 5) random Consequently, diffusion equation for correlated motion in a cubic crystal is given as [5] D 1 6 zcw r f 2 =. (1. 6) In the atom jump process involved in diffusion, the diffusing atom passes through an unstable high-energy state. Determination of the energy of this state is required for obtaining the activation energy and hence calculating the atom jump frequency. Different theories were developed that attempt to define the jump or migration process. In the section below some of the well-established theories are described in some detail. 1.2 Theories describing migration Eyring s reaction rate theory In the field of chemistry, reactions are explained on the basis of an activated complex that forms between the reactants, which disassociates to form products. The rate at which the reaction occurs is given by a rate constant k defined by Eyring [6], in the nineteen thirties, as a product of the ratio of concentration of the activated complex C TS to the concentration of the reactants C IS multiplied by the associated velocity of the reactants for a forward reaction to form the activated complex, ν. This is expressed as 4

24 k = ν C TS C IS. (1. 7) Absolute rate theory described above approximates that the activated complex is in equilibrium with the reactants. This concept has been extended by Glasstone [7] in 1941 to describe thermal activation of defects in solids with rate constant k representing the jump frequency w, the frequency of vibration of the system in the direction of the vacancy represented by the velocityν and concentration of the different species representing their partition functions. FIG. 1.1 illustrates the original and activated states. Wert and Zener [8] were the first to treat this equation and define the jump frequency w of Eq. (1.6) as = ~ G exp m w ν. (1. 8) k BT G m is the free energy required for the diffusing atom in its original state to move to the activated state and ν ~ is the vibrational frequency in the direction of the vacant site of the diffusing atom in its original state (characteristic vibrational frequency). 5

25 (a) (b) FIG Figure illustrating a plane of close-packed atoms as solid spheres (a) when the tracer atom (shaded) is in normal lattice position with an adjacent vacant site (b) when tracer atom is in transition to the adjacent vacant site. 6

26 1.2.2 Vineyard s Transition State Theory (TST) Einstein frequency was obtained by treating atom migration as a one-body problem, in which motion of only the diffusing atom is considered while all other atoms are considered fixed in their normal lattice positions. Though it is mainly one atom involved in the elementary jump, the diffusing atom is surrounded by other atoms with which it interacts. Thus in 1957, Vineyard [9], by treating migration as a many-body problem, derived more complete expression for atom jump frequency in a system. In a system with N atoms and 3N-dimensional configuration space, illustrated in FIG. 1.2, the jump frequency is given as: w = 3N 6 i= 1 3N i= 1 ν i exp( H 7 ν where enthalpy of migration, i m k T), (1. 9) B H m, is the enthalpy difference between the state with all atoms in their equilibrium positions and the state with the diffusing atom at the saddle point and remaining atoms in their equilibrium positions. From equations (1.8) and (1.9) we can see that the Vineyard theory puts forth the expression below, terming it the * effective frequency ν : 3N 6 ν i ~ i= 1 ν exp( S m k B ) =. (1. 10) 3N 7 ν i= 1 i 7

27 frequencies Explicitly, effective frequency is defined as the quotient of product of the phonon ν i of the original state to the product of the phonon frequencies ν i of the activated state. It must be noted that the imaginary frequency of the unstable vibration mode of the activated state due to negative curvature of the energy-diagram for motion in the direction of the vacancy, is excluded from the product in the denominator of Eq. (1.10). This theory involves further approximations to those from absolute rate theory such as: (i) there are no quantum effects (ii) the vibrations are assumed to be harmonic, following the theory of small vibrations Later theories In 1958, Rice [10] formulated a dynamical model for diffusion, describing diffusion in solids by a method that proceeds directly from the consideration of lattice dynamics, rather than using thermodynamic functions to define an unstable state. Later, the dynamical theory and transition state theory were shown by Glyde [11] and Fiet [12] to be nearly equivalent in their final definition for jump frequency, though the physical concepts involved in obtaining them were different. Later in the seventies, Van Vechten [13] developed the ballistic model, describing migration in terms of the absolute rate theory. According to this model, the atoms obstructing the movement of the diffusing atom to the vacant site move apart with a time period of vibration τ m. The tracer atom should possess a velocity to travel a distance equal to its interatomic distance in half the timeτ m to make the jump. From this model, 8

28 the migration barrier is defined from the kinetic energy of the tracer while moving as a free particle into the adjacent vacant site. As seen above, expressions involving thermodynamic property changes describing the migration process in diffusion have been introduced by these theories to define atom jump frequency. To determine the thermodynamic quantities in terms of the results from first-principles calculations i.e. ground state energies and the phonon frequencies, theories involving approximations in treating the potential have been developed. Some of the most commonly used are described below. 9

29 FIG N dimensional energy diagram [9] showing constant potential energy hypersurfaces (solid lines) about two equilibrium lattice positions A and B. Atom in site A to pass through saddle point P to move to site B. When in transition state, the atom is constrained in its motion to the hyper-surface S that is perpendicular to the direction of motion. M is a representative point on hyper-surface S. 10

30 1.3 Approximations Harmonic Approximation According to the conventional harmonic approximation (HA) [14] system energy is approximated based on potential considered nearly parabolic i.e. the potential energy is limited to quadratic terms of interatomic distance. This approximation considers all the atom interactions with each other, giving a 3Nx3N (ignoring the rigid body translational and rotational modes) force-constant or dynamical matrix, for a system with N atoms. Within harmonic approximation, the equation for Helmholtz free energy given by Maraduddin et al. [14] is as follows: F( T ) = F c + k T B 3N The first term on the right side of the equation q 6 hν j ( q) ln 2sinh. (1. 11) = 1 2kBT j F c represents the free energy of the system due to the atoms having a specific configuration, called configurational free energy or potential energy. The second term is the contribution from the vibrational degrees of freedom of the system, called the vibrational free energy or kinetic energy. Temperature dependence of total free energy within harmonic approximation, in which temperature dependence of volume is not taken into account, is described from the contribution of the vibrational degrees of freedom (DOF). Besides, different wave vectors in the reciprocal space are characteristic of wave motion of the atoms in real space for each of which the system exhibits a unique set of phonon frequencies. Hence sampling the space with high number of wave vectors gives a more precise value of the vibrational free energy. Enthalpy and entropy within HA can be derived from Eq. (1.11) as: 11

31 12 = + = q N j B j j c T k q h q h E T H ) ( )coth ( 0.5 ) ( ν ν, (1. 12) = = q N j B j B j B j B T k q h T k q h T k q h k T S ) ( 2sinh ln 2 ) ( coth 2 ) ( ) ( ν ν ν. (1. 13) Local Harmonic Approximation In this case [15], the effect of the diffusing atom on the neighboring atoms is considered negligible. The approximation considers the diffusing atom in an effective force field and hence the force constant matrix or the dynamical matrix is a 3x3 matrix with only diagonal terms being non-zero. Hence the free energy equation from this approximation would involve summation of j over 3 frequencies itself, instead of 3N-6 within HA, in Eq. (1.11) Quasi-harmonic Approximation To account for the non-harmonic nature of the potential, an approximation which is not entirely anharmonic, but which considers harmonic approximation at different volumes, and extrapolates the volume dependence of the microscopic quantities, is termed quasi-harmonic approximation (QHA). Free energy description from such an approximation has additional volume dependence as represented in the following equation + = q j B j B c T k V q h T k V F T V F 2 ), ( 2sinh ln ) ( ), ( ν (1. 14)

32 Enthalpy and entropy following QHA can be obtained from equations similar to (1.12) and (1.13). These approximations have a further simplified form when considered for application at high temperatures, called the classical limit [14], valid mostly for temperatures above the Debye temperature of the system. In this limit, k B T >> hν, application of which simplifies the equations above: sinh(x) is replaced by x as value of x is small. 1.4 Other Computational Methodologies Calculation of self-diffusion coefficients, as described by Eq. (1.2), can be performed (i) following either static or dynamic simulations (ii) using either inter-atomic potential or pseudo-potentials. Each simulation technique has its advantages and limitations in terms of input that is required, computational costs that would be incurred and output it can generate. In section 1.2, the different theories used for describing diffusion from static calculations have been presented. Computational methodologies that have been followed by other researchers are briefly presented below. Molecular dynamics, kinetic Monte Carlo and ab-initio molecular dynamics are dynamic techniques, which simulate system behavior as a function of time. Molecular dynamics (MD) simulation is realistic as the system naturally evolves under inter-atomic forces and the defect has the freedom to choose the most favorable diffusion mechanism. As the defect takes long times to actually diffuse, this method is computationally expensive even when simulating a system at high temperature. Additionally, though the inter-atomic potentials can be generated more accurately using models including many- 13

33 body interactions such as embedded-atom (EAM) [16], they are available for limited alloys systems due to limitations in extending the model to different types of bonding [17]. Kinetic Monte Carlo (KMC) on the other hand is an extremely fast and accurate technique as every jump attempt results in a transition and millions of Monte Carlo steps per atom can be accumulated, giving highly accurate diffusion coefficients. The technique needs a catalogue of jump frequencies for all possible atomic permutations around the vacancy atleast upto a few coordination shells around the vacancy, which is an extremely overwhelming task. It also cannot predict the mechanism for diffusion. Ab-initio molecular dynamics (AIMD) is a better technique to MD as it is an electronic structure relaxation method. The pseudo-potentials needed for the simulation are available for almost all elements in the periodic table, which can be simply concatenated together to represent different elements in the system. But the accuracy comes at the cost of high computational expense. Also, the pseudo-potentials are limited in their applicability depending on the covalent or ionic nature of the bonds. Molecular statics that function on the basis of inter-atomic potentials and static first-principles [18] that use pseudo-potentials are static techniques, which simulate a system configuration at a specific point of time. While, molecular statics suffers with similar limitation of lack of accurate force-fields for a wide range of systems as MD, static first-principles calculations has the added advantage of affordable computational costs compared to AIMD, while lacking the capability to predict diffusion mechanism and predicting diffusivity in concentrated alloys. 14

34 The results from static techniques can be used to calculate diffusion coefficients taking the help of theories that describe diffusion, the commonly used of which have been described in section 1.2. Also, in the case of static first-principles technique, a description to link the microscopic quantities obtained to macroscopic diffusion parameters needs to be adopted. Most commonly used approximations for this have been discussed in section 1.3. Though static first-principles calculations suffer from some limitations, the extensibility of the approach and reliability of the results obtained through this procedure creates interest for further research in this area. 1.5 Thesis Overview So far, the basic equations guiding diffusion, different theories describing diffusion process and approximations involved in linking the microscopic quantities to macroscopic quantities and the goal of the present work to study different approaches for diffusion calculations from static first-principles, have been presented. In chapter 2, the computational methodology followed for the calculation, theory guiding the methodology, details of implementation will be presented. Chapters 3, 4 and 5 will involve application of the procedure for calculating self-diffusion coefficients, impurity diffusion of s, p metals, impurity diffusion of d block elements, in fcc systems, respectively. Chapter 6 will present calculation of self and impurity diffusion coefficients in bcc and hcp elements. Chapter 7 will list conclusions from the current work and scope for future work. 15

35 Bibliography [1] J. R. Manning, Diffusion kinetics for atoms in crystals (D. Van Nostrand Company, Inc., Princeton, 1968). [2] A. Einstein, Ann. Phys. 17, 549 (1905). [3] N. L. Peterson, J. Nucl. Mater , 3 (1978). [4] A. D. Le Claire, Acta. Met. 1, 438 (1953). [5] A. D. Le Claire, in Treatise on Physical Chemistry (Academic Press, New York, 1970), Vol. 10, p [6] H. Eyring, J. Chem. Phys. 3, 107 (1935). [7] S. Glasstone, K. J. Laidler, and H. Eyring, (McGraw-Hill Book Company, Inc., 1941), p [8] C. Wert and C. Zener, Phys. Rev. 76, 1169 (1949). [9] G. H. Vineyard, J. Phys. Chem. Solids 3, 121 (1957). [10] S. A. Rice, Phys. Rev. 112, 804 (1958). [11] H. R. Glyde, Rev. Mod. Phys. 39, 373 (1967). [12] M. D. Feit, Phys. Rev. B 5, 2145 (1972). [13] J. A. Van Vechten, Phys. Rev. B 12, 1247 (1975). [14] A. A. Maradudin, E. W. Montroll, and G. H. Weiss, Theory of Lattice Dynamics in the Harmonic Approximation (Academic Press, New York, 1971). [15] R. Lesar, et.al., Phys. Rev. Lett. 63, 624 (1989). [16] S. D. Murray, Baskes, M.I, 29, 6443 (1983). [17] Y. Mishin, Farkas, D., Mehl, M.J., Papaconstantopoulus, D.A., 59, 3393 (1999). [18] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). 16

36 2 COMPUTATIONAL METHODOLOGY 2.1 Introduction In this chapter, the general methodology followed for this thesis on computing diffusion coefficients is presented. The theory underlying the techniques used is detailed, followed by the implementation of the methodology in the usage of softwares. We have seen in the previous chapter that the process of vacancy-mediated diffusion mainly involves two steps: (i) formation of vacancy adjacent to the diffusing species (ii) migration of the diffusing species. As also seen in section 1.3, for calculating thermodynamic property changes during these processes one needs to calculate: (i) ground state energy (ii) phonon frequencies. To determine the enthalpy or entropy change on formation of vacancy, the properties of the system configurations with and without vacancy are to be determined. Similarly, free energy change when the tracer atom migrates from original equilibrium position to the vacant site is estimated from the properties of the initial and transition states. Thus, overall there are three configurations of interest: (i) system configuration with no vacancy (ii) with vacancy in the system (iii) system with the tracer atom in activated state. Hence ground state energy and phonon frequencies of these three system configurations need to be calculated (i) for a single volume; thermodynamic properties 17

37 are obtained as a function of temperature from their vibrational degrees of freedom i.e. within harmonic approximation (ii) for several volumes; temperature dependence of the calculated properties include volume thermal expansion of the system i.e. following quasi-harmonic approximation. For calculations of the ground state energy at T=0K, electronic structure relaxations are conducted, as implemented by highly efficient first-principles code - VASP [1]. Getting the right configuration of the system when the tracer atom is in the transition state is not easy, as the system geometry needs to be relaxed to get its minimum energy configuration while maintaining its high energy state with the diffusing atom atop the barrier. To find the right saddle point and at the same time obtain the relaxed state of this configuration, nudged elastic band (NEB) [2] method is used. The supercell method [3] implemented by ATAT [4] is used for obtaining the phonon frequencies. For calculations within quasi-harmonic approximation, the phonon calculations are repeated for different volumes, implemented by ATAT. In the coming three sections, details on how interaction of charges within the system are treated and the approximations involved in defining the interaction energies are presented for (i) density functional theory of first-principles calculations (ii) supercell method of phonon calculations and (iii) nudged elastic band method giving the saddle point along the migration path. In section 2.5 details on the implementation of the methodology are presented and conclusions are stated in section First-principles Density Functional Theory 18

38 The fundamental basis of density functional theory (DFT) [5] is that ground state energy of a many electron system is a functional of the electron density n(r) and can be obtained by energy minimization with respect to n(r). The Hohneberg-Kohn-Sham [5] theory simplifies DFT through a one electron approximation in which the very complex many-body problem of interacting electrons is replaced by an equivalent, but much simpler problem of solving the wave function of a single electron interacting with other electrons through an effective potential. A Schrodinger-like equation of one electron wave function is defined, the potential energy term in which, is the effective potential that is a function of charge density. This Kohn-sham equation is solved self-consistently to get the ground state charge density, from which the ground state energy of the system with interacting electrons is obtained Ground state energy The total energy of electrons in a many electron system is the sum of kinetic energy of the electrons E k, the electron-electron interaction energy E ee, and E ext the interaction energy between the electrons and nuclei. E [ n(r) ] = E k [ n(r) ] + E ee [ n(r) ] + E ext [n (r) ] (2. 1) The total energy of the system of atoms fixed in their motion, identified by its potential energy E p, is then defined by the equation: E p = E [ n(r) ] ZiZ j r r r (2. 2) i, i j i j j 19

39 where the second term on the right side of the equation is the nuclei-nuclei interaction energy. If V ext (r) represents the potential describing the interactions between the nuclei and the electrons, then E ext is given by the equation: E ext r r r = n(r)v (r)dr (2. 3) ext A variety of approximations such as muffin-tin potential, different pseudo-potentials etc., are available for describing the crystal potential V ext (r). While E ext could be readily expressed as a functional of n(r), the problem in expressing E k and E ee as functionals of n(r) is solved differently in different approaches. Once this is done effectively, the problem of finding the ground state energy is reduced to minimizing E [ n(r) ] with respect to n(r). The classical interaction energy between charges of density n(r), called Hartree energy, is defined as E H r r 1 n(r)n(r ) r = r 2 r - r r r drd (2. 4) If E ee were identified with E H, the electron-electron interaction energy would be overestimated for three reasons: (i) Owing to the Pauli exclusion principle the electrons are kept out of each other s way. This leads to the lowering of electron-electron interaction energy by the so-called exchange energy (ii) Mutual electrostatic repulsion of electrons also keep the electrons apart. This repulsion, termed as the correlation energy, further lowers the electron-electron interaction energy (iii) Interaction of an electron with itself is also included in E H which needs to be excluded. The sum of the corrections that need to be added to the Hartree energy to correctly define E ee is called the exchange and 20

40 correlation energy, E xc, i.e. E ee = E H + E xc. Some approaches have self-interaction energy included in the correction while some do not. Within the density functional theory, the kinetic energy E k [ n(r) ] is considered to be the functional of electron density of the non-interacting case o E k [ n(r) ]. The difference between them is usually assumed to be included in the exchange and correlation energy. Thus the total energy of the system is E [ n(r) ] = E [ n(r) ] + o k 1 r r n(r)n(r ) r r r drdr r 2 - r r r + E xc [ n(r) ] + n( r)vext(r)d (2. 5) Considering the system to be a fictitious system of non-interacting electrons, by defining an effective potential V eff (r), the above equation reduces to: E [ n(r) ] = o E k [ n(r) ] + r r r n( r)veff (r)d (2. 6) The requirement that the ground states of the system with interacting electrons and that with non-interacting electrons have the same charge density leads to an equation for determining V eff (r) by applying the variational principle to Eqs. (2.5) and (2.6): r r r n( r)veff(r)dr 1 r r n(r)n(r ) r r 2 r - r = r r drdr r r r + E xc [ n(r) ] + n( r)vext(r)d (2. 7) To solve the above equation for V eff (r) one needs to know the charge density n(r). For a system with non-interacting electrons the charge density can be written as r n(r) = 2 N i= 1 r r Ψi (r) Ψi (r) r where Ψ (r) are one-electron wave functions that correspond to the N lowest occupied i states; the factor of two arises due to spin degeneracy. The variational condition of Eq. (2. 8) 21

41 (2.6) then leads to the Schrödinger-like equation called Kohn-Sham equation [6] for the fictitious system of non-interacting electrons: 1 2 r r r r Ψi (r) + Veff (r) Ψi (r) = ε iψi (r) (2. 9) 2 r Solutions of the above equation are the wave functions Ψ (r) and eigenenergies ε i. Hence starting with an initial guess for the charge density n(r) one can obtain a value for the effective potential V eff (r) from Eq. (2.7) with suitable pseudo-potential definition of V ext (r) and appropriate description of the exchange and correlation energy E xc. Solving Eq. (2.9) and using Eq. (2.8) a better estimate for the charge density is obtained. This process is reiterated until the ground state charge density of desired accuracy is obtained. The iterative process of self-consistently solving the one-electron Kohn-Sham equation to obtain the charge density of the ground state is illustrated in the flow diagram illustration of FIG.2.1. The energy of the fictitious system can be expressed in terms of the eigenenergies as i o E k [ n(r) ] + r r r n( r)veff(r)d = 2 i= N 1 ε (2. 10) i From Eq. (2.10) the kinetic energy o E k [ n(r) ] corresponding to the ground state is obtained. After this, using Eq. (2.5) ground state total energy of the system with interacting electrons can be obtained. The difficulty in this process comes down to finding a suitable description for the nonlocal exchange-correlation energy E xc as a functional of n(r) in Eq. (2.7). For this reason, 22

42 additional approximations such as the local density approximation (LDA), generalized gradient approximation (GGA) have been introduced. 23

43 FIG Flow chart [7] illustrating the iterative scheme for the ground state electron density from density functional theory using Kohn-Sham approximation [6]. 24

44 2.2.2 Exchange-correlation energy Different approximations are made to define the exchange-correlation energy as a functional of charge density. Most commonly followed approximations are explained here. In the framework of local density approximation [8] (LDA), the exchangecorrelation energy is assumed to be a functional of local electron density. The functional is then expressed as: [ E xc r r r r n(r) ] = n(r) (n(r))dr. (2. 11) ε xc r where ε (n(r)) is the exchange and correlation energy per electron at a point r due to the xc electron density n(r) at r. A common approximation is to employ the exchange- r correlation energy ε (n(r)) as a function of uniform charge density in place of non- xc uniform density, as the former can be calculated, with high accuracy, from jellium models [9]. LDA has been demonstrated successful when employed to solids. The reason for this is that the most important effect of exchange and correlation is formation of an exchange-correlation hole near each electron that contains a positive charge equal in magnitude to that of an electron. The shape of the hole in general is not spherically symmetric, as it depends on the surrounding environment. In the case of uniform electron gas approximation, though the shape is assumed symmetric, the charge is still equal to that of an electron with a positive sign, explaining the suitability of LDA. 25

45 The definition of the exchange and correlation energy functional in generalized gradient approximation[10] (GGA) accounts for the gradient of charge density in addition to the charge density at r and is termed generalized gradient approximation, expressed as [10]: [ E xc r r r r n(r) ] = f(n(r), n(r))dr. (2. 12) This approximation is more accurate in its description compared to local density approximation from the shape of the electron-correlation holes that change from the influence of the associated electrons taken into account by means of the effect at r of the changing electron density surrounding it. Mainly the distinguishing features of various methods of total energy calculations are the choice of basis function used to reproduce wave function from the periodicity of the crystal and crystal potential V ext (r) that describes the interactions between the nuclei and the electrons. The choice is made depending on the type of the system being studied, such as simple metals, transition metals, semiconductors etc. There are different types of crystal potentials such as full, muffin-tin, pseudo-potentials etc.[11] Similarly there are different types of basis functions of which the commonly used are: (i) plane-wave like which are de-localized basis functions (ii) atomic-like that are localized in the vicinity of individual atoms. The atomic potential due to the electrons outside the core is replaced by pseudopotential [12] that considers the inner electrons frozen. The valence electrons move in the field of the pseudo-potential. Pseudo-potentials are used in conjunction with plane-wave type basis functions. There are different kinds of plane-waves [13] such as 26

46 orthogonalized plane waves, augmented plane waves, projector-augmented plane waves (PAW) [14] etc. The PAW plane waves have been proven to be best suitable basis functions for metals [1]. 2.3 Supercell method Direct supercell approach [3] is a method for calculating phonon frequencies from full real space inter-atomic force constant matrices. It is based on the observation that limited set of planar force constants, which can be easily and accurately determined using supercells, for some high symmetry directions can be used to construct the full three dimensional force constant matrix, also called the dynamical matrix. The method has an advantage over other methods of phonon frequency calculation by its ability to study the anharmonic effects. The planar force constants are obtained by evaluating the Hellmann-Feynman [15,16] forces from the change in the total energy, in the presence of displacement of an equivalent plane of atoms. Within harmonic approximation, the force on atom α in the nth layer, F α (n), is defined as the product of the collective displacement of atom β of the mth layer u β (m), and the planar force constant matrix λ αβ (n) for the chosen direction: α αβ β F ( n) = λ ( n m) u ( m) (2. 13) m, β 27

47 The planar force constant matrix can be mapped from projected force constants for transverse and longitudinal displacements. From this matrix, the dynamical matrix can be obtained as αβ αβ λ ( n) = D ( R) (2. 14) R, eˆ ( R+ τ αβ ) = d n where ê is the unit vector normal to the atomic layer, d n is the distance between the displaced layer and the layer where force is being evaluated, and τ αβ = τ α - τ β is the vector connecting atoms α and β in the basis. Repeating the procedure for other high symmetry directions, linear set of equations for the dynamical matrix is obtained. The crystal symmetry reduces the number of independent variables in the dynamical matrix and thus the full dynamical matrix can be obtained from knowing sufficient planar force constant matrices. Summation over R is limited to a sphere of radius R max, which is carefully chosen from convergence study of the planar force constants. The dynamical matrix is solved as an eigen-value problem; eigen values resulting are the phonon frequencies. 2.4 Nudged elastic band method Nudged elastic band (NEB) method is an improved method to the elastic band method for finding the minimum energy path (MEP) between the initial and final state of a transition [17]. Configurations with diffusing atom at intermediate positions along the transition path called images are considered connected by springs, the forces due to which hold the images together like a band, in elastic band theory. NEB method includes 28

48 between images only the component of the spring force parallel to the tangent and only the perpendicular component of the true forces while relaxing the system for the MEP. This force projection referred to as nudging is more efficient in (i) not allowing interference of the spring forces in relaxation of the elastic band to the MEP (ii) ensuring the true forces against distorting the sampling of the images to give the right saddle point. This is attained by maintaining equivalent distances between the images. In implementing the method, N+2 images are considered, where N images are from intermediate states along the transition path and the other two are the initial and final states of migration. At first the tangent at an image is estimated from bisecting the position vectors of its two adjacent images, ensuring equidistance between the images. The force acting on the image is then calculated from the sum of the spring force parallel to the tangent and true force perpendicular to the tangent. A minimization algorithm is applied, moving the images in the direction to minimize the forces, to give a final relaxed energy path, as illustrated in FIG Kinks could form in the energy path between the images, but typically the estimation of the saddle point region and its energy and not affected [17] by them. The method is widely used for estimating transition rates [18] within the harmonic transition state theory (TST) in conjunction with electronic structure calculations, in particular in the plane wave based density-functional theory as implemented in VASP. 29

49 FIG Schematic illustrating the minimum energy path on the potential energy diagram between two equilibrium positions (solid lines) [17]. Solid dots represent the images positions and dotted line the minimum energy path obtained from NEB relaxation. 30

50 2.5 Implementation In this section, the details of the system setup parameters in the input files needed for using the softwares VASP and ATAT to obtain the ground state energy and phonon frequencies are presented. Further, the procedure followed in implementing the methodology presented in section 2.1 is put forth. From the knowledge of the crystal structure of the material, the position coordinates in a unit cell are known. Other main input parameters to VASP are the accuracy defining parameters such as energy cut-off and size of the k-point mesh size, the extent of relaxation combinations of volume, shape, position changes and the minimization technique. At first, tests are conducted to decide on suitable system size for predicting the properties of interest to the desired accuracy. Supercells of different number of lattice sites are built from unit cells stacked in three dimensions and the convergence tested. Testing for the size of the supercell is specifically important in case when there are defects in the system, so that periodic repetition of the supercell does not increase the defect concentration significantly from interactions between them, causing deviation from the equilibrium value of defects from experiments. Similar tests are conducted to check convergence of all the desired properties such as entropy of vacancy formation, migration barrier etc. for different energy cut off values and k-point mesh size individually. The three individual configurations are then relaxed with respect to all their internal degrees of freedom to get accurate ground state energy. For the relaxed configuration of the transition state, relaxation is conducted via the NEB method. 31

51 Same settings of k-point and energy cut-off are used for phonon frequency calculations. Using ATAT, a volume with no strain imparted to the relaxed equilibrium configuration at 0K is used for HA calculation, or several volumes each with certain incremented strain level on the equilibrium volume from ground state is setup for QHA calculation. For QHA, the number of volumes and the amount of strain on each volume is guided by the thermal expansion coefficient of the material. A constant volume relaxation is performed on each of the volumes. Then, the amount of perturbation to be given to the atom is to be decided. The perturbation should not be so small that it would be overwhelmed by the noise factor and at the same time should not be too large that it would break the bonds. Convergence of the phonon density of states (PDOS) is checked to find suitable perturbation magnitude. A static relaxation (no ionic relaxation is allowed) is performed on the perturbed configurations generated by ATAT for each of the volumes, to get the forces on the atoms. For generating the dynamical matrix, as described by the supercell method in section 2.3, the cutoff radius R max needs to be input. This is usually taken to be half the dimension of the supercell used for the phonon calculations. (Note: The supercell size for vibrational calculations needs to be tested for convergence of the force constants to get a good estimate for R max. Due to insufficient computational resources, in the present work the same supercell size as that used for ground state relaxation is used.) Phonon frequencies obtained from harmonic approximation are directly used along with the ground state energy to give the free energy of the system, following Eq. (1.11). Phonon density of states of each volume from the quasi-harmonic approximation, are similarly used to calculate free energies corresponding to that volume. Using the data 32

52 of free energy at different volumes, the equation of state (EOS) below is solved for the fitting parameters at a specific temperature. 1/ 3 2 / 3 1 E ( V ) = a + bv + cv + dv (2. 15) Using these parameters, the equilibrium volume can be calculated using the following equation [19]: V e c 9bcd + ( c 3bd )(4c 3bd ) = (2. 16) 3 b Equilibrium volume data at different temperatures can be used within the EOS to yield free energies as a function of temperature within the quasi-harmonic approximation. 2.6 Conclusion In this chapter, we have seen how the quantities required for obtaining diffusion coefficients are calculated by: (i) first showing how to set up the system to closely resemble a realistic system size (ii) and then detailing on the implementation procedure following varied approximations. It is also explained how the charge interactions in the system are treated within the first-principles calculations and how supercell method gives the accurate phonon frequencies and nudged elastic band method the right saddle configuration. To summarize the pros and cons of the procedure followed in this work: (i) Pros (a) the procedures follows a very accurate approach for ground state calculation, its energy at 0K and calculation of vibrational properties (b) this accuracy is obtained within affordable computational expense (c) the procedure is very straightforward in implementation using VASP and ATAT. (ii) Cons (a) suitable only for metallic systems 33

53 (b) in the case higher defect concentration, larger supercell sizes are needed which will need huge computational power. Bibiliography [1] G. Kresse and J. Furthmuller, Comp. Mat. Sci. 6, 15 (1996). [2] H. Jonsson, G. Mills, and K. W. Jacobsen, in Classical and Quantum Dynamics in Condensed Phase Simulations, edited by B. J. Berne, Cicotti, G., Coker, D.F. (World Scientific, 1998). [3] S. Wei and M. Y. Chou, Phys. Rev. Lett. 69, 2799 (1992). [4] A. Van de Walle, M. Asta, and G. Ceder, CALPHAD 26, 539 (2002). [5] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). [6] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). [7] V. Vitek, (University of Pennsylvania, 2003). [8] D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980). [9] C. Fiolhais and J. P. Perdew, Phys. Rev. B 45, 6207 (1992). [10] J. P. Perdew, K. Burke, and Y. Wang, Phys. Rev. B 54, (1996). [11] R. M. Martin, Electronic structure: Basis theory and practical methods, cambridge, 2004). [12] D. Vanderbilt, Phys. Rev. B 41, 7892 (1990). [13] G. Kresse and J. Furthmuller, Phys. Rev. B 54, (1996). [14] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999). [15] H. Hellmann, Einfuhrung in die Quantumchemi, 1937). [16] P. R. Feynman, Phys. Rev. 56, 340 (1939). [17] G. Henkelman and H. Jonsson, J. Chem. Phys. 113, 9978 (2000). [18] G. Mills, H. Jonsson, and G. K. Schenter, Surf. Sci. 324, 305 (1995). [19] S. Shang and A. J. Bottger, Act. Mater. 53, 255 (2005). 34

54 3 SELF-DIFFUSION IN FCC METALS 3.1 Introduction In this chapter, at first, the theory underlying the equations defining self-diffusion is presented. Equations for diffusion via di-vacancy mechanism derived based on those from mono-vacancy mechanism are put forth. Then the setup parameters for the calculation of self-diffusion coefficient in specific systems are presented. In this work self-diffusion in pure fcc Al, Cu, Ni and Ag elements are dealt. Determination of the atom jump frequency constituent of the diffusion equation is done following different theories in literature and the results compared. Individual quantities involved in diffusion equation along with the resulting self-diffusion coefficient obtained as a function of temperature are compared to experimental and other theoretical predictions. Finally, conclusions on the cause for curvature in the diffusion plot of fcc systems being either from the anharmonic effects of the diffusion parameters D 0 diffusion pre-factor and Q activation energy or from di-vacancies contributions at higher temperatures is discussed. 3.2 Theory In fcc systems, diffusion can occur by both mono-vacancy and di-vacancy mechanisms, with di-vacancy mechanism making a dominating contribution at high temperatures (temperatures close to melting) [1]. Di-vacancies i.e. two vacancies that are 35

55 adjacent to each other, stabilize at high temperatures due to higher vacancy-vacancy binding. Diffusion via this mechanism occurs by jumps that leave the two vacancies bound to each other. Total diffusion in the system is then given as the sum of via both mechanisms [1]: where D 2 = D1 v + D v (3. 1) D 1 v is the diffusion due to mono-vacancy jumps and D 2 v is the diffusion that occurs as a result of jumps involving di-vacancies. Application of the diffusion equation described in Eq. (1.6) to describe diffusion by the two mechanisms is explicated in the coming two sub-sections Mono-vacancy mechanism Diffusion by mono-vacancy mechanism in a pure element involves only one jump frequency for any system configuration and hence self-diffusion equation by such mechanism is described as: 1 = (3. 2) 6 2 D1 v z1 vc1v w1 v r f1 v For face centred cubic (fcc) structured elements, z 1, the number of possibilities of v formation of a vacancy surrounding the tracer atom, is equal to 12 and the jump distance r = a 2 where a is the lattice parameter. From this, self-diffusion in fcc is described by the equation: 2 D v f1 va C1 vw1 v 1 = (3. 3) 36

56 As mono-vacancy is the common mechanism for diffusion in cubic crystals, most of the times in the literature subscripts ( 1v used here) are not used to represent the quantities involved in describing diffusion by this mechanism (Eq. (3.3)). Similar representation is followed in this thesis. Also, wherever reference to mono-vacancy is being made it is usually referred to as vacancy. Sub-sections below annotate the definitions of the quantities that need to be determined: correlation factor f, vacancy concentration C and atom jump frequency w Correlation factor In the case of random walk, where the jumps are uncorrelated, correlation factor f =1. Mean square displacement from N jumps in an observation 2 X is expressed as [2] ( x + x + x + x ) 2 2 X =... N (3. 4) For the case when the jumps are uncorrelated the mean square displacement reduces to X 2 random = N i= 1 x 2 i From Eq. (1.5) and the above two equations, we see that correlation factor f is (3. 5) f N 1 N i= 1 j= i+ 1 N = x x x (3. 6) i j i= 1 2 i In the numerator of the second term on the right side of Eq. (3.6), the product of two different jump vectors includes cosine of the angle between the two jump directions. In the case of uncorrelated motion the value of the numerator is zero as all jump directions are equally probable and cosine of the angles between different jump vectors 37

57 cancel each other, causing f =1. In the case of correlated motion, the numerator guided by symmetry is a negative value, with the value of the fraction being less than one. This sets the minimum for the correlation factor to be zero. More correlated the jumps in a system are, more negative is the value of the numerator and lower is the value of f. Correlation factor is a function of the structure of the system and the mechanism of diffusion. So, for known diffusion mechanism and crystal structure of pure elements, the value of correlation factor can be determined based on the symmetry. In the case of cubic crystals that have two or threefold symmetry along their jump directions, Eq. (3.6) reduces to [3] f = 1+ 1 cosθ cosθ (3. 7) where θ is the angle between any two successive jumps. Compaan and Haven [4] have evaluated the values of cos θ for diffusion by vacancy mechanism in fcc structure among other crystal structures and obtained the correlation factor to be equal to Equilibrium vacancy concentration A vacancy is formed when an atom escapes to the surface from the bulk of the crystal. The energy required for removing an atom in the bulk H change that the system undergoes as a result of vacancy formation f and the entropy S f together yield the free energy change on vacancy formation G f through G f = H f T S f. The probability of formation of vacancy adjacent to a tracer atom describes the concentration of vacancies for tracer jump: 38

58 C = S k B f H f exp k BT exp (3. 8) In Eq. (3.8), the entropy includes both vibrational (or thermal) and configurational (mixing) entropies of vacancy formation, v S f and c S f, respectively. In a system with N+n lattice sites with n vacancies and N atoms, configurational entropy is [2] S c f = k B N n N ln + n ln (3. 9) N + n N + n From this, the free energy of formation of n vacancies is n G f v N n = n( H f T S f ) + k BT N ln + n ln (3. 10) N + n N + n Equilibrium vacancy concentration is obtained from equating to zero the derivative of free energy of vacancy concentration of the system with respect to n, which gives the following equation: v n S f H f = exp exp (3. 11) N + n k B k BT As n is very small compared to N, the above equation can be equated to n / N from which the equilibrium vacancy concentration is C eq v S f H f = exp exp (3. 12) k B k BT In the case of pure elements where configurational entropy is negligible, equivalent. Ceq and C are Atom jump frequency 39

59 Atom jump frequency definition, as explained in section 1.2, is derived from Eyring s reaction rate theory [5]. The absolute reaction rate theory assumes the transition state to be in thermodynamic equilibrium with the equilibrium state, based on which the atom jump frequency w is defined as [6] w Z TS = ν (3. 13) Z IS where ν is the frequency of vibration in the direction of diffusion and Z TS is the partition function of the transition state and Z IS that of the initial equilibrium state (this state which is also the state of the system with a vacancy and all atoms in their equilibrium positions is referred to as initial state (IS) in this document), both with contributions from their 3N-6 vibrational degrees of freedom (DOF), for a system with N atoms. It is well known that the transition state has an unstable mode with respect to vibration in the direction of the vacant site. This frequency is imaginary from the negative force constant due to the downward curvature of the energy diagram in the direction of diffusion. In an attempt to facilitate the evaluation of the rate constant [6] Glasstone took the following steps: (i) the partition function of the transition state Z is TS written as a product of partition function from the unstable phonon mode Z ~ TS and the partition function from the remaining (3N-7) vibrational modes Z (ii) the frequency of vibration in the diffusion direction ν is approximated to be equal to the frequency of the diffusing atom when at the saddle point * TS ~ ν TS. From the harmonic definition of the partition function, Z ~ TS is expressed as BT h TS k ν ~, resulting in the equation of rate constant as: 40

60 w k T h Z Z * = B TS (3. 14) IS In the above paragraph, the implementation of (i), i.e. splitting partition function of a state as a product of partition functions of partial contributions, assumes that the vibrational modes are independent i.e. the fact that the vibrations are actually due to coupled motions of the atoms is not considered. From this assumption it follows that the vibrational frequency of the system in diffusion direction is equal to that of the diffusing atom vibrating in the diffusion direction independently of the rest. This forms the basis for the application of (ii). Wert and Zener [7] realized that in order to define the difference of thermodynamic properties between the transition state and the initial state, with the two states in reversible equilibrium, must have properties defined from contributions of equal and similar degrees of freedom. Hence, the partition function of the initial state was split as Z = Z ~ Z, separating the partition function of the vibration mode in the diffusion IS IS IS direction Z ~ IS from the rest. Again, following harmonic approximation this partition function is expressed in terms of the vibrational frequency of the diffusing atom in its equilibrium position in the diffusion direction vibrational frequency) as [6] ~ ν IS (also referred to as characteristic Z ~ = k T h ~ ν (3. 15) IS B IS From substituting Eq. (3.15) in Eq. (3.14) and from the definition of partition function in terms of the free energy Z = exp( G k T ) [8], the definition for jump frequency given by Wert and Zener (Eq. (1.8)) is obtained, which is represented here as: B 41

61 ~ * IS TS IS w =ν exp( ( G G ) k T ) (3. 16) B Di-vacancy mechanism For the case of fcc structure, it has been found [9] that it is enough to consider two vacancies to be bound if they are first or second nearest neighbors to each other, as their interaction is weak for the third neighbor which is farther away from the second. Therefore considering two vacancies that are first or second nearest neighbors of each other, before and after the jump, there are four possible jumps: (i) when two vacancies are initially adjacent to each other: (a) jumps that cause them to remain first-nearest neighbors- w 11 (b) jumps that cause them to be second nearest neighbors- w 12 (ii) when two vacancies are second nearest neighbors initially: (a) jumps that cause them to remain second nearest neighbors- w 22 (b) jump that leaves the two vacancies adjacent to each other- w

62 w 12 w 21 w 11 FIG Figure illustrating di-vacancy jumps in pure fcc system [1]. In an fcc structure, for any two lattice sites there are four nearest neighboring sites that are common to the two. Hence jumps of type w 11 are possible. In this structure, jumps of type w 12 and w21 are also possible. But as jumps occur only to first nearest neighboring sites, jumps of type w 22 are not possible. From these different kinds of jumps contributing to diffusion by di-vacancy mechanism, diffusion equation by this mechanism is given as [1]: D 1 2 1n 1n 2n = f 2va ( z11c2vw11 + z12c2vw12 z21c2v 21) (3. 17) 12 2v + w where n C 1 2 v is the concentration of di-vacancies of first nearest neighbors and n C 2 2 v is the concentration of di-vacancies of second nearest neighbors and f 2 v is the correlation factor for diffusion via di-vacancies. Representation similar to that used for the different 43

63 jump frequencies is followed for z. Their values for the different jumps are discussed in the sections below. Modification in the definition of vacancy concentration of divacancies is presented in the immediately following sub-section taking the case of jump w11 as an example. The equations resulting from applying this to other jumps are presented in the next sub-section Total jump frequency of jump w 11 1n Total jump frequency Γ 11 of jump w 11 is Γ 11 = z11c2v w11. For the jump w 11, the two vacancies have to be nearest neighbors to each other (1n di-vacancy) and also to the tracer atom, so that after jump by tracer the vacancies are still first nearest neighbors. For this jump, the number of ways in which di-vacancy can be formed surrounding the tracer atom z 11 = 24. (Note: In the document nearest neighbor stands for first nearest neighbor) This is because, for a site occupied by the tracer and a vacancy in one of its nearest neighboring sites there are four nearest neighboring sites that are common to both that the second vacancy could occupy. Like this, for each of the 12 nearest neighbors of the tracer atom there are 4 different possibilities of a di-vacancy, giving a total of 48, and thus a net of 24 unique possibilities after eliminating the ones from double-counting. For divacancy in any of the 24 cases, there are 2 different jumps of the tracer that leave the two vacancies adjacent to each other. Hence there are a total of 48 possibilities for jump w 11. Therefore Γ 11 can be written as: Γ (3. 18) 1 11 = 48C n 2vw11 44

64 Vacancy concentration as given by Eq. (3.8) is defined as the Boltzmann probability of the energy required to form a vacancy. Similarly, the di-vacancy concentration C 1 is defined in terms of the formation energy of 1n di-vacancy n 2 G 1 as: n v f C 1n 2v 1n G f = exp( ) (3. 19) k T B This has been expressed by Howard and Lidiard [10] as: C 1n 2v 1v 2v 2 G f Gb = exp( ) (3. 20) k T B where v G 1 f is the free energy of mono-vacancy formation and v G 2 b is the 1n di-vacancy binding energy. The equation implies that = n G 1 f 2 1v 2v G f Gb i.e. the free energy for formation of two vacancies in the system v 2 G 1 f minus the energy gained when these two vacancies come adjacent to each other, in the case of 1n di-vacancy, gives the free v G 2 b energy of formation of 1n di-vacancy. Energy of formation of two vacancies in a system is said to be equal to twice the energy of formation of a single vacancy, when the two vacancies are non-interacting. This can be understood by first considering the case of formation of a single vacancy in a perfect system, as done for mono-vacancy. In such case energy of formation of the first vacancy 1v(1) G f is given by where 1v(1) 1v N 1 G f = GIS GPS (3. 21) N v G 1 IS is the free energy of initial state of the system with one vacancy and GPS is the free energy of the perfect state. Energy for formation of another vacancy, in the system with a vacancy, 1v(2) G f is given by 45

65 where G 1v(2) f v v G 1 IS = G 1v+ 1v IS ( G 1v IS 1 G N PS ) (3. 22) 1 + is the energy of the system with two non-interacting vacancies and 1v 1 ( GIS GPS ) is the energy of the system before formation of the second vacancy with N contribution from the same number of atoms as that after formation of the second vacancy. Sum of the energies in Eq. (3.20) and Eq. (3.21) gives the energy for formation of two non-interacting vacancies in a system. G 1v+ 1v 1v+ 1v N f = GIS 2 G N PS Energy of the system with two non-interacting vacancies can be expressed as: (3. 23) v G 1 IS G = 2 G G (3. 24) 1v+ 1v 1v IS IS PS 2 is the energy of system with 2N-2 atoms and 2 non-interacting vacancies, assuming the system (N) is big enough and the energy is scalable. Subtracting the energy of N atoms vacancies G PS from this gives the energy of system with N-2 atoms and 2 non-interacting v v G 1 IS 1 +. Substituting Eq. (3.24) in Eq. (3.23) gives: Thus G G 1v+ 1v f 1v+ 1v f = 2( G = 2 G 1v IS 1v(1) f N 1 G N PS ) (3. 25). From the above understanding along with knowing that 2v 2v 1v+ 1v G = ( G G ) we can also obtain the definition for 1n di-vacancy concentration b to be: IS IS C 1n 2v 2v N 2 GIS GPS = exp( N ) (3. 26) k T B 46

66 This equation is very similar to the case of mono-vacancy and can be directly used for obtaining 1n di-vacancy concentration from the free energy of the initial state with 1n divacancy and free energy of the perfect state. Jump frequency is given in similar form as for mono-vacancy: w 11 * ' 2v 2v ~ GTS GIS =ν 11 exp( ) (3. 27) k T B Total jump frequency of w and 12 w 21 In the case of jump w 12, the two vacancies are nearest neighbors to each other but only one of them is adjacent to the tracer atom. The second vacancy should be such that after jump by the tracer the two vacancies are second nearest neighbors to each other. For such a jump to occur there are 12 possible ways in which di-vacancy can be formed adjacent to the tracer atom. Each of the 12 nearest neighboring positions has 2 positions where the second vacancy can be formed such that it is second nearest neighbor to the initial tracer position. As there exist no possibilities of double-counting in this case, a total of 24 unique possibilities exist for forming a di-vacancy suitable for this jump i.e. z 12 = 24. There is only one possibility for the tracer jump as only one of the vacancies is adjacent to it. Hence the overall jump frequency is then given by Γ (3. 28) 1 12 = 24C n 2v w12 For jump w 21, a 2n di-vacancy is initially present such that the two vacancies are nearest neighbors to the tracer atom. The number of different possibilities of di-vacancy formation for the case of this jump is 12 times 7. After accounting for double-counting 47

67 we have z 21 = 42. The two vacancies being nearest neighboring to the tracer atom, there are again 2 possibilities of jump for each possibility of di-vacancy. Hence a total of 84 possibilities exist for tracer jump by di-vacancy mechanism according to this jump, giving the total jump frequency defined as: Γ (3. 29) 2 21 = 84C n 2v w Diffusion equation Definitions from the previous sections put together in Eq. (3.17) gives: D 1 2 1n 1n 2n = f2va (48C2v w C2vw12 84C2v 21) (3. 30) 12 2v + w Knowing the definitions of the equilibrium defect concentration and jump frequency, we can see that, 1n 2n C2v w12 = C2v w21, (3. 31) as the two jumps being reverse jumps of each other have the same transition state. Substituting this in Eq. (3.30) we have: D 2v 2v 2v ( 11 + w12 2 1n = f a C 4w 9 ) (3. 32) Correlation factor for di-vacancy mechanism has been plotted as a function of the ratio of the jumps w 12 and w21 by Mehrer [11]. Contribution from di-vacancy mechanism to diffusion obtained from Eq. (3.32) is added to the mono-vacancy diffusion to describe the total diffusion in the system. 3.3 Present work 48

68 For the first-principles calculations, pseudo-potentials in conjunction with projector augmented wave [12] (PAW) function and the local density approximation [13] (LDA) for the exchange-correlation are used. Tests indicate that a Monkhorst-Pack [14] k-point mesh size of 11x11x11 and an energy cutoff of 300 ev are suitable to yield migration barriers converged within 0.01eV compared to experiments, for the systems fcc Al, and fcc Cu. Similar convergence of the energetics is obtained, when calculated using supercells with 32 and 64 lattice sites, for fcc Al. Hence, a supercell with 32 lattice sites (2x2x2 conventional fcc cells) was employed. Similar settings are also applied to Ag and Ni, with the energy cut-off set to be about 1.3 times the energy maximum listed in the potential file of the element, as it is claimed to be a pretty good upper limit value by the makers of VASP. With this, the set up of two files: KPOINTS file (with k-point mesh size) and POSCAR file (with position coordinates of atoms in the supercell), for first-principles calculation with VASP is complete. For an element there could exist more than one potential file depending on the valence electrons considered interacting i.e. not part of the frozen core. From the recommendation in the manual, the potential file POTCAR for a specific element is chosen. For Al and Cu both PAW GGA and PAW LDA have been used and the results compared. Based on the results, for Ni and Ag only PAW LDA has been chosen. Unless otherwise mentioned all calculations are completely relaxed with respect to internal coordinates, volume and shape VASP setting ISIF=3. Conjugate gradient algorithm is chosen for the relaxations (IBRION=2), as it is a good relaxation technique for difficult relaxation problems, as suggested in the VASP manual. The precision of the 49

69 calculation is set to accurate. The maximum number of ionic steps can be set using the parameter NSW. Other parameters are set at values suggested in the manual for a specific system. For example, the manual suggests different smearing values (ISMEAR) for different elements. The format of INCAR, KPOINTS, POSCAR files for a sample calculation are posted in APPENDIX. To quantitatively determine the transition state, the nudged elastic band (NEB) method [15] is used. Full relaxation of the system configuration before and after the jump is performed. These relaxed configurations are used for setting up a desired number of images (image is the system configuration with the diffusing atom at different positions between its initial and final positions) along the diffusion path for determining the minimum energy path. The directory setup for conducting a NEB relaxation is followed from the description in the VASP manual. The additional parameters settings in the INCAR file such as the spring constant (ISPRING) are also taken from the manual. A sample INCAR file for NEB relaxation is also shown in APPENDIX. It is very important to note that for obtaining a completely relaxed configuration, full relaxation using (ISIF=3) with either regular VASP relaxation or using NEB is done twice. Explicitly, CONTCAR which is the file with relaxed configuration, after convergence of full relaxation for the first time, is copied for the POSCAR file and the full relaxation process repeated. This ensures that the obtained configuration is completely relaxed. Also, for the ground state energy a static calculation is to be performed with the completely relaxed configuration as POSCAR. Static calculation allows no relaxation (NSW=0), and also the parameter determining how the partial occupancies are set for each wavefunction, ISMEAR, is set to -5 to use the tetrahedron 50

70 method with Blochl corrections. This setting yields accurate total energy for bulk materials. The normal phonon frequencies are calculated using the direct force-constant approach [16] as implemented in the Alloy Theoretic Automated Toolkit (ATAT) [17] package. Similar energy cut-off, k-point mesh size and supercell size used for the total energies are used for the vibrational calculations. To maintain the same supercell size as that of the relaxed configuration, the parameter er used to define the size of supercell generated by fitfc module of ATAT, should be set to a value equal to the smallest dimension of the relaxed supercell. The perturbation size from convergence tests has been found to be 0.05Å for Al with lattice parameter 4.05Å. In the case of harmonic approximation, the perturbed configurations generated of the equilibrium configuration of 0K are relaxed with NSW=0 (no movement of atoms; only electrons relaxed) and ISMEAR=1 to get the right forces on the atoms. Also, for phonon calculations, to obtain precise force constants, EDIFF is set to 1e-6. For fitting the force constants the cut-off radius is chosen to be slightly more than half the dimension of the supercell i.e. in the case of Al 2x2x2 supercell with dimensions of er about 8.1 Å, the value is chosen to be about 4.5 Å slightly greater than er/2 but less than the next higher nearest neighboring distance to er/2. Phonon density of states (PDOS) obtained as a result of such calculation is illustrated in FIG PDOS from the three configurations are also compared in this figure. The phonon frequencies and the ground state energies are thus obtained for the three configurations (i) perfect state system without vacancy; properties of this state are represented with subscript PS (ii) initial state all atoms in equilibrium and a vacant site; 51

71 subscript IS is used to indicate properties of this state (iii) transition state atom at the saddle point during its motion to the vacancy, represented by subscript TS. For including temperature dependence due to volume expansion, strained configurations of the fully relaxed configurations are generated by fitfc with the input of the number of additional configurations (ns) and the strain on each (ms). A sample command is included in APPENDIX. Constant volume relaxation of each strained configuration ( E c (V ) ) followed by static relaxation of perturbed configurations of each volume, when fit for force constants yields phonon frequencies ( ν ( q, V ) ) that include the effect of volume expansion. Using equations similar to those listed in section 1.3, enthalpy and entropy within QHA are obtained from: i H ( V, T ) = E ( V ) c h hν i( q, V ) ( q, V )coth 2k T ν (3. 33) i q m B hν i( q, V ) hν i ( q, V ) hν i( q, V ) S( V, T ) = k B coth ln 2sinh (3. 34) q m 2kBT 2kBT 2kBT In these equations m represents the stable vibrational degrees of freedom. This implies that the high negative frequency, as output by ATAT for the unstable phonon mode, is eliminated within the summation. 52

72 FIG Figure illustrating the distribution of the phonon frequencies. The negative frequencies in PDOS of TS are due to the unstable mode. The high peak at low frequencies for the IS are from the weak bonds surrounding the vacancy. The magnitude of low frequencies is high for TS due to weakening of several bonds when the diffusing atom moves to the saddle point. The distribution is smeared from the different neighbors being affected to a different extent. The diffusing atom actually gets closer to some of its neighbors. This is shown by the small peak of high frequencies for this state. 53

73 From the definitions of vacancy concentration Eq. (3.8) and jump frequency Eq. (1.8) in the previous section, self-diffusion equation Eq. (3.3) can be represented as: D = S + S fa ~ f exp k B m H f + H exp k BT 2 ν (3. 35) m The enthalpy and entropy of vacancy formation can be defined as [18]: N 1 H f = H IS H N N 1 S f = S IS S PS N PS (3. 36) By taking (N-1)/N fraction of the properties of the perfect state with N atoms, for the difference, it is ensured that the degrees of freedom included for contribution for the perfect state (3N-3) is equal to that for the initial state with N-1 atoms i.e. (3(N-1)). For clarity of representation 3N-6 vibrational degrees of freedom of a system with N atoms is represented in the above sentence as 3N. For calculating the enthalpy and entropy of migration, one needs to have enthalpy and entropy of the initial and transition states from equal degrees of freedom. But the results from Eq. (3.31) and Eq. (3.32) for the enthalpy and entropy of the initial and transition states do not have equal degrees of freedom. How this problem is dealt is presented in the next section. Besides, equations (3.31)-(3.32) do not include the contributions from electronic density of states [19]. For fcc systems dealt in this work these contributions have been found to be negligible. Equations for electronic enthalpy and entropy are presented in Chapter 6 describing diffusion in bcc systems where their contributions are significant. Also, among the fcc systems dealt in this work, significant contribution from di-vacancies at temperatures above ( 2 /3)Tm is observed experimentally 54

74 only for fcc Ag. Hence attempt to predict diffusion coefficient via di-vacancy mechanism is conducted only for fcc Ag. 3.4 Results and discussion In this section the results of this work are presented, not only in terms of data obtained but also from the perspective of novel procedure formulated. The procedure is explained and the results are compared with other works, taking aluminum as the working example. Then the results from the procedure applied to other systems: Cu, Ni and Ag is demonstrated. Other aspects of physical understanding are put forth and finally conclusions are drawn FCC Aluminum For the case of aluminum, results using both the local density approximation (LDA) and the generalized gradient approximation (GGA) for the exchange correlation are examined and compared. Within each approximation, calculations for both harmonic and quasi-harmonic approximations are conducted, to elucidate the effects of thermal expansion on the various factors entering into the diffusion coefficient Equilibrium vacancy concentration Upon creating a vacancy, a small amount of effective surface is created in the material. The commonly-used GGA and LDA functionals underestimate this surface 55

75 energy, and a method has recently been demonstrated [20] to compute the corresponding correction terms. The calculated surface correction terms of 0.15 ev for the vacancy formation energy and 0.05 ev for the migration energy have been determined for Al within the GGA [20, 21]. Similarly, correction terms of 0.06 ev and 0.02 ev [20, 21] respectively, have been determined for the vacancy formation and migration energy calculated within the LDA. These corrections are added to the enthalpy and entropy of formation results from first-principles for harmonic and quasi-harmonic approximations. The temperature dependence of the errors is very small. Hence the same value is used for all temperatures and within both approximations. From the obtained enthalpy and entropy of vacancy formation results, using Eq. (3.12) equilibrium vacancy concentration of Al, as shown in FIG. 3.3, is obtained. The agreement with experiments is excellent, particularly for GGA calculations. The good match of the current simulation results of equilibrium vacancy concentration with experiments is a strong proof that the system size is sufficient to closely represent a big system used for experiments. 56

76 FIG Equilibrium vacancy concentration from first-principles (both LDA and GGA harmonic and quasi-harmonic calculations) plotted in comparison with experimental data [22-24]. The slightly different slope at higher temperatures of measured data was suggested by experimentalists [24] to be from contributions of di-vacancies. The result from the current work is from mono-vacancies only. The experimental results of Seeger [23] is data for only mono-vacancies. 57

77 Atom jump frequency As mentioned in the last paragraph of section 3.4.1, there exists a difficulty in calculating the enthalpy and entropy of migration, with the enthalpy and entropy of the IS and TS having contributions from unequal DOF. In the sub-sections below different ways of handling this issue to obtain atom jump frequency is presented Vineyard s TST (TST-V) Atom jump frequency definition in Eq. (1.9) can be written as: w = S exp k B H exp k BT ~ν m m (3. 37) In the right hand side of this equation, the product of the first two terms is an effective frequency given as [7]: = ~ S ν exp k B * m ν (3. 38) This effective frequency as defined by Vineyard [25], described in section 1.2, is the quotient of the product of the phonon frequencies ν i of the initial state to the product of phonon frequenciesν i of the transition state after ignoring the frequency of the unstable mode. From the fitfc module of ATAT using a single k-point (kp=1) for fitting the force constants, 3N phonon frequencies of the system are attained. From these eliminating the three small frequencies for the initial state and additionally the large negative frequency for the transition state, the effective frequency is obtained from the current work following the equation: 58

78 59 = = = * N i i N i i ν ν ν (3. 39) According to TST-V, that assumes harmonic approximation in the classical limit and ignores wave-vector dependence, Eq. (1.11) for free energy reduces to [26]: + = m B i B c T k h T k E T F ν ln ) ( (3. 40) From such an approximation, the enthalpy equation in Eq. (3.33) and entropy equation in Eq. (3.34) reduce to: T mk E T H B c + = ) ( (3. 41) = m B i B T ek h k T S ν ln ) ( (3. 42) Eq. (3.42) can also be written as: = m B i B T ek h k S ν exp. (3. 43) One can see how from this definition of entropy the definition from TST-V Eq. (1.10) is derived, from the following sequence of equations: = = = = + = * * ~ ' ~ ) ~ exp( ~ N i i N i i B m B i m B i IS TS h T k T k h T k h k S S S ν ν ν ν ν ν ν ν. Enthalpy of migration is the difference between the enthalpy of the initial and transition states with equal DOF. Hence from enthalpy of the transition and initial states defined following Eq. (3.41), enthalpy of migration is simply the difference of the ground state energy of the transition state to the initial state. The vibrational contributions to the

79 enthalpies cancel each other, as within the classical approximation and from ignoring wave-vector dependence, the vibrational contribution to the enthalpy of the transition and initial states from equal degrees of freedom is equal. Though entropy of migration cannot be still defined, as can be seen from Eq. (3.42), the alternative is to calculate the effective frequency ν *, which is sufficient to * obtain jump frequency w. Using Eq. (3.38), ν can be calculated, from the single k-point (Γ point) phonon frequencies, which can be obtained for both HA and QHA. Jump frequencies are thus calculated in a very straightforward manner, following Vineyard s TST for the first-time, in the current work. To obtain the individual quantities in Eq. (3.38) along with their temperature dependences yielding temperature dependent * ν, doublewell approach described below is proposed in the current work Double-well (DW) approach This approach for calculating jump frequency overcomes the shortcomings of the approach using Vineyard s TST, by using Eq. (3.33) and Eq. (3.34) for the enthalpy and entropy of the two states involved in migration. The vibrational contributions from the transition and initial state are not equal and thus more correct temperature dependences of the atom jump frequency are obtained. In addition ignoring wave vector dependence also yields inaccurate value of effective frequency from TST-V, which is improved in this approach. Thus this approach yields more accurate temperature dependences of atom jump frequency, obtained from calculation of temperature dependences of the individual properties of migration. 60

80 The problem of unequal number of stable vibrational DOF makes it difficult to define properties that consider equilibrium between the two states. Wert and Zener [7] eliminate contribution of the diffusing atom in its initial state in the diffusion direction from the full partition function to resolve this issue. Here we implement this concept, by calculating the partition function contribution along the diffusion direction and eliminating the enthalpy and entropy contribution of this mode from the results of initial state attained using Eq. (3.33) and Eq. (3.34). As described in section the frequency of this mode also gives the characteristic vibrational frequency. The potential of the vibration mode in the diffusion direction is a double-well function as illustrated in FIG To obtain the double-well potential from first-principles, system geometries with diffusing atom at intermediate positions along the diffusion path are relaxed with respect to volume and shape (atomic positions are not relaxed in order to avoid system decomposition to equilibrium positions). The potential thus obtained is scaled to the barrier height from NEB to compensate for the incomplete relaxation. The resulting double-well potential is identical to the potential diagram from NEB for the path between the two equilibrium positions, as can be seen in FIG The double-well potential along with the kinetic energy gives the Hamiltonian of this vibration mode. This equation is numerically solved to get the eigen-energies, ε i, a procedure followed in a previous work to treat phonon instabilities in bcc Zr [27]. Fortran code written for solving a Hamiltonian to give the eigen-energies is used. Eigen-value problem solvers for are freely available online that can be used for the purpose. Potential energy, mass of the diffusing atom and temperature range of interest are provided as input to the solver. From the eigen-energies Z ~ is obtained from the equation: 61

81 ~ = ε exp( k T Z i ) i B (3. 44) The enthalpy and entropy contributions H ~ and S ~ help define enthalpy and entropy of migration as: * ~ H m = H TS ( H IS H ) S = S ( S S ~ ) m * TS IS (3. 45) Characteristic vibrational frequency ν ~ is obtained from Z ~ using Eq. (3.15). As quantum effects of atomic vibrations are not taken into account in this work, the results are valid for temperatures above Debye temperatureθ D. The calculations also do not consider re-crossing effects, longer correlation lengths etc. that would be dominant at and above melting temperaturet M. Hence the results are valid upto temperatures close to melting. The values of the individual quantities calculated from this double-well approach for temperatures varying from K for Al are listed in Table 3.1 in comparison with data from other theoretical works and experiments. An excellent match with results from dynamic simulation methods [21] and also with experimental data is seen. In order to compare the difference in the predictions by the two approaches TST * and double-well, FIG. 3.4 illustrates the plots of effective jump frequency ν, enthalpy of migration (migration barrier) H m 3.4 (a) that (i) there exists a difference in the value of and atom jump frequency w. It can be seen from FIG. * ν from HA due to the ignored wave vector dependence in TST-V and (ii) the QHA result from TST is almost temperature independent similar to HA due to the classical limit approximation in deriving the quotient of the products formula (Eq. (3.37)). Similar influence of the 62

82 approximations in TST-V is evident from the results of enthalpy of migration in FIG. 3.4 (b). It should be noted that migration barrier within QHA from TST exhibits temperature dependence though the vibrational contributions are not included due to the inclusion of the effect of volume expansion on the configurational energy. It can also be seen that while temperature dependences from vibrational DOF are important within harmonic approximation, within quasi-harmonic approximation temperature dependences of potential / configurational energy are more dominant. FIG. 3.4 (c) clearly illustrates the effect on atom jump frequency and its temperature dependences, of using the double-well approach compared to TST, which is not very evident in the usual form where logarithm of jump frequency is plotted as a function of inverse temperature (FIG. 3.4 (d)). There is not data available to compare our jump frequency results. But, as the individual quantities tabulated in Table 3.1 and the resulting diffusion coefficients (FIG. 3.9) from double-well approach match well with experimental data compared to the diffusion results from TST, it can be said that the atom jump frequency result from double-well approach is more accurate. Comparing the results of harmonic and quasi-harmonic approximation the effect of volume expansion on the jump frequency is evident from: (i) differing values of effective vibrational frequency and migration barrier at θ D (ii) higher effect of temperature on these quantities. While the effect (ii) is obvious to be a result of the additional temperature dependence included in the quasi-harmonic calculations, effect (i) can be understood from the following explanation. As the volume used for predicting the vibrational frequencies and energy of the system in harmonic approximation is lower for Al than its equilibrium volume of θ D, which is used for quasi-harmonic calculation, the 63

83 frequencies from harmonic approximation will be higher and the energy lower (lesser stable / lesser negative), specifically for the initial state, compared to the results from quasi-harmonic approximation. Thus the higher value of * ν and lower value of H m at θ D from harmonic approximation compared to quasi-harmonic (see FIG. 3.4 (a) & (b)) can be understood. 64

84 FIG Potential representing the motion of the diffusing atom from an equilibrium lattice position to the adjacent vacant position. The potential of pure Al system is a symmetric double-well. The discrete energy states of this potential are illustrated in the figure. The potential between the two equilibrium positions matches well with the NEB potential, as illustrated. 65

85 Table 3. 1 First-principles calculated quantities entering calculation of self-diffusion coefficient. QHA results from current work for T from K, using GGA and LDA (with surface corrections as described in the text) are compare to other calculated and experimental data. Data for H m in experimental column is deduced from experimental values for Q from [28-32] using H f =0.67eV. Similarly, using the value of entropy of formation of 1.1k B from experiments, and value of ν ~ of 2.1 THz from the current work, value of entropy of migration is deduced from pre-exponential factor D 0 of experimental data [28-32]. Results of Sm from BM referring to ballistic model and cbω model are listed. EAM / MD data in the last column refers to the dynamic simulation results using embedded atom potential fitted to first-principles data (GGA). Thermodynamic Computational quantity GGA LDA H f (ev) S f (k B ) H m (ev) S m (ev) [20] 0.76[20] [20] 1.2[20] ± 0.02[21] BM 2.68 [42] cbω 2.99 [46] ν ~ (THz) Experimental 0.67 [24] 0.67±0.03 [33] 1.1 [24] 0.7 [33] 0.81, 0.64, 0.61, 0.66, 0.58± , 2.39, 2.96, 2.71, 1.34± 0.4 * ν EAM / MD 22.6 [21] 66

86 (a) (b) (c) (d) FIG Figure comparing the results from the transition state theory (TST) to the doublewell (DW) approach for fcc aluminum. The jump frequency results from the two approaches differ only slightly. The difference between HA and QHA results of jump frequency is more evident for TST than that from DW. 67

87 Simplified approach Further, an approach has been discovered that has the advantage of yielding jump frequency results similar to the double-well approach without the need for obtaining the double-well potential. Although, this approach suffers from the disadvantage of not yielding the individual quantities: enthalpy and entropy of migration and characteristic vibrational frequency. This procedure is formulated from the understanding obtained of the definitions of the migration terms and the characteristic frequency presented in the previous sub-section. From substituting these definitions in Eq. (3.37) we have: w = ~ ~ k BT H S ( H exp( )exp( )exp h k BT k B * TS ( H k T B IS ~ H )) ( S * exp * TS ( S k B IS S ~ )) (3. 46) From this we get the equation for jump frequency as: * k BT ( H TS H w = exp h k BT IS ) ( S exp * TS S k B IS ) (3. 47) This definition for jump frequency (Eq. (3.47)) has the advantage of not requiring the properties of unstable phonon mode to yield the jump frequency result one would obtain by using the best suitable approach to obtain the unstable phonon mode properties. In summary, using Eq. (3.47) accurate atom jump frequencies as a function of temperature can be attained from static first-principles, with minimum computational expense. This formulation for atom jump frequency could contribute greatly to the area of diffusion modeling, not only through aiding calculation of self and impurity diffusion coefficients with dilute impurity concentration, as shown in the present work, but also via contributing to calculation of inter-diffusion coefficients in concentrated alloys; the highly efficient Kinetic Monte Carlo when employed requires a catalog of jump 68

88 frequencies for several atomic permutations (as detailed in Section 1.4). Jump frequencies prediction using Eq. (3.46) needs only determining the vibrational properties of the transition state of different jumps and different initial states to be considered in a system, and hence very appropriate for the purpose Anharmonic effects In this section, the factors that contribute to the anharmonic effects in equilibrium vacancy concentration, atom jump frequency and finally self-diffusion coefficients, within the limits of harmonic and quasi-harmonic approximation, are discussed. As can be seen from their definitions, in addition to the temperature term, temperature dependences of the enthalpy and entropy of vacancy formation and (or) migration and (or) characteristic vibrational frequency and (or) lattice parameter, is the cause for the net anharmonic effects on C, w and D. It is seen that, from the cases considered herein, all the thermodynamic and microscopic quantities are nearly constant with temperature in harmonic approximation. Even within quasi-harmonic approximation, where the thermodynamic and microscopic quantities are temperature dependent (especially using GGA), their temperature dependences nearly cancel within the equations defining C, w and D. Figures illustrating this behavior are presented below. This leaves the temperature dependence of the quantities C, w and D to mainly be a consequence of the temperature term in their definitions. 69

89 FIG Figure illustrating trends followed by the (a) enthalpy and (b) entropy of vacancy formation with changing temperature using within HA and QHA, using LDA and GGA potentials. 70

90 FIG Figures illustrating trends followed by the (a) enthalpy and (b) entropy of atom migration with changing temperature using within HA and QHA, using LDA and GGA potentials. 71

91 FIG. 3.8 Derivatives of enthalpy and entropy of vacancy formation and atom migration with respect to temperature from GGA within QHA. In the harmonic approximation, from the equations of enthalpy and entropy shown in Eq. (3.41) and Eq. (3.42), one can see that enthalpy and entropy difference between two configurations would be constant with temperature i.e. dh f / dt, ds f / dt, dh m / dt and ds m / dt are almost zero at all temperatures. This is demonstrated from the results from the double-well approach of H f, H m, S f and S m using both LDA and GGA approximations, shown in FIGS From such a behavior, vacancy concentration, atom jump frequency (see FIG. 3.5(c)) and diffusion coefficients are 72

92 quantities that exponentially increase with temperature as a consequence of the temperature term in their definitions. The enthalpies and entropies from QHA are seen to change with temperature. The enthalpy and entropy of vacancy formation are seen to increase and that of atom migration are seen to decrease with increase in temperature, as shown in FIG. 3.6 and FIG. 3.7 respectively. As within quasi-harmonic approximation, the thermal expansion coefficient of the system is taken into account, and the three configurations (perfect state, initial state and transition state) have different dependences on volume with temperature, the vibrational frequencies and configurational free energy that are function of volume (see Eqs ) cause temperature dependences of enthalpy and entropy that differ between the three configurations. Hence the difference between enthalpy and entropy of two configurations giving the quantities H f, m H, S f and S m is not constant with temperature. This is illustrated in FIGS. 3.6 and 3.7. But as can be seen from FIGS. 3.6 and 3.7 and further explicitly from FIG. 3.8, the temperature dependence of H f is canceled by T times the temperature dependence of S f that exists in the equation for free energy of vacancy formation, to give vacancy concentration from QHA that is almost similar to the result from HA (see FIG. 3.3). Similar is the case with atom jump frequency and selfdiffusion coefficient as illustrated in FIG. 3.5 (d) and FIG. 3.9 respectively. The difference between the results from HA and QHA, for instance in the case of C v, arises from the fact that d G f / dt is equal to d H f / dt Td S f / dt S f with the temperature dependence of S f, which is not constant from QHA, contributing towards the 73

93 74 temperature dependence of vacancy concentration. Difference in the value of entropies and ν ~ between HA and QHA causes difference of intercept in their v C plots (see FIG. 3). To illustrate clearly the origin for possible difference between HA and QHA diffusion results the following set of equations are presented. Eq. (3.35) can be written as: + + = T k H H H k S S S h T k fa D B m f B m f B ~ exp ~ exp 2 (3. 48) Slope of the diffusion plot lnd vs 1/T is B k Q / from the Arrhenius form of diffusion equation ) / exp( 0 T k Q D D B =. Considering Eq. (3.48) to be in the Arrhenius form activation energy Q is H H H m f ~ +. Considering Eq. (3.48) to be an equation of a curve (not a straight line) slope of the diffusion plot lnd vs 1/T is obtained to be: ( ) ( ) ( ) ( ) ( ) = = = B m f B m f B B m B f B B m B f B m f B m f B k H H H T a T d d Q k H H H k H T d d k H T d d k H T d d T k S T d d k S T d d k S T d d T T d d a T d d T k H H H T d d k S S S T d d T T d d a T d d h k f T d d T d D d ~ ) ( ln ) (1/ 2 ~ ~ ) (1/ ) (1/ ) (1/ 1 ~ ) (1/ ) (1/ ) (1/ ln ) (1/ ln ) (1/ 2 0 ~ ) (1/ ~ ) (1/ ) ln( ) (1/ ) ln( ) (1/ ) ln( ) (1/ ) (1/ ln 2 (3. 49) In such case Eq. (3.48) cannot be expressed in the Arrhenius form of equation. While it is true that due to the temperature dependences of the individual quantities diffusion equation is not really a straight line, the temperature dependences are so low compared to the temperature dependence that is imparted to D due to the exp (1/T) dependence of Eq. (3.48) that the equation can be assumed to be in the Arrhenius form with nearly constant

94 D0 and Q. In the case of Al enthalpy of formation and migration have been seen to have nearly equal and opposite temperature dependences (see FIG. 3.8) and hence their sum is almost constant with temperature, not causing much difference between HA and QHA. In the case of most other solids too even though the temperature dependences of formation and migration are not opposite, the change in activation energy with temperature is seen to be small. The temperature dependence of H ~ is almost similar to that of T k B and hence cancel each other in the equation. It is mainly the thermal expansion of the system that causes difference between the slope of HA and QHA diffusion results. As the temperature dependence of thermal expansion is of small order (almost linear) for most metals, slope of the diffusion plot is almost constant, reproduced equivalently by both HA and QHA results. Only for the case of solids whose thermal expansion coefficient changes as a higher order of temperature visible curvatures could be introduced into the diffusion plot from the temperature dependences obtained accurately in such cases from QHA. Attempting to understand the trend in the thermodynamics of formation and migration, we know configurational energy E c is a function of the volume, shape and position of atoms in the system. Phonon frequencies are a function of the bond strengths and mass of the atoms i.e. frequency of vibration is directly proportional to the bond strength or force constant of the bond and inversely proportional to the mass of atoms forming the bond [26]. Such a definite relationship cannot be given for configurational energy dependence on its functional parameters. From the dependence of phonons we can say that the system with vacancy has a set of weaker bonds and hence have a higher number of low frequencies compared to perfect system, as illustrated in FIG Hence 75

95 entropy of initial state is higher than entropy of perfect state and we get positive entropy of vacancy formation. With increase in temperature, the rate of increase in entropy of initial state is less relative to that of perfect state due to lesser decrease in frequencies of the weak bonds compared to that of these bonds in system without vacancy. Hence the difference between the higher entropy of the initial state and the lower entropy of the perfect state decreases with increase in temperature as can be seen in FIG. 3.6 (b). It is seen that the enthalpy of initial state is higher than that of perfect state with same number of atoms as initial state, due to additional energy supplied to the initial state to move an atom to the surface. Physically, the volume of the system with vacancy is smaller than the perfect system due to local contraction surrounding the vacancy. This change in volume, lowering the stability, results in higher configurational energy (less negative) of the initial state. The vibrational contribution to enthalpy is small in relation to the configurational contribution. So, with increase in temperature, as the system with vacancy has higher enthalpy and also its volume increases at a slower rate than perfect state, enthalpy of vacancy formation decreases with rise in temperature. When an atom moves from its equilibrium position to the saddle point, there is net softening of bonds, due to the diffusing atom moving away from seven of its nearest neighbors, while moving closer to four others. Hence the entropy of the transition state is higher than the initial state and entropy of migration is positive. Similarly, the lower total energy of the unstable transition state yields positive enthalpy of migration. It is observed that with increase of temperature, the system tends to relax the stress at the saddle position, causing the vibrational frequencies of the TS to decrease and thus entropy to increase at a faster rate compared to IS, resulting in increasing entropy of migration with 76

96 temperature. Also, though the rate of change of volume of saddle configuration is almost same as that of initial configuration, change in the shape of the relaxed configuration from a slightly tetragonal to almost cubic shape, causes increase in the configurational energy at a rate faster than the initial state, leading to increasing enthalpy of migration with temperature. As the temperature dependence of characteristic vibrational frequency is small, logarithm of jump frequency as a function of inverse temperature has a linear dependence, as shown in FIG. 3.4 (d) Self-diffusion coefficient Using atom jump frequency definition Eq. (3.47), diffusion equation Eq. (3.35) simplifies to the equation below, which does not require calculation of properties of initial state: D = a 2 ( H kbt f exp h * TS N 1 H N k T B PS ) ( S exp * TS N 1 S N k B PS ) (3. 50) Results from both LDA and GGA within HA and QHA for fcc Al are plotted in FIG. 3.9 in comparison to experimental and other theoretical works available in literature. From the description in the previous section, expressing diffusion equation Eq. (3.35) in the Arrhenius form of diffusion equation D = D 0 exp (-Q / k B T), we have: D 0 kbt = h Q = H * TS 2 S fa exp * TS + ( N 1 N ) H PS ( N 1 N ) k B S PS (3. 51) 77

97 Q and D 0 values within both HA and QHA are listed in Table 2, in comparison with other experimental data. FIG First-principles results of self-diffusion coefficients for Al. Results are shown for both the HA and QHA using both LDA and GGA. Calculated results are compared with experimental results [28-32] and theoretical work [21], showing excellent agreement for the GGA (with surface correction included) calculated self-diffusion coefficient. 78

98 Table 3. 2 The frequency factor D0 and activation energy Q of self-diffusion in aluminum from double-well approach in comparison with experimental data. The temperatures represent the ranges over which measured diffusion coefficients were fit to extract D 0 and Q. Studies D 0 (m 2 /sec) Q (ev) T (K) Current GGA HA Current LDA HA Current GGA QHA Current LDA QHA Exp [28] Exp [29] Exp [30] Exp [31] Exp [32] In this work, for the first time, self-diffusion coefficients have been calculated completely from static first-principles. Previously, attempts have been made for a parameter-free calculation of self-diffusion coefficients. Though the works in literature showed that the vacancy formation enthalpy and entropy can be calculated from static abinitio methods, calculation of the characteristic vibrational frequency and entropy of migration either involved some approximations from static first-principles methods or required dynamic simulation techniques for accurate predictions of the effective frequency ν * as discussed below. 79

99 Frank et al. [18] calculated the enthalpy of formation and migration at 0K and entropy of formation of lithium, using static first-principles methods, but assumed the temperature dependence of these quantities and the entropy of migration to be zero. The characteristic vibrational frequency was approximated as a fraction of the Debye frequency of the system. When studying the process of self-diffusion in silicon, Blöchl et al. [34] calculated the jump frequency and its temperature dependence using ab-initio molecular dynamics. Also, in obtaining the vacancy concentration, entropy of vacancy formation was calculated assuming local-harmonic approximation [35]. Sandberg et al. [21] calculated the enthalpies of formation and migration of aluminum (Al) using static * first-principles methods while the entropy of formation, the pre-factorν, and their anharmonic effects were calculated by means of molecular dynamics using semiempirical embedded atom potentials. Also for the first time, in this work, different approaches of calculating atom jump frequency have been compared, with calculations performed from static first-principles. The results from either approach do not involve any crude approximations. The simplified approach presents a straightforward procedure to obtain accurate temperature dependent atom jump frequencies. In addition, transition theories that existed for several decades, have been implemented for the first time in this work, by determining the double-well potential of the diffusion mode, yielding the entropy and enthalpy of migration and characteristic vibrational frequency as a function of temperature. The double-well procedure is applied to fcc Cu too, to compare the results of individual diffusion parameters with large amount of experimental data available for this element. Only simplified approach is applied to fcc Ni and Ag, as discussed below. Only 80

100 harmonic approximation is used to obtain the self-diffusion coefficients, due to the cancellation of temperature dependences that occurs within the diffusion equation FCC Copper For Cu, the system is setup similar to that mentioned in the previous section. Calculations are performed using double-well approach to get description of individual diffusion parameters along with diffusion coefficients. Both GGA and LDA are used for comparing their performance within harmonic approximation. Quasi-harmonic calculations have been conducted with LDA and the anharmonic effects due to thermal expansion are seen to be negligible, in the case of Cu too similar to Al. Comparison of the calculated quantities within HA to other data from literature is presented in Table 3. Due to the lack of surface correction terms predicted for Cu, the energetics of the defect-containing systems as obtained from the calculations have been used for obtaining self-diffusion coefficients. The results of copper self-diffusion thus attained are shown in FIG. 3.9, in comparison with measured data. Without including surface correction terms, the first-principles results from LDA yield values of individual quantities and selfdiffusion coefficients are in excellent agreement with experimental measurements. It is expected that the cancellation of errors within the correlation function of local-density approximation, which considers total energy a function of the electronic charge density, results in negligible or small errors in the predicted energetics of vacancy-containing system unlike GGA, which considers total energy a function of gradient of the electronic charge density. 81

101 Table 3. 3 First-principles calculated quantities entering calculation of self-diffusion coefficient for fcc Cu. HA results from current work for T = θ D using GGA and LDA (without surface corrections) are listed, and compared to other calculated and experimental data. From the value of entropy of formation of 2.5k B and entropy of activation of 4.15k B, obtained from analyses of different experimental works, value of entropy of migration listed in the experimental column is deduced. Thermodynamic Computational quantity GGA LDA Other H f (ev) S f (k B ) H m (ev) S m ( k B ) ν ~ (THz) [38] [36] [36] ± 0.01 [41] [42] 0.90 [43] 2.92 [42] 1.72 [46] Experimental 1.19 [24] 1.15 [37] 3 [24] 2.5 [37] 1.5 ± 0.5 [39] 2.6 ± 0.5 [40] 0.71 [44] 0.76 ± 0.04 [45] 1.65 [39, 47, 48] * ν (THz)

102 FIG First-principles calculation of self-diffusion coefficients for Cu showing results for HA using both LDA and GGA. Calculated results are compared with experimental data [48-53] showing excellent agreement for the LDA calculated diffusion coefficient without surface correction. 83

103 Table 3. 4 Table comparing frequency factor D0 and activation energy Q of self-diffusion in copper within HA from LDA and GGA, with experimental data. Studies D 0 (m 2 /sec) Q (ev) T (K) Current GGA Current LDA EAM [41] Exp [49] Exp [47, 50] Exp [48] Exp [53] ± ± Exp [52] Exp [51] FCC Nickel Spin and non-spin polarized calculations are conducted for fcc Ni. Experimental data is mainly available for temperatures greater than 700K where Ni is para-magnetic. Hence the diffusivity predictions from non-spin polarized calculation are compared with this data, illustrated in FIG As temperature dependence is only linear above the Neil temperature of fcc Ni i.e. for T > 700 K, quasi-harmonic approximation calculations are not conducted for the case of fcc Ni too. The self-diffusion coefficient results are in good agreement with experiments. Electronic contributions are found to be negligible. 84

104 FIG Entirely first-principles, non-spin polarized calculation results within HA using LDA in comparison with experiments [53-62]. Table 3. 5 Table comparing frequency factor D0 and activation energy Q of self-diffusion in nickel within HA from LDA, with experimental data. Studies D 0 (m 2 /sec) Q (ev) T (K) Current LDA EAM [39] ±0.05 Exp [53] Exp [63] Exp [55]

105 3.4.4 FCC Silver For silver, LDA calculations within HA are conducted and the self-diffusion results obtained are shown in FIG It is predicted from experiments that silver has contribution from di-vacancies at high temperatures and hence has different slope of diffusion plot at these temperatures. FIG compares self-diffusion in silver upto temperatures about 800 K where di-vacancies contribution is not dominant. Vacancy concentration prediction for the system with dilute concentration of vacancies (monovacancy) is compared with experimental data in FIG FIG Self-diffusion coefficient of Ag from LDA using HA in comparison with experimental data [66-69]. 86

106 FIG Equilibrium vacancy concentration from fcc Ag with mono-vacancy in comparison with experiments[66, 70] Di-vacancy mechanism 2x2x2 supercell is also used for calculating di-vacancy contributions. Silver is believed to have di-vacancy contributions at high temperatures. Hence calculation of divacancy contributions is attempted for this element. Following the description of diffusion parameters in section 3.2.2, the jump frequencies of the different jumps, their corresponding defects equilibrium concentration are calculated and diffusion contribution from di-vacancies obtained. As expected diffusion from di-vacancies has a larger slope than diffusion from mono-vacancies, indicating lower diffusion rates at low 87

107 temperatures and dominating diffusion rates after a certain temperature, relative to monovacancy diffusion. This is illustrated in FIG As can be seen in the plot the prediction of di-vacancies contribution to diffusion from the present calculation does not suggest that di-vacancy contribution will be higher to mono-vacancy at temperatures indicated by experimental measurements. The cause for discrepancy is suspected to be the insufficiency of LDA without surface correction for di-vacancies, as the di-vacancy binding energy predicted is small (see Table 3.6), thus representing smaller concentration of di-vacanies as can be seen in FIG It could also be the pseudo-potential of Ag from LDA that could be not completely reliable. So, attempting calculation on another fcc system would be recommended. Table 3. 6 Other parameters obtained in the calculation are compared with data from theoretical works and experiments. Studies di-vacancy binding energy (ev) di-vacancy migration energy (ev) Activation energy (ev) Current LDA EAM [41] Exp [44] Exp [65] 2.35 ±

108 FIG Figure illustrating silver diffusion coefficient from mono-vacancy and divacancy contributions predicted from LDA using HA in comparison with other data in literature [66-69]. The di-vacancy contribution predicted from the present work is seen to have similar slope to experimental measurements by Lam [68]. 89

109 FIG Di-vacancy concentration from the present work in comparison with experiments [71]. 3.5 Conclusions From the current work a major dilemma of the cause for the curvature of diffusion plots in fcc systems being either from temperature dependence of the individual diffusion parameters or due to contribution from di-vacancy mechanism has been resolved by illustrating that anharmonic effects to diffusion coefficient are minimum and only divacancy mechanism can cause dominant curvatures. Also precise definitions of temperature dependent diffusion parameters D0 and Q from only the thermodynamic 90

110 properties of transition and perfect state and thermal expansion of the system, is presented in this work. As the computational expense to obtain self-diffusion coefficients using TST-V is the same as that for simplified approach, the latter procedure that yields more precise results is recommended. Also for information on the migration properties and their temperature dependences, double-well approach is suggested. Prediction of diffusion contributions from di-vacancies is yet to be proven feasible using this approach. Bibiliography [1] H. Mehrer, J. of Nucl. Mater , 38 (1978). [2] N. L. Peterson, J. Nucl. Mater , 3 (1978). [3] A. D. Le Claire, in Treatise on Physical Chemistry (Academic Press, New York, 1970), Vol. 10, p [4] K. Compaan and Y. Haven, Trans. Faraday Soc. 52, 786 (1956). [5] H. Eyring, J. Chem. Phys. 3, 107 (1935). [6] S. Glasstone, K. J. Laidler, and H. Eyring, (McGraw-Hill Book Company, Inc., 1941), p [7] C. Wert and C. Zener, Phys. Rev. 76, 1169 (1949). [8] M. Dole. [9] H. Mehrer, J. Nucl. Mater , 38 (1978). [10] R. E. Howard and A. B. Lidiard, Rep. Prog. Phys., 161 (1964). [11] H. Mehrer, (University of Stuttgart, 1973). [12] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999). [13] D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980). [14] G. Kresse and J. Furthmuller, Phys. Rev. B 54, (1996). [15] G. Henkelman and H. Jonsson, J. Chem. Phys. 113, 9978 (2000). [16] S. Wei and M. Y. Chou, Phys. Rev. Lett. 69, 2799 (1992). [17] A. Van de Walle, M. Asta, and G. Ceder, CALPHAD 26, 539 (2002). [18] W. Frank, U. Breier, C. Elsasser, and M. Fahnle, Phys. Rev. Lett. 77, 518 (1996). 91

111 [19] Y. Wang, Z.-K. Liu, and L.-Q. Chen, Acta Mater. 52, 2665 (2004). [20] K. M. Carling, G. Wahnstrom, T. R. Mattsson, N. Sandberg, and G. Grimvall, Phys. Rev. B 67, (2003). [21] N. Sandberg, B. Magyari-Kope, and T. R. Mattsson, Phys. Rev. Lett. 89, (2002). [22] R. O. Simmons and R. W. Balluffi, Phys. Rev. 117, 52 LP (1960). [23] A. Seeger, D. Wolf, and H. Mehrer, Phys. Stat. Sol. B 48, 481 (1971). [24] T. Hehenkamp, J. Phys. Chem. Solids 55, 907 (1994). [25] G. H. Vineyard, J. Phys. Chem. Solids 3, 121 (1957). [26] A. Van de Walle and G. Ceder, Rev. Mod. Phys. 74, 11 (2002). [27] Y. Wang, R. Ahuja, M. C. Qian, and B. Johansson, J. Phys.-Condes. Matter 14, L695 (2002). [28] F. Y. Fradin and T. J. Rowland, Appl. Phys. Lett. 11, 207 (1967). [29] R. Messer, S. Dais, and D. Wolf, Magn. Reson. and related phenomena 18, 327 (1974). [30] T. S. Lundy and J. F. Murdock, J. App. Phys. 33, 1671 (1962). [31] T. G. Stoebe, R. D. Gulliver II, T. O. Ogurtani, and R. A. Huggins, Acta. Met. 13, 701 (1965). [32] T. E. Volin and R. W. Balluffi, Phys. Status Solidi 25, 163 (1968). [33] P. Erhart, P. Jung, H. Schultz, and H. Ullmaier, Atomic Defects in Metal (Springer-Verlag, 1991). [34] P. E. Blochl, E. Smargiassi, R. Car, D. B. Laks, W. Andreoni, and S. T. Pantelides, Phys. Rev. Lett. 70, 2435 (1993). [35] R. Lesar, et.al., Phys. Rev. Lett. 63, 624 (1989). [36] N. Sandberg and G. Grimvall, Phys. Rev. B. 63, (2001). [37] J.-E. Kluin, Philos. Mag. A 65, 1263 (1992). [38] T. Hoshino, N. Papanikolaou, R. Zeller, P. H. Dederichs, M. Asato, T. Asada, and N. Stefanou, Comput. Mater. Sci. 14, 56 (1999). [39] R. O. Simmons and R. W. Balluffi, Phys. Rev. 129, 1533 (1963). [40] S. Mantl and W. Triftshauser, Phys. Rev. B 17, 1645 (1978). [41] J. B. Adams, J. Mater. Res 4, 102 (1989). 92

112 [42] J. F. Wager, Philos. Mag. A-Phys. Condens. Matter Struct. Defect Mech. Prop. 63, 1315 (1991). [43] T. W. Dobson, J. F. Wager, and J. A. VanVechten, Phys. Rev. B 40, 2962 (1989). [44] R. W. Balluffi, J. Nucl. Mater. 69/70, 240 (1978). [45] A. S. Berger, S. T. Ockers, and R. W. Siegel, J. Phys. F. 9, 1023 (1979). [46] P. A. Varotsos and K. D. Alexopoulos, Thermodynamics of Point Defects and Their Relation with Bulk Properties (North-Holland, 1986). [47] D. Bartdorff, (Tech. Universitat Berlin, 1972). [48] K. Maier, C. Bassani, and W. Schule, Phys. Lett. 44A, 539 (1973). [49] H. Mehrer and A. Seeger, Phys. Stat. Sol. 35, 313 (1969). [50] S. J. Rothman and N. L. Peterson, Phys. Stat. Sol. 35, 305 (1969). [51] H. G. Bowden and R. W. Balluffi, Phil. Mag. 19, 1001 (1969). [52] M. Beyeler and Y. Adda, J. Phys. 29, 345 (1968). [53] M. Weithase and F. Noack, Z. Phys. 270, 319 (1974). [54] R. E. Hoffmann, F. W. Pikus, and R. A. Ward, Trans. Metall. Soc. AIME 206, 483 (1956). [55] J. R. MacEvan, J. U. MacEvan, and L. Yaffe, Can. J. Chem. 37, 1623 (1959). [56] K. Monma, H. Suto, and H. Oikawa, J. Jpn. Inst. Met. 28, 188 (1964). [57] A. R. Wazzan, J. Appl. Phys. 36, 3596 (1965). [58] I. G. Ivantsov, Phys. Met. Metallogr. 5, 77 (1966). [59] H. Bakker, Phys. Stat. Sol. 28, 569 (1968). [60] M. B. Bronfin, G. S. Bulatov, and I. A. Drugova, Fiz. Met. Metalloved. 40, 363 (1975). [61] M. Feller-Kniepmeier, M. Grundler, and H. Helfmeier, Z. Metallkd. 67, 533 (1976). [62] K. Maier, H. Mehrer, E. Lessmann, and W. Schule, Phys. Stat. Sol. B 78, 689 (1976). [63] A. B. Vladimirov, V. N. Kaigorodov, S. M. Klotsman, and I. S. Tracktenberg, in DIMETA 82, Proceedings of International conference on diffusion in metals and alloys, edited by F. J. Kedves and D. L. Beke (Trans. Tech. Publications, Switzerland, Tihany, Hungary, 1982), p

113 [64] M. P. Macht, V. Naundorf, and R. Dohl, in Proceedings of International conference on diffusion in metals and alloys, edited by F. J. Kedves and D. L. Beke (Trans. Tech. Publications, Switzerland, Tihany, Hungary, 1983), Vol. Defect and diffusion monograph series No. 7. [65] G. Neumann and V. Tolle, Phil. Mag. A 54, 619 (1986). [66] G. Rein and H. Mehrer, Phil. Mag. A 45, 467 (1982). [67] J. Bihr and K. Maier, Phys. Stat. Sol. A 50, 171 (1978). [68] N. Q. Lam, S. J. Rothman, H. Mehrer, and L. J. Nowicki, Phys. Stat. Sol. B 57, 225 (1973). [69] J. J. Burton, Phil. Mag. 29, 121 (1974). [70] R. O. Simmons and R. W. Balluffi, Phys. Rev. 119, 600 (1960). [71] M. Doyama and J. S. Koehler, Phys. Rev. 127, 21 (1962). 94

114 4 IMPURITY DIFFUSION IN FCC METALS 4.1 Introduction Impurities are present in alloys used for commercial purposes. Extensive experimental works have been conducted over past decades to determine diffusion coefficients of impurities in various elements. Efforts in estimating the diffusion kinetics in dilute alloys aid in understanding the influence of impurities on the mechanical properties of the materials. Impurity diffusion coefficients are also helpful in understanding the mechanism that causes a specific material phenomenon. For eg: addition of small amount of Sn remarkably influences the formation of G.P. zones and precipitate phase in Al-Cu alloys [1]. Knowledge of diffusion coefficient of Sn in Al- 2%Cu system is useful for the analysis of this observation. In this chapter, the model describing the diffusion mechanism in an fcc structure with dilute impurity concentration called the five-frequency model is presented. Impurity diffusion equation and the definition of the quantities involved (diffusion parameters) in the equation is presented. The system setup parameters for the fcc systems dealt in this work: Mg, Si and Cu in Al are specified. The results obtained for impurity diffusion coefficients along with that of individual diffusion parameters as a function of temperature are compared to experimental and theoretical predictions in literature. The trend in diffusivities of the different impurities are discussed. 95

115 4.2 Theory Diffusion coefficient of impurity is the diffusion coefficient of the impurity / solute that is present in smaller concentrations in the solvent. In the case of dilute impurity concentration where the impurities are non-interacting, there exists only one jump frequency with which impurity jumps. Jump frequency of the solvent atoms with impurity in its neighborhood will be different from the jump frequency of the solvent atoms with no impurity in its neighborhood. Different jump frequencies exist for solvent atoms with different environments, from the non-influence to being influenced by the impurity either before or after the jump. These different jump frequencies of solvent atoms surrounding the vacant site influences the rate of diffusion of the solute atom adjacent to the vacancy. To decide the number of significantly different jump frequencies of the solvent, there needs to be appropriate assumptions made as to the radius or the nearest neighboring shell up to which the influence of the impurity can be felt. The distances of nearest neighboring shells depend on the crystal structure. Thus suitable models for different crystal structures are developed. Five-frequency model initially developed by le Claire [2] is best suited for dilute fcc alloys and hence used in the current work Five-frequency model FIG. 4.1 illustrates the five jump frequencies in an fcc structure. According to this model five jump frequencies describe the different jumps that take place in dilute fcc alloys i.e. only the first nearest neighbors to the impurity are considered to be influenced 96

116 by the presence of the impurity. The model is most suitable for fcc structure elements as the second nearest neighboring distance in fcc structure is reasonably farther compared to the first. The five frequencies are unique based on the (i) diffusing species (ii) environment (species surrounding it) before the jump (iii) and environment after the jump. In the case of a system without impurities, considered for the case of calculating self-diffusion, there is only one diffusing species and any diffusing atom has the same environment before and after the jump. Hence, only a single jump frequency can describe volume diffusion in pure systems. In the presence of impurity, there will be four jumps with frequencies different from the jump frequency of the solvent unaffected by the impurity, considering the effect of impurity to be screened by its first-nearest neighbors. The five jump frequencies in fcc with dilute impurity and vacancy concentrations are: (i) jump of the parent atom to its adjacent vacant site when it does not have the impurity in its nearest neighboring site before and after the jump w 0, (ii) jump of the parent atom to its adjacent vacant site when it has impurity in its nearest neighboring site before and after the jump w 1, (iii) jump of the impurity atom to its adjacent vacant site with the same environment before and after the jump w 2 ; this jump frequency determines the diffusion coefficient of impurity in the system (iv) jump of the parent atom to its adjacent vacant site when it does not have impurity in its nearest neighboring site before the jump, but has it after the jump w 3 ; this jump causes disassociation of the impurity atom with the vacancy and hence is called disassociation jump (v) jump of the parent atom with impurity in its nearest neighboring site to a vacant position that does not have 97

117 impurity in its nearest neighboring site w 4 ; this jump leaves the vacancy associated with the impurity atom and is called association jump. Five-frequency model defines impurity diffusion coefficient D 2 from selfdiffusion coefficient D 0 and the five jump frequencies described above as: D 2 f 2 w2 w4 w = 1 (4. 1) D 0 f 0 w 1 w 0 w 3 where f 2 represents the correlation factor for impurity diffusion and f 0 the correlation factor for self-diffusion. With the knowledge of self-diffusion coefficient, impurity diffusion coefficient can be obtained by calculating the jump frequencies in the impuritycontaining system and the impurity correlation factor. Attempting to extrapolate the diffusion prefactor D 0 and activation energy Q of impurity diffusion, equation Eq. (4.1) is not straightforward. To resolve this, in the next section we try to understand the impurity diffusion coefficient in the Einstein form of equation, similar to the self-diffusion equation. The connection between such form of impurity diffusion equation and Eq. (4.1) is also presented. 98

118 w 0 w 4 4 Solvent atom Solute atom Vacancy 1 w 1 w 3 1 w 3 w 3 w 4 3 w 2 w 4 2 FIG Five-frequency model illustrated for fcc with dilute impurity concentration. The arrows indicate the direction of the vacancy jump and the numbers n stand for the n th nearest neighboring site to the impurity. In the case of w 1, w2 and w 3 jumps the vacancy position is indicated by solid box and the position of the solvent atom to which the vacancy jumps to is indicated by filled circle. For jumps w0 and w 4 the vacancy position is indicated by dotted box and the solvent position of the jump is indicated by open circle. 99

119 4.2.2 Impurity diffusion equation For the Einstein s diffusion equation (Eq. (1.6)) D = fr 2 zcw 6 representing impurity diffusion, C is the concentration of vacancy adjacent to the impurity and w is the jump frequency with which impurity jumps successfully ( w 2 in the current form of representations). z is the number of possible sites to which impurity atom can jump i.e. the number of positions at which vacancy can be created around the impurity atom, equal to 12 for fcc. Thus, impurity diffusion coefficient for jump of an impurity atom to its nearest neighboring site in fcc with jump distance of a/ 2, can be expressed using a sub-script 2 to represent all parameters related to impurity diffusion (following convention) as: D =, (4. 2) 2 2 f 2a C2w2 where a is the lattice parameter. We can see that by dividing Eq. (4.2) by Eq. (1.6) we get the diffusion equation from five-frequency model Eq. (4.1) (wherein self-diffusion parameters are represented using a subscript 0) from satisfying the equation: C 2 w = 4 (4. 3) C 0 w 3 This equation is justified in the sections below after the definition of vacancy concentration adjacent to impurity and the implication between the jump frequencies involved are presented. 100

120 Diffusion equation Eq. (4.2) can be expressed in terms of the vacancy formation and atom migration properties as: D S exp + S i H f + H exp kbt i i i m = f a 2~ i f m 2 2 ν (4. 4) kb where superscript i is used to represented the properties of system containing impurity. The main difference in the nature of the diffusion parameters between impurity diffusion and self-diffusion is the temperature dependence of the impurity correlation factor f 2 as opposed to the constant value of self-diffusion correlation factor f 0. Due to the temperature dependence of f 2, the Arrhenius diffusion parameters appear as: i i ~ Q = H f + H m H kbd ln f2 / d(1/ T ) i i ~ k T S f + Sm S 2 B k D = f a exp( )exp 0 2 h kb B d ln f 2 / d(1/ T ) kbt (4. 5) Knowledge of impurity diffusion coefficient expression in the form of Eq. (4.4) is very useful: (i) in its ability to express the impurity diffusion coefficient in terms of activation barrier and diffusion pre-factor, (ii) in understanding the equation from fivefrequency model and its explicit form of assumptions, (iii) in understanding the physics underlying diffusion from information of temperature dependences of individual parameters constituting Q and D 0. The definitions of these individual impurity diffusion parameters are described below Vacancy Concentration 101

121 Consider the energy of 5 systems with N sites: (i) G PS for N atoms of component one and no vacancy, (ii) G v for N-1 atoms of component one and a vacancy (ii) G i for an impurity (component two) in component one i.e. N-1 atoms of component one and 1 atom of component two, (iii) G (i+v) with a vacancy formed adjacent to the impurity in component one i.e. N-2 atoms of component one and 1 atom of component two and 1 vacancy, (iv) G (i,v) with a vacancy formed in component one with an impurity, such that the vacancy and impurity atom are non-interacting. The impurity and vacancy binding energy denoted by G b can be expressed as: G b = -(G (i+v) G (i,v) ) (4. 6) For sufficiently large N, G (i,v) can be expressed as the sum of G i + G v G PS. Vacancy concentration C 2 is defined from the free energy of formation of vacancy adjacent to impurity [3], similar to Eq. (3.8) as: G fi C 2 G fi = exp( ), (4.7) k T B The free energy of vacancy formation adjacent to the impurity is defined as [4]: G fi = G (i+v) (G i G PS /N) (4. 8) Writing equation (4.8) in terms of the binding energy using Eq. (4.6) and also inserting the expression for G (i,v) mentioned above, we have: G fi = - G b + (G v G PS (N-1)/N ) = - G b + G f (4. 9) where G f is the free energy of vacancy formation in system with component one and no impurity. The vacancy concentration equation (4.7) can then be written as C 2 G f Gb = exp( ), (4. 10) k T B 102

122 From the definitions of vacancy concentrations C 2 and C 0 i.e. equations (4.10) and (3.8) respectively, we have the expression: C C 2 0 Gb exp( ) k T = (4. 11) B Atom jump frequency The jump frequency equation in the same for the two cases: jump in impuritycontaining system and jump in system with no impurity. Hence the different jump frequencies in dilute fcc alloy are defined from the expression for jump frequency described earlier in section (Eq. (3.16)). Thus jump frequency of impurity atom is: w 2 i ~ i Gm = ν exp( ), (4. 12) k T B where i is the free energy required for migration of the impurity atom and ~ ν is the i G m vibrational frequency of the impurity atom in the direction of the vacant site. The initial states for jumps w 1, w 2 and w 3 are the same, with the vacancy being adjacent to the impurity site before the jump. Jump w 4 has a different initial state with the vacancy in either the second, third or fourth nearest neighboring sites to the impurity. Also, w 0 has a different initial state as it is the jump frequency in a system with no impurity. There are three types of w 3 and w 4 jumps i.e. a w 3 jump can cause the vacant site adjacent to the impurity atom to jump to either second, third or fourth nearest neighboring site to the impurity atom. Similarly a w 4 jump could cause a vacancy that is 103

123 either second, third or fourth nearest neighboring to the impurity atom to move adjacent to it. Each of these three types of jumps would have a different jump frequency due to the varying distance of the vacancy from the impurity atom before or after the jump. Fivefrequency model assumes the three types of jumps to be equivalent and thus have the same jump frequency. Besides, w 3 and w 4 conceptually being reverse jumps of each other have the same transition state and same characteristic vibrational frequency ν ~. Therefore, from Eq. (4.6), it can be seen that: w w 4 3 Gb = exp( ) (4. 13) k T B as the free energy of initial state of w 3 jump is G (i+v) and that of w 4 jump is G (i,v). The assumption of five-frequency model that vacancy in the second, third or fourth nearest neighboring to the impurity is considered to be non-interacting with the impurity is evident in this equation. Further it can be seen how equations (4.11) and (4.13) together justify Eq. (4.3). It should also be noted that in the case of jumps w 2 and w 0, the double-well potential of diffusion is symmetrical. The other three jumps have assymetrical doublewell potentials due to the effect of impurity on one half of the potential well, which is missing on the other half. Irrespective of this Eq. (4.11) holds for the different jump cases Impurity correlation factor The impurity diffusion correlation factor, f 2, related to the probability of the impurity atom making a reverse jump back to its previous position, taking into account 104

124 the probability of the vacancy returning to its current position (function F) after disassociation by w 3 jump, as described by Le Claire[5] is: f ( w3 / w1 ) F( w4 / w0 ) = (4. 14) 1+ ( w / w ) + 3.5( w / w ) F( w / w ) The definition of F, after explicitly considering the probabilities of the vacancy returning from second, third and fourth nearest neighboring positions to the initial vacant position, was given by Manning [6] to be: x x + 927x x) = x x + 254x + 597x F (4. 15) ( 2 where x stands for w 4. w Literature Study Several models that theoretically describe system interactions such as those given by le Claire [7], Neumann-Hirschwald [8] etc. were put forth [9] to yield diffusion parameters similar to results from experiments. From such models different potentials such as the Thomas-Fermi potentials [10] from the model by le Claire, oscillating potentials [11] from model by Friedel [12] have been generated. Though these potentials suited well for some systems they failed to describe systems such as 3d transitions metals in aluminum. Better descriptions came in the form of semi-empirical methods such as embedded atom method (EAM) by Daw and Baskes [13], which is based on densityfunctional theory, but built on interatomic potentials. Adams et al. [14] predicted 105

125 diffusion coefficients in fcc metals from five-frequency model using embedded atom potentials. In their work, interatomic potentials are generated for different systems of interest by fitting to their bulk properties, thus evaluating diffusion parameters. Such an approach though rigorous suffers from inaccuracies in predicting finite temperature nonequilibrium properties, which exist due to the potentials fit to 0K equilibrium properties. Over the past decade studies have been conducted to predict impurity diffusion coefficients from parameter free calculations using density-functional theory. Blochl et al. [15] calculated diffusion coefficients of hydrogen in silicon from first-principles determining contributions from entropy to free energy from local harmonic approximation using Morse potentials. Later, Milman et al. [16] calculated impurity diffusion coefficients from ab-initio molecular dynamics (AIMD). Since dynamical simulations are practical only for temperatures close to melting, thermodynamic integration is used to calculate the free energy of migration at finite temperatures. The force on the diffusing atom that is constrained to specific locations along the diffusion path is obtained from dynamical runs conducted at different temperatures. Integration of forces on the atom at different positions gives the free energy of migration. From this and the enthalpy of migration calculated from static calculations, entropy of migration is obtained. Problems in such an approach would be the uncertainty in knowing the right diffusion path and getting the right saddle point, especially in the case of asymmetric jumps. While Janotti et al. [17] and Kremar et al. [18] used only first-principles for predicting diffusion coefficients of 4d and 5d elements in fcc Ni, the effective frequency * ν was approximated to be the same for all solute elements, assumed to be some approximate value. The entropy terms were not explicitly calculated, thus yielding 106

126 inaccurate diffusion prefactors. Recently, Sandberg et al [19] predicted 0K migration barriers of different impurities in Al from first-principles. Impurity diffusion coefficients were not calculated in their work. 4.4 Present work In this work impurity diffusion coefficients have been calculated from firstprinciples without major approximations as those involved in the previous works. The approaches presented in this work are general in their application and require only minimum input such as the atomic mass and the crystal structure. Depending on the level of physical understanding required and the accuracy desired one of the three approaches presented in Chapter 3 for calculating atom jump frequency have been adopted. For these calculations, similar to self-diffusion, the projector augmented wave (PAW) potentials [20] as implemented in the highly efficient Vienna ab- initio simulation package (VASP) [21] have been used. Results using both the local density approximation (LDA) [22] and the generalized gradient approximation (GGA) [23] for the exchange correlation are examined and compared. Convergence tests again indicate that a Monkhorst-Pack k-point mesh of 11x11x11 and an energy cutoff of 300 ev are suitable to yield converged impurity migration barriers within 0.01eV for these systems. Similar convergence of the energetics calculated using supercells with 32 and 64 lattice sites is attained. Table 4.1 illustrates the convergence tests conducted. Hence, supercell with 32 lattice sites (2x2x2 conventional fcc cells) of the same size as for self-diffusion is employed. For dilute impurity concentration, one atom in each supercell is substituted by impurity atom. 107

127 POSCAR file is prepared similar to that for self-diffusion. One of the lattice positions is chosen for the impurity. A sample POSCAR file for the case with impurity is listed in APPENDIX. The settings for VASP relaxation listed in the INCAR and KPOINTS files are the same as for the case of self-diffusion. POTCAR file is a combination of two potential files, one of the solvent element and other of the solute element, joined in the order they are listed in the POSCAR file. This implies that the position coordinate of the impurity atom should be listed at the beginning or at the end of the POSCAR file, irrespective of the position coordinate assigned to it, as shown in APPENDIX. 108

128 Table 4. 1 Tabulated values are for impurity jump w2 from full relaxation using GGA. System Kpoint mesh size ; Energy cut-off 300eV Size Migration Barrier (ev) 32 lattice sites; Kpoint mesh size Energy cut-off (ev) Migration Barrier (ev) 32 lattice sites; Energy cut-off 300eV Kpoint mesh size Migration Barrier (ev) Al-Mg Al-Si Al-Cu

129 In the case of impurity diffusion coefficient calculation, five jump frequencies need to be calculated. For this, as mentioned in the previous section, a total of 8 configurations: 4 transition states, 3 initial states and 1 perfect state, have to be relaxed and their ground state configurations determined. As mentioned earlier, the solvent jumps in the impurity-containing system are assymetrical and hence for obtaining the saddle point nudged elastic band (NEB) method is necessary. In the case of tracer jump in system without impurity or impurity jump in system with impurity, due to symmetry the saddle point for the tracer atom in transition state is about half way between its initial and final equilibrium positions. Hence starting with this initial guess, a VASP relaxation is mostly enough to get the right saddle point, in these cases. The more computationally expensive NEB method is not necessary. Again, for the ground state energy all configurations are completely relaxed with respect to internal coordinates, volume and shape of the supercell. Vibrational calculations also have very similar settings to those of self-diffusion. The file with relaxed configuration (str.out / str_relax.out) has a tag attached to the position coordinates indicating the element type. This is the only means by which information of the two elements present in the system is provided to the simulation (as no POTCAR file is supplied to the program). In the case of a non-standard potential being used for a specific element mention is made of this through the tag SUB_ATOM listed with the description of the potential given here. An example of this command is also listed in the INCAR file for impurity diffusion presented in APPENDIX. It should also be noted that due to the low symmetry of transition state for the jumps w 1, w 3 and w 4, several perturbed configurations result, based on the principle of functioning of super-cell 110

130 approach that considers all non-symmetry equivalent perturbed configurations for generating the force constant matrix. It has been seen from the current work that different types of w 3 and w 4 jumps are not identical in their jump frequencies and hence when the jumps w 3 and w 4 are not reverse jumps of each other, a significant deviation of the predicted impurity diffusion coefficient from experimental data results. Hence for the purpose of this work similar kind (jump to or from the same nearest neighboring site to impurity) of w 3 and w 4 jumps are considered such that they are reverse jumps of each other i.e. their motion along jump direction is described by the same double-well, causing them to have the same transition state and characteristic vibrational frequency. In this work, impurity diffusion coefficients are calculated following harmonic approximation. As anharmonic effects are seen to nearly cancel in aluminum, quasiharmonic calculations have not been conducted for impurity-containing aluminum systems. For each configuration, using harmonic equations of (3.33) and (3.34), temperature dependent total enthalpy and entropy are obtained from the total energy from VASP relaxation and phonon frequencies determined by ATAT. From this the five jump frequencies can be calculated using jump frequency definition of simplified approach Eq. (3.46) for determining the impurity correlation factor (Eq. (4.14)). For determining the migration properties of impurity atom jump, its double-well potential is obtained following procedure similar to that mentioned in section The five jump frequencies are also calculated from Vineyard s TST following equations in section These results are listed in the next section 4.5. The impurity jump frequency and its constituent parameters * i ν and i H m from Vineyard s TST are compared to the result from the double-well approach, which is also presented in 111

131 section 4.5. In section 4.6, the individual thermodynamic properties of vacancy formation and impurity migration, correlation factor, characteristic vibrational frequency of the impurities are compared and discussed. All the results presented in the next section are from PAW LDA, without the surface correction. 4.5 Results and discussions Five jump frequencies (TST-V) The jump frequency parameters and the frequencies calculated from TST-V are tabulated in Tables 4.2, 4.3 and 4.4. One can see that the Al system with Cu impurity has a * low ν, Mg atom has the lowest migration barrier and the barrier of Al self-diffusion is the highest. The theory underlying the trends of these quantities along with that of the vacancy formation parameters listed in Table 4.5 are discussed in the next section. From Eq. (4.14) it can be said that higher the frequency of the w 2 jump and lower the frequencies of w 1 and w 3 jumps, higher would be the correlation of impurity atom movement. This behavior can be observed from the results tabulated in Table 4.4, giving a very low correlation factor for diffusion of Mg impurity in Al system and very high correlation factor for Cu diffusion in Al. Table 4. 2 TST-V results of * ν (THz) for the 5 jumps of different impurities. Eq. (3.39) is used for calculating this quantity for each jump from their respective initial and transition states. 112

132 Impurity w 0 w 1 w 2 w 3 w 4 Mg Si Cu Table 4. 3 Migration barriers H m (ev) of 5 jumps of different impurities from TST-V. Impurity w 0 w 1 w 2 w 3 w 4 Mg Si Cu Table 4. 4 Five jump frequencies (Hz) for different impurities in Al. Results for T=400K using ν * from Table 4.2 and H m from Table 4.3. Impurity w 0 w 1 w 2 w 3 w 4 f 2 Mg Si Cu Table 4. 5 Enthalpy and entropy of formation of a vacancy adjacent to impurity atom H f, S f and quantities involving the correlation factor (T=400K) for the Arrhenius diffusion parameters. System Enthalpy of vacancy formation (ev) Entropy of vacancy formation (k B ) k B d ln f 2 / d (1/ T ) f2 kbd ln f2 / d(1/ T ) exp( ) k T Al Al-Mg Al-Si Al-Cu Impurity diffusion data (DW) B 113

133 Double-well potential is determined only for the impurity jump. As jump frequencies calculated from double-well approach are exactly same as the result from simplified approach, jump frequency results from simplified approach are tabulated below in Table 4.6. Table 4. 6 Five jump frequencies (Hz) for different impurities in Al. Results for T=400K using Eq. (3.46). Impurity w 0 w 1 w 2 w 3 w 4 Mg Si Cu Table 4. 7 Quantities describing the impurity diffusion coefficient D 2 (see Eq. (4.4)) calculated from double-well approach are listed. r is the jump distance in angstroms for the impurity atom to reach the vacant site, which is not exactly equal to a / 2. Impurity r (A) f ~ ν 2 (THz) H f (ev) S f (k B ) H m (ev) S m (k B ) -k B lnf 2 /d(1/t) (ev) T = 400K Mg Si Cu T = 900K Mg Si Cu It can be observed from the results for impurity-containing systems listed above that, similar to the case of self-diffusion, within HA the enthalpies and entropies of vacancy formation and atom migration and characteristic vibrational frequency are almost constant. Temperature dependence of impurity correlation factor in contrast to that 114

134 of self-diffusion is evident from the values listed in the table, especially for the case of Mg diffusion in Al. The value of r is used in place of a / 2 to improve the accuracy of the result, as the change in the jump distance for the impurity to its neighboring vacant site due to local contraction on formation of vacancy is not entirely reflected in the lattice dimensions Comparing data from TST-V and DW Table 4. 8 Comparing the * ν (THz) for the impurity jump from double-well approach to Vineyard s TST, of different impurities. For clarity, Eq. (3.37) is used for calculating quantities in column 2 from the gamma point frequencies and Eq. (3.36) is used with the quantities involved calculated within harmonic approximation from double-well. Impurity Mg Si Cu TST-V Double-well T=400K T=900K From these results we see the limitation of TST-V in making accurate predictions of values representative of high temperatures. It can be seen that the change in * ν with temperature for temperature rising from 400 to 900K is minimal, because above Debye temperature ( θ D ) all the vibration modes are already excited and hence within HA the changes with further increase in temperature are less. It can also be observed that the result of * ν from TST-V is significantly lower than the result from DW. This is because 115

135 * below θ D the change in ν would be large due to the change in entropies that go to zero * for 0K and TST-V yields ν representative of low temperatures T << θ D. Table 4. 9 Comparing the H m (ev) for the impurity jump from double-well approach to Vineyard s TST, of different impurities. The difference between the two results is that the vibrational contributions to enthalpies of the IS and TS are included in double-well approach and enthalpy of migration from TST is the difference between the ground state energies. Results from TST-V are constant at all temperatures. Impurity Mg Si Cu TST-V double well T=400K T=900K For the same reason explained above the change in migration barrier with temperature increasing above Debye temperature upto temperatures close to melting is minimal. As the vibrational contributions to the enthalpy of migration are small compared to the configurational, and as within HA the configurational energies are constant (due to constant volume), the value of migration barrier from TST-V is very similar to that from DW. Table Comparing the impurity jump frequency w 2 (Hz) from double-well approach to Vineyard s TST, of different impurities. Using the results from the respective approaches * from Table 4.8 and Table 4.9 in equation w =ν exp( H k ), the jump frequencies obtained are tabulated below. 2 m BT Impurity Mg T=400K T=900K TST DW TST DW 116

136 Si Cu 1.86e6 2.63e5 2.22e6 4.40e5 1.31e e9 1.54e e9 As the migration barrier within the exponential term in the definition of jump frequency is almost same for both the approaches, the result of jump frequency does not vary significantly. Even in the case of impurity diffusion, within HA, results from TST-V are not quite different from the DW results (see FIGS ). At the same time the result from TST-V is not accurate enough as can be seen from the impurity diffusion results Comparison with experiments Comparing to experiments the migration energy results from GGA and LDA for the different impurities, it can be found in this case too that without surface-correction LDA yields a good match with measured data. Table 4.11 illustrates comparison of (1) Enthalpy o H f and entropy o S f of vacancy formation and migration barrier o H m of pure Al (2) enthalpy H b and entropy Sb of solute-vacancy binding and migration barrier H m of Mg, Si, Cu in Al. (Enthalpy H f and entropy S f of vacancy formation adjacent to solute in system with impurity is tabulated in the form of their binding components G = G o ( f f b G ) for comparison with experiments ). Table Calculated results from LDA and GGA (without surface-correction) to experimental data. The measured data of enthalpy and entropy of solute-vacancy binding listed are the critically assessed data by Balluffi and Ho [24]. The migration barrier of pure Al is from the difference of assessed o Q =29 kcal/mol value of Peterson et 117

137 al. [9] and the assessed value of enthalpy of vacancy formation (listed below) of Erhart et al. [25]. From the activation energies obtained by the studies referenced in measured column of migration barrier and the value of enthalpy of vacancy formation in impurity system, the migration barrier values listed in measured column are deduced. System Enthalpy of vacancy formation (ev) Entropy of vacancy formation (k B ) Migration barrier (ev) LDA GGA Measured LDA GGA Measured LDA GGA Measured Al ± ± 0.03 [9] Al-Mg ± ± ± 0.07 [26] Al-Si ± 0.03 [27] Al-Cu ± ± ± 0.15 [9] 118

138 FIG Diffusion coefficient with HA of Mg in fcc Al from simplified approach using LDA and GGA and from TST-V using LDA (without including surface corrections), in comparison to experimental data [26, 28-33]. 119

139 FIG Diffusion coefficient with HA of Si in fcc Al from simplified approach using LDA and GGA and from TST-V using LDA (without including surface corrections), in comparison to experimental data [28, 34-36]. 120

140 FIG Diffusion coefficient with HA of Cu in fcc Al from simplified approach using LDA and GGA and from TST-V using LDA (without including surface corrections), in comparison to experimental data [9, 30, 36-38]. 121

141 Table Table listing the diffusion pre-factor and activation energy values from the current work using LDA for the temperature ranging from K, in comparison with experiments and other theoretical calculations. System Present Experimental data Other Computations [39] Mg ± [26] [32] [14] ± [33] D 0 (m 2 /sec) Si ± [28] [34] ± [35] [14] Cu [9] [38] [14] Mg [39] 1.35 ±0.05 [32] 1.29 ± [33] 1.35 [14] 1.2 [19] Q (ev) Si ± 0.03 [28] 1.33 [34] 1.28 [35] 1.28 [14] 1.0 [19] Cu ± 0.01 [9] 1.35 ± 0.07 [38] 1.4 [14] 122

142 From Table 4.12 we can see that the experimental data of the Arrheius s parameters are scattered. Though the activation energies and diffusion pre-factors from the current work do not match very well with experiments, the diffusion coefficients (FIGS ) predicted from the current work are in the same magnitude as that from experimental data. 4.6 Discussion on trends FIG Diffusion coefficients of Mg, Si and Cu in Al and Al self-diffusion. It is seen that Al self-diffusion is slower than the impurity diffusion in Al. 123

143 As seen in Table 4.12 the activation energy of Mg is higher than that of Cu, but from FIG. 4.5 it can be seen that diffusion coefficient of Mg is still higher to Cu. From this the significance of the role of diffusion pre-factor in diffusion coefficients is depicted. A connection between the microscopic quantities to the macroscopic diffusion parameters is established to explain the trends, in the section below. From Table 4.11 we see that the enthalpy of vacancy formation adjacent to impurity is in the order of H < Al Si f H < Al Cu f H Al Mg f. Vacancy formation energy consists of bond-breaking energy to form a vacancy and binding energy of the impurity atom to the vacancy, as shown in Table These quantities would differ depending on the following properties of the impurity atom: (i) size, which determines the nearest neighboring (nn) distances to the impurity, (ii) excess valence i.e. the difference in valence of the impurity atom to the host atom, which influences the bond breaking energy and, (iii) unscreened nuclear charge, which affects the atom-vacancy binding, as shown in Table Table Table lists the (i) distances of the first nearest neighboring atoms to the impurity atom obtained from the fully relaxed structures of the Al system with impurity (no vacancy) (ii) energy required for bond breaking, which is defined as the energy of the unrelaxed system with vacancy minus the energy of the relaxed system without vacancy, (iii) total binding energy, which is negative of the difference between energy of relaxed system with vacancy and energy of the unrelaxed system with vacancy. Energies listed are in ev. 124

144 Diffusing Atom r (Å) nn distance (Å) Excess valence Bond breaking Nuclear charge Total Binding H f Al Mg Si Cu Enthalpy of vacancy formation is calculated as the difference between the enthalpy of system with vacancy and system without vacancy. To understand the trend of the difference between the enthalpies of vacancy formation, we first try to understand the trend of the enthalpies before and after the formation of vacancy. Table 4.14 lists the enthalpies of the two configurations involved in vacancy formation. Table Ground state energies without vibrational contributions and volume after relaxation of perfect state (PS) with no vacancy and initial state (IS) with a vacancy adjacent to the impurity atom. r (Å) V PS (Å 3 ) V IS (Å 3 ) E PS (ev) E IS (ev) H f (ev) Al Mg Si Cu Ground state energy is a function of shape, volume and, interactions from internal atom positions of the system configuration. From the results in Table 4.14 it can be seen that smaller the impurity size lower is the volume of the system. This can be understood as being the consequence of the atoms surrounding the impurity moving closer to it or away from it depending on its size, as it also evident from the nearest neighboring distances to the impurity listed in Table From Table 4.14 it is also observed that Al 125

145 system with Si and Cu impurity atoms allow greater interaction between the atoms and hence are more stable than pure Al system. While Mg atom being larger makes the system less stable, Si atom having higher valence causes more stability than Cu atom. This indicates that an impurity that has a negative size misfit and positive excess valence compared to the host imparts higher stability to the system. The first criterion is more influential than the other for causing stability. When there are two systems both with negative excess valence, one higher negative than the other and the system with lower negative excess valence has a positive size misfit (eg: Al-Mg) almost equivalent to the negative size misfit of the other system (eg: Al-Cu), the system with negative size misfit is more stable than the system with lesser negative excess valence. Enthalpy of formation is larger when the change in energy of the system on formation of vacancy is larger. It is difficult to quantitatively conclude on how the trend of the vacancy formation energy would vary with size or its misfit with the host atom size from the cases considered in the present study. But from the present results it can be observed that Mg atom with positive size misfit causes a greater change in energy on formation of vacancy adjacent to it due to its greater motion towards the vacant site (~0.12 Å) obstructing its neighbors adjacent to the vacancy from moving much towards the vacant site. While Si atom allows almost uniform motion of all surrounding atoms including itself (~0.04 Å) causing lesser change in energy, Cu atom moves away from the vacancy (~0.03 Å) causing its neighboring atoms adjacent to the vacancy to move to a greater extent towards the vacancy preventing motion of the remaining atoms towards the vacancy, as illustrated in the Table The overall high change in nn distances in the case of Al system with Mg impurity causes higher change in energy of formation. From 126

146 this we can conclude that greater the size misfit greater would be the vacancy formation energy. Table Displacements (in Å) of the nearest neighboring atoms towards the vacancy position (0.25, 0, 0.25). The nearest neighboring positions of (0.25, 0, 0.25) includes the impurity at (0, 0, 0). The nearest neighboring atoms to the vacancy (8 of them present in the supercell of the current work) are categorized based on their distances from the impurity atom: 1 st nearest neighbors to the impurity atom i.e. (0.25, 0.25, 0) and (0, 0.25, 0.25), 2 nd nearest neighbors to the impurity atom i.e. (0.5, 0, 0) and (0, 0, 0.5), 3 rd nearest neighbors to impurity atom i.e. (0.5, 0.25, 0.25) and (0.25, 0.25, 0.5) and, 4 th nearest neighbor i.e. (0.5, 0, 0.5). Impurity atom 1 st nn 2 nd nn 3 rd nn 4 th nn Mg Si Cu From the values listed in this table the greater influence of the first nearest neighbors compared to others can also seen. The displacement of the 1 st nn atoms to Cu atom by ~0.1 Å has greater influence in decreasing the loss in energy on bond breaking than about the similar distance ~0.09 Å moved by the 4 th and 2 nd nns to Mg atom. The 1 st nn atoms to the Mg atom are farther away (see Table 4.13) and continue to be so on vacancy formation, which does not facilitate much compensation of the loss of interactions that occur on formation of vacancy. Hence resulting in a high change in energy of vacancy formation in Al system with Mg impurity. 127

147 Now turning towards understanding the trend of entropy of formation we first look at the trends of entropies of the PS and IS configurations, listed in the Table We also know that phonon frequency of vibrations in a system is proportional to the bond strengths and inversely proportional to the square root of mass [40] and that system with high vibrational frequencies has less entropy. Table Entropies (k B ) of PS and IS configurations for T=400K. force constant listed is in ev. Diffusing Mass (m) Force constant atom (amu) (k) k / m S PS S IS S f (k B ) Al Mg Si Cu In perfect configuration (PS), Si with high excess valence and nuclear charge and shorter nn distance is expected to have high force constant. As its mass is higher than Mg, its frequency could be lower than that of Mg. Mg atom with a huge size is expected to have lower force constant than Si that could be overridden by the higher mass of Si, leading to its higher frequency than Si. Cu with high mass is expected to have low frequency of the three. For instance, the force constant listed in Table 4.16 (taken from Table 4.12), which is representative of the bond strength in the system before formation of vacancy, is in the order of Al-Si > Al-Mg > Al-Cu. The result that the system with Mg has higher force constant than that of Cu indicates that the force constant is high for impurity with higher valence and unscreened nuclear charge, unlike the case of total energy where size and the consequent interactions due to the nearness of the atoms caused higher stability of the 128

148 system (Al-Cu > Al-Mg, see Table 4.14). These when divided by square root of mass of the respective diffusing atom, results in quantities indicative of the order of phonon frequencies: Al-Mg > Al-Si > Al-Cu. Hence the entropies for the perfect configurations of different impurity-containing systems are in the order of S > S > S. PS Cu PS Si PS Mg In the presence of vacancy, the bond strengths of the Mg atom are lowered significantly, due to the large nearest neighbor distances (see Table 4.15), leading to very low force constants, which are not overridden by the higher mass of Si. This leads to lower frequencies and higher entropy of Mg containing system compared to that of Si. Cu with its nn atoms enclosing towards it (as seen in Table 4.15) experiences a significant increase in force constants on vacancy formation. Its considerably high mass still causes lower frequencies and hence higher entropy. Hence for the initial state, the entropies are in the order of S > S > S. IS Cu IS Mg IS Si Migration barrier is the difference between the total energy of the two states with vacancy but with the diffusing atom in its equilibrium position (IS) and the other in which it is at the saddle point along the path towards the vacant site (TS). We have already understood the trend of system stability from the energies (higher negative the energy more stable is the system) of the initial state being in the order: E Al Si IS > E > E. On motion of the diffusing atom to the saddle point, the same Al Cu IS Al Mg IS order is followed, from having more interactions increasing the stability for system with smaller size and higher excess valence, with the former effect being more influential than the latter. Hence the order of stability of the TS of the systems would be: Al-Si > Al-Cu > Al-Mg, as can be seen in Table

149 Table Energies (ev) (no vibrational contributions included), volumes (Å 3 ) of the two configurations are tabulated. b/c is the ratio of the dimensions of the unit cell, indicating the tetragonality of fcc in its relaxed state with a vacancy and the impurity atom in TS and IS. E IS E TS V IS b/c IS V TS b/c TS H m Al Mg Si Cu On motion of the diffusing atom to the saddle point, the distances to the impurity atom of the atoms surrounding the vacancy decrease. These changes for movement of Mg impurity to transition state is less, especially of the 1 st nn atoms to the impurity, as can be seen in the table below. The decrease in the distance for Cu atom is higher than Si due to the 2 nd and 4 th nearest neighboring atoms which were away from the Cu atom in IS coming a lot closer to Cu atom at the saddle point. Hence the migration barrier is in the order of H > H > H. Al Cu m Al Si m Al Mg m Table Change in distances (in Å) to the impurity atom, from the atoms surrounding the vacancy, when the impurity atom moved from its initial equilibrium position to the saddle point at (0.125, 0, 0.125). (For information on the positions the different nearest neighboring (nn) atoms stand for refer to caption of Table 4. 15) 1 st nn 2 nd nn 3 rd nn 4 th nn Al Mg Si Cu

150 The trend of the migration barrier can be interpreted as the larger the size of the impurity atom compared to that of the host atom, the lower is the energy required for the impurity atom to reach the saddle point and vice versa. The force constant when diffusing atom is in the transition state is expected to be of the same order as when the diffusing atom is in the initial state. Hence the order of entropies in TS is expected to be S > S > S, as illustrated in Table TS Cu TS Mg TS Si Table Entropies (k B ) of IS and TS configurations for T=400K. The quantity listed in parenthesis in column 1 is the mass of the diffusing species in amu. S IS S TS S m (k B ) ~ ν (THz) Al(26) Mg(24) Si(28) Cu(63.5) We see that entropy of migration and characteristic vibrational frequency follow the same trend: increase with increasing impurity size. To understand the trend of these quantities resulting from the vibrational frequencies in the system, we study the force constants in these systems. Force constants are obtained in the present work for the first and second nearest neighbors only. It is seen that the change in force constants of the impurity atom with first nearest neighboring Al atoms, when it is at the saddle point compared to when it is at its equilibrium position, yields frequency change 1nn 1nn 1nn 1nn 1nn 1nn ( ν = ln( ν ν ) ) [40] to be of the order ν (0.82) > ν (0.65) > ν (0.34). The TS IS change in frequencies due to force constant change of Mg with its second nearest neighboring Al atoms ( ν 2nn (1.37) ) is significantly high compared to that of Cu Mg ( ν 2nn (0.35) ). This causes the overall frequency change (ν ) to be of the Cu Cu Si Mg 131

151 orderν > ν > ν, thus explaining the trend in entropy of migration values for Mg Si Cu different solutes. The study helped in understanding that effect of factors such as size, mass, excess valence, solubility etc of the impurity in the host element together reflect in the change in nearest neighboring distances for the energetics and change in force constants towards the phonon frequencies to determine its migration properties. Tables can be used for understanding the relaxed configuration of systems different impurity atoms without vacancy, with vacancy adjacent to the impurity and when impurity is at the saddle point. To summarize, it is observed that on formation of vacancy all the atoms surrounding the vacancy position move towards it. A larger diffusing atom moves a greater distance towards the vacancy. This causes its nearest neighbors from moving much distance towards the vacancy. Similarly, when the diffusing atom is significantly small, the host nearest neighbors move greater distance towards the vacancy, inhibiting the small impurity atom to move towards the vacancy. Crystal symmetry guides the extent of motion of the other atoms surrounding the vacancy; larger impurity causes greater closing in on the vacancy by these atoms and vice versa. This effect of the size of the impurity atom affects the trend of the migration properties. When the diffusing atom is at the saddle point, as expected, a larger impurity causes its neighbors to be pushed apart to a greater extent. Table Distances (in Å) of the first nearest neighboring atoms of the atom at position (0.25, 0, 0.25) (before vacancy is formed) to the vacancy position. 1 and 2 represent 1 st nn to the impurity atom i.e. (0.25, 0.25, 0) and (0, 0.25, 0.25). 3 and 4 represents the 2 nd nn i.e. (0.5, 0, 0) and (0, 0, 0.5). 5 and 6 represent atoms 3 rd nn to impurity atom i.e. (0.5, 132

152 0.25, 0.25) and (0.25, 0.25, 0.5). 7 represents its 4 th nn atom i.e. (0.5, 0, 0.5) and 8 represents the impurity atom position (0, 0, 0). Impurity Al Mg Si Cu Table Distances of the nearest neighboring atoms of the vacancy position (0.25, 0, 0.25) to the impurity atom position (0, 0, 0) in system with no vacancy. Here 8 represents the atom at position (0.25, 0, 0.25) which is also 1nn to the impurity atom. Impurity Al Mg Si Cu Table First nearest neighboring distances (in Å) to position (0.25, 0, 0.25) after formation of vacancy. 8 here represents the impurity atom position. Impurity Al Mg Si Cu Table Distances from the impurity atom of the nearest neighboring atoms to the vacant position (0.25, 0, 0.25) after vacancy is formed. Here 8 represents the position where vacancy is created. The column listed itself is the distance the impurity atom moves after formation of vacancy from its previous position at (0, 0, 0). 133

153 Impurity itself Al Mg Si Cu Table Distances of the nearest neighbors of vacant site to the vacancy position (0,25, 0, 0,25) after movement of the impurity atom to the saddle position. Impurity Al Mg Si Cu Table Distances of the nearest neighbors of the vacant site to the impurity atom at the saddle position (0.125, 0, 0.125). Impurity Al Mg Si Cu

154 4.7 Conclusions As seen from the results and discussions sections, physical understanding of the underlying phenomena of impurity diffusion coefficients has been obtained from such first-principles works. In comparing the diffusion coefficients of different impurities, it is important to note that the impurity diffusivities do not correlate entirely with size as claimed by Janotti et al. [17], while the migration barriers do. This reflects on the importance of the pre-exponential factor. The diffusion coefficients are neither in the order of their solubilities as suggested by some researchers or follow the trend of their valence but are a result of all these factors, as described in the text. The simplified approach gives very accurate impurity diffusion coefficients in comparison with experiments and at the same time is a very computationally efficient procedure. As observed from the results of impurity diffusion coefficients of normal metals Mg, Si, Cu in Al, the diffusion coefficients are higher compared to pure Al. In the next chapter, the anomalous behavior i.e. very low diffusivities compared to selfdiffusion, of 3d transition metals diffusion in Al is presented. 135

155 Bibiliography [1] J. R. Manning, Diffusion kinetics for atoms in crystals (D. Van Nostrand Company Inc., Princeton, New Jersey, 1968). [2] A. D. Le Claire, J. Nucl. Mater , 70 (1978). [3] R. E. Howard and A. B. Lidiard, Rep. Prog. Phys., 161 (1964). [4] N. Sandberg, B. Magyari-Kope, and T. R. Mattsson, Phys. Rev. Lett. 89, (2002). [5] A. D. Le Claire, in Treatise on Physical Chemistry (Academic Press, New York, 1970), Vol. 10, p [6] J. R. Manning, Phys. Rev. 136, A1758 (1964). [7] A. D. Le Claire, Phil. Mag. 7, 141 (1962). [8] G. Neumann and W. Hirschwald, Phys. Stat. Sol. B 55, 99 (1973). [9] N. L. Peterson, Rothman, S.J., Phys. Rev. B 1, 3264 (1970). [10] V. Vitek. [11] G. Erdelyi, Phil. Mag. B 38, 445 (1978). [12] A. P. Blandin and J. Friedel, J. Phys. Radium 21, 689 (1960). [13] M. S. Daw and M. I. Baskes, Phys. Rev. B 29, 6443 (1984). [14] J. B. Adams, J. Mater. Res 4, 102 (1989). [15] P. E. Blochl, C. G. Van de Walle, and S. T. Pantelides, Phys. Rev. Lett. 64, 1401 (1990). [16] V. Milman, M. C. Payne, V. Heine, R. J. Needs, J. S. Lin, and M. H. Lee, Phys. Rev. Lett. 70, 2928 (1993). [17] A. Janotti, M. Krcmar, C. L. Fu, and R. C. Reed, 92, (2004). [18] M. Krcmar, C. L. Fu, A. Janotti, and R. C. Reed, 53, 2369 (2005). [19] N. Sandberg and R. Holmestad, Phys. Rev. B 73 (2006). [20] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999). [21] G. Kresse and J. Furthmuller, Phys. Rev. B 54, (1996). [22] D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980). [23] J. P. Perdew, K. Burke, and Y. Wang, Phys. Rev. B 54, (1996). 136

156 [24] R. W. Balluffi and P. S. Ho, in Diffusion (Americal Society of Metals, Metals park, Ohio, 1973), p. 83. [25] P. Erhart, P. Jung, H. Schultz, and H. Ullmaier, in Atomic Defects in Metals, edited by H. Ullmaier (Springer-Verlag, Berlin, 1991), Vol. 25. [26] S. J. Rothman, N. L. Peterson, L. J. Nowicki, and L. C. Robinson, Phys. Status Solidi B 63, 29 (1974). [27] D. Bergner, Neue Hutte 29, 207 (1984). [28] S.-I. Fujikawa and Y. Takada, Defect Diffus Forum , 409 (1997). [29] Y. Minamino, T. Yamane, T. Miyake, M. Koizumi, and Y. Miyamoto, Mater. Sci. Technol. 2, 777 (1986). [30] S. Fujikawa and K. Hirano, Mater. Sci. Eng. 27, 25 (1977). [31] K. Hisayuki and T. Yamane, Z. MetaIlkd. 90, 423 (1999). [32] G. Moreau, J. A. Cornet, and D. Calais, J. Nucl. Mater. 38, 197 (1971). [33] J. Verlinden and R. Gijbbels, Adv. Mass Spectrom. 8A, 485 (1980). [34] A. H. Beerwald, 45, 789 (1939). [35] D. Bergner, Neue Hutte 29, 207 (1984). [36] R. H. Mehl, F. N. Rhines, and K. A. Vonden Steinen, Met. and All. 13, 41 (1941). [37] M. S. Anand and R. P. Agrawala, Phil. Mag. 26, 297 (1972). [38] J. B. Murphy, Acta Metall. 9, 563 (1961). [39] S. Fujikawa and K. Hirano, Mater. Sci. Forum 13-14, 539 (1987). [40] A. Van de Walle and G. Ceder, Rev. Mod. Phys. 74, 11 (2002). 137

157 5 DIFFUSION OF 3d TRANSITION METALS IN Al 5.1 Introduction 3d transition metals in Al (Al-3d) are important commercial alloys. Over the past quarter century several experimental studies [1-17] have been conducted to measure reliable data for diffusion coefficients of 3d transition elements in Al. The experimental data is scattered due to experimental difficulties such as: (i) low solubility of these elements in Al (ii) competitive diffusion from grain boundary and other short-circuiting diffusion mechanisms and (iii) unavoidable effects from oxide in Al surfaces. To validate the results, researchers worked on finding reasons for the diffusivities being high or low in comparison to Al self-diffusion, the possible mechanism responsible for such a diffusion rate and suitable reasoning validating such mechanism. It has been concluded that high diffusion pre-factor D 0 and high activation barrier Q cause these anomalously low diffusivities, which is believed to be the effect of segregation of solute atoms to the dislocations [1, 6, 16, 18, 19] due to their low solubilities in the host element. Though some researchers [3, 18] proved that low solubilities is not the cause, reasoning as to why dislocation mechanism should be the dominant mechanism or a clear proof for dislocation mechanism being the cause for anomalous behavior does not exist yet. Normal metals (s, p block elements) diffuse in Al at rates that are within an order of magnitude of Al self-diffusion. Fully filled d-shell elements also exhibit normal behavior [20]. Vacancy-mediated diffusion describes diffusion of normal metals in Al 138

158 [12]. Normal diffusion can be explained from leclaire-lazarus theory [21] i.e. using the Thomas-Fermi potential this model describes, within the framework of electrostatic interactions, diffusion of substitutional impurities in Al, to differ within an order of magnitude of the self-diffusion coefficient. Al system with normal metal impurities is non-magnetic [22]. The experimental data for diffusion coefficients in these systems are consistent. The results from this thesis work, presented in the previous chapter, on impurity diffusion coefficients of normal metals Mg, Si, Cu in Al, also reflects all these behavioral features expected of such diffusion. In the coming sections we introduce the different theories that have been proposed in the past to explain the diffusivities of Al-3d systems based on the electronic interactions in the system and magnetic nature of the solutes. We will present the approach we have followed in obtaining diffusivities in these systems. From such a calculation, we conclude on the predominant diffusion mechanism and the possible cause for low diffusivities of 3d elements in Al. We compare our results in terms of 3d solute diffusivities and activation energies with experiments. Having attained good description of the system interactions using the Hubbard U parameter [23], we report the magnetic moments and the partial density of states from d-orbital of 3d elements in Al. From these results we present the theories our results agree with and those that are in disagreement. Deducing the enthalpy of formation and migration and activation entropy to evaluate the physical behavior leading to high D0 and Q in these systems, have also been areas of research interest [6, 10]. Besides this issue, the temperature dependence of solute diffusion correlation factor that has been attempted determination from isotope-effect experiments, anharmonic nature of the impurity diffusivities from either contributions 139

159 from di-vacancies or temperature effects of the system thermodynamics have also been of significant interest. These different aspects are systematically studied in the current work for all the 3d transition elements in Al, the results from which are presented and discussed in this chapter. 5.2 Background study Friedel-Anderson described existence of virtual bound state [24] in Al alloy with 3d transition elements (Al-3d), from resonance between the similar energy states of partially filled d shell of transition impurities and the conductive states of aluminum, causing scattering of the conduction electrons of the host element. As a consequence, electron density of the localized d states of the transition element and the residual resistivity of the Al-3d system increase [25]. As a matter of evidence to this theory, it has been experimentally observed [26] that the excess local charge around the impurity screens the excess nuclear charge of the transition element, causing perturbation in the potential that is not felt beyond its first-nearest neighboring atom. Hence the solute diffusivities are seen to be independent of concentration to about ~10%. Further, this phenomenon could also alter the local magnetic moment of the transition element i.e. it could cause magnetic solutes to be non-magnetic or non-magnetic solutes to be magnetic in Al. In an attempt to understand the underlying electrostatic interactions causing anomalous behavior, few experimentalists developed models [27-29]; the model developed by Blandin and Deplante [29], for the screened potential around transition 140

160 impurities in normal metals including the effect of oscillating charge distribution described by Friedel [30], resulted in impurity diffusion data that agreed with experiments. Although the theoretical basis of this model was better than other models, it is not satisfactory in its consequence of disagreement of the predicted activation energy values with measurements [12]. Further, study of magnetic properties of dilute Al-3d alloys has been of interest to researchers over the past several years. Different theories [31, 32] were proposed to explain the magnetic behavior of Al system with 3d elements, taking support from the experimental results. Later Heeger [33] analyzed these models using results from direct and indirect measurements and concluded that the dilemma of the magnetic nature of 3d elements in Al is unresolved. In the past decade first-principles calculations based on the density functional theory (DFT) have become more prevalent, although few first-principles works [34-37] have been conducted on Al-3d systems. Sandberg et al. [34] attempted to predict migration barriers of the 3d elements in Al using pseudo-potentials - PAW GGA and LDA. Sensitivity of the migration barriers and local magnetic moments of the impurities to the relaxed configuration and the geometry of the nearest neighboring atoms to the impurity is reported [34, 36]. In the recent past researchers [35-37] have specifically investigated the magnetic property of Fe in Al using ab-initio procedures. They concluded that the magnetic nature varies depending on the extent of relaxation of internal degrees of freedom i.e. system is magnetic when unrelaxed and non-magnetic when relaxed, with the extent of relaxation for which the system becomes non-magnetic being dependent on the lattice parameter with which one starts the calculation. 141

161 5.3 Procedure and System setup Impurity diffusion coefficient equation Eq. (4.2), described in the previous chapter is used for the calculation. For calculating the impurity diffusion correlation factor using Eq. (4.13) the five jump frequencies are calculated using the simplified approach Eq. (3.46). Vacancy concentration adjacent to impurity atom is calculated using Eq. (4.9). Thus for calculating the 3d impurity diffusion coefficient and the individual quantities entering the diffusion equation Eq. (4.4), the temperature dependent enthalpy and entropy of the pure solvent and of the initial and transition states of the five jumps in impuritycontaining system are calculated. Again, it must be noted that from five-frequency model, initial states of jumps w 1, w 2 and w 3, and transition states of w 3 and w 4 jumps are the same. In the present work, first-principles calculations are performed using the projector-augmented wave (PAW) method [38, 39], as implemented in the highly efficient Vienna ab initio simulation package (VASP) [40]. The local density approximation with the Hubbard U (LDA+U) [41] is used for the exchange-correlation potential. Convergence tests were conducted. A Monkhorst-Pack k-point mesh of and energy cutoff of ev depending on the valence description of the pseudopotential yielded converged migration barriers within 0.01eV. Similar convergence of the energetics was obtained using supercells with 32 and 64 lattice sites. Hence, a supercell with 32 lattice sites (2x2x2 conventional fcc cells) was employed. Unless otherwise mentioned, all calculations are completely relaxed with respect to internal coordinates, 142

162 volume and shape. The transition state is determined by the nudged elastic band (NEB) method [42] implemented by VASP. The phonon frequencies are similarly calculated using the direct force-constant approach [43], as implemented in the Alloy Theoretic Automated Toolkit (ATAT) [44] package. The contributions to the free energy from the normal phonon frequencies are calculated through the standard thermodynamic relations [45]. Again, no quasi-harmonic calculations are performed for the transition metals in aluminum systems. Calculations are performed within the harmonic approximation [46] at the fixed 0-K equilibrium volume for the initial and transition states. In the case of magnetic systems, spinpolarized calculation (ISPIN=2) implemented by VASP that calculates the magnetization density from the difference between the up and down spins, is performed. 5.4 Methodology In the present work, while using PAW LDA, problem was encountered while relaxing the configuration of the activated state, especially for the cases of Cr, Mn, Fe, Co, Ni impurities, when the spin-polarized calculation exhibited magnetic moment irregularly, thus causing huge deviations (~ 0.5eV) in the resulting energetics. Allowing constrained relaxation (relaxing only the atom positions without changing the volume or shape of the system) starting with a non-spin polarized fully relaxed structure (a procedure similar to that followed by Sandberg et al. [34]) caused significant improvements in the resulting activation energy of some impurities in comparison to experiments, while that of some others remained almost unaffected. Thus using PAW or USPP LDA and performing full or constrained relaxation did not yield activation 143

163 energies that vary systematically from experimental results. Thus there exists a complication in handling these Al systems with 3d solutes within density function theory. The probable cause for this behavior could be non-uniform distribution of charges in the system, which in the case of Al-3d system could be due to localization of charges around the transition metal elements. To treat such a case where both band-like and localized behavior exists in dilute alloys Hubbard [23] proposed a model to include the electronelectron correlation and exchange energies of the d state of transition impurity within the Hamiltonian of the system through U and J parameters. In this work the LDA+U potential [41], from including Hubbard parameter U with PAW LDA, is used for modeling these metallic systems. Initially for a single U-J [47] value (U-J=3) diffusion coefficients of all 3d elements has been calculated. Then from comparison of predicted diffusivities with experimental data, different U-J values are chosen and both spin polarized and non-spin polarized relaxations are conducted, and the unique U-J value and spin-relaxation type for each Al-3d system is determined. INCAR file for the case of +U calculation is listed in APPENDIX for reference. ISPIN is the function that is used to set the calculation to be spin-polarized. When ISPIN is set to 2 a spin-polarized calculation is conducted by VASP. In the calculations value of J parameter has always been set to 1eV. Only the value of U parameter has been varied determining the U-J value for the calculation. Static relaxations are done after VASP relaxations by setting the LORBIT=11 to obtain the density of states from different principal orbitals of all elements in the system (DOSCAR). In the current calculations NEDOS is set to Static relaxation conducted setting LCHARG=TRUE yields 144

164 charge density of the system (CHGCAR file). A format of INCAR file is included in APPENDIX. 5.5 Results and Discussions U parameter According to Heeger s analysis [33] if the value of U (ev) is greater than the width of the virtual bound state W (ev) of the transition impurity, there would be splitting of the d-band, resulting in the system being magnetic; if U is equal to the width then there would be local spin fluctuations and the system would exhibit magnetism above its Kondo temperature (temperature up to which resistivity decreases in the system and increases there up); on the other hand if U is smaller than W the system would be non-magnetic. Hence, the U parameter value determines the ground state energy and magnetic moment of the system. As mentioned previously, the values of U have been chosen by verifying the diffusion coefficients from the current work to match data from reliable experiments. To ensure correctness of the U parameter one has to: (i) pick experimental diffusion data that are consistent from different measurements (ii) ensure that the diffusion coefficients do match well with the experimental results and finally (iii) compare the U value with value suggested for this element by other works. FIG. 1 indicates the value of U suited for different TM elements in Al. An increasing trend of U values is observed as we move across the periodic table. This follows from the d-d interaction energy U increasing with the total number of d electrons. 145

165 From this it can be concluded that more the number of valence d electrons (without the d- shell being completely filled), greater are the d-d interactions and higher is the charge localization, requiring larger value for U-J. Similar trend for U parameters of 3d elements was predicted by a recent work using density functional theory [48], as shown in FIG

166 FIG Trend of Hubbard U parameter for 3d elements in Al in comparison with values of U parameters calculated for 3d elements from the work of Aryasetiawan et al. [49]. 147

167 5.5.2 Activation energies For comparison of the activation energies from the current work to experimental data, assessed values of Du et al. [50] along with data consistent over different works of measurements are chosen, tabulated in Table 5.1. From the results it is seen that spinpolarized calculation does not change the energetics of the system with Sc, Ti, Cu or Zn impurity in Al (see Table 5.1), besides giving a zero magnetic moment on full relaxation, indicating the non-magnetic nature of these dilute Al alloys. It is also found that an LDA+U calculation of activation energy in these alloys yields almost the same value as that from LDA, indicating low localization effects. This could be due to lower d-d interactions for Sc and Ti with lower number of d electrons and for Cu and Zn with completely filled d-shell. For the case of solutes V to Ni, a significant change is seen in the energetics of the system on conducting a spin-polarized calculation (see column 2,3 of Table 5.1). The magnetic moments from the spin-polarized calculation for the different diffusion states of these Al-3d alloys, illustrated in Table 5.2, show that Fe, Cr and Mn in Al are magnetic, Ni and Co are non-magnetic and V in Al is magnetic only when V is at the saddle point in transition state. Comparing the LDA+U energetics from spin polarized calculation for the magnetic V, Fe, Cr and Mn solutes in Al and non-spin polarized calculation for the nonmagnetic Al systems with Ni and Co to energetics from LDA (see Table 5.1), a significant influence of U parameter for the Al-3d systems from V to Ni is seen. From the plot of the activation barriers illustrated in FIG. 5.2, it can be seen that the results from the current 148

168 work systematically match the experimental data. The activation barrier plot for elements across 3d series has a maximum at V. Table 5. 1 Activation energies for 3d elements in Al obtained using LDA and LDA+U, with and without spin-polarization. The U-J values that went into the LDA+U calculations are listed. Activation energy of the Al-3d systems from type of potential and relaxation suitable to it is bolded. Assessed experimental data for activation energy is tabulated for comparison. All the values listed in the table are in ev. 3d element U-J Q (LDA+U + spin) Q (LDA+U + non-spin) Q (LDA + spin) Q (LDA + non-spin) Q (Exp) Sc [51] Ti [52] V [52] Cr [50] Mn [50] Fe [7] Co [50] Ni [50] Cu [50] Zn [50] 149

169 FIG Systematic match in the trend of activation energies between current work (LDA+U) and experimental data. Sandberg s [34] results using PAW LDA are also plotted for comparison. 150

170 We attempt to understand the trend followed by the activation barriers of the 3d elements in Al. The size of the 3d elements in fcc lattice [53] varies across the series with a minimum at Fe. Further the activation barrier is also seen varying in an order that is not similar to that of the solute excess valence or their solubilities. It is expected that the activation energy of 3d solute in Al-3d alloy can be estimated from the bonding strength of the 3d element with its nearest neighboring Al atoms - a combination of: (i) the bonding strength in perfect state when there is no vacancy adjacent to the transition element, which determines the energy of vacancy formation and, (ii) the bonding strength of the transition element in initial state with vacancy adjacent to it, which determines the energy of migration. In this respect, Morinaga et al. [54] also described the trend in activation barriers of 3d transition metal solutes in Al to be a function of the bond order. Therefore, we analyze the strength of bonding of each transition metal impurity with its nearest neighbors before and after the formation of vacancy. To estimate the bonding strength of the 3d element with Al, we look at the partial density of states from the d orbital (d-dos) of transition metal impurity. It is found that the partial DOS from the d orbital of the 3d element at the Fermi energy (E F ) of the Al-3d system varies across the 3d period with a peak at V, for the sum of the fully relaxed perfect and initial states, as illustrated in FIG This behavior can be understood to be a consequence of virtual bound state (described earlier) of the 3d elements with Al. Greater the d-dos at the system Fermi energy level higher will be the electronic density of states (edos) around the impurity atom due to resonance, resulting in stronger bonding of the transition element with its nearest neighbors. Highest d-dos at the Fermi level is observed for V in both its perfect and initial states (see FIG. 5.3), which causes it to have 151

171 the highest activation barrier among all 3d elements. The trend in activation energies of the other 3d elements cannot be quantitatively explained by the trend in the d-dos at Ef, but only qualitatively, as can be seen by comparing FIGS. 5.2 and 5.3. According to Friedel-Anderson s theory of virtual bound state, greater the resonance higher would be the electrical resistivity. While the trend matches between the prediction for resistivity trend from the current work matches that from experiments, it should be noted that there exists a discrepancy in the trend between Al-V and Al-Cr, as illustrated in FIG The reason for this discrepancy is not known. But, to verify the higher edos surrounding V compared to Cr before and after formation of vacancy, charge density difference of Al-V to Al-Cr structures is plotted in FIG Positive value of the difference is observed proving higher charge density around V compared to Cr, which supports the result of higher d-dos at E F of V compared to Cr. 152

172 FIG Partial DOS from the d electrons of the transition element in Al at the Fermi level (E F ) from non-spin polarized, fully relaxed perfect (PS) and initial state (IS). Residual resistivity values are taken from experimental work of Steiner et al. [55]. 153

173 FIG Difference in charge density between un-relaxed Al-V and Al-Cr structures: (a) without vacancy (b) with vacancy adjacent to the solute. The brown colored lines for the charge density difference Al-V to Al-Cr indicate a positive value. 154

174 Partial DOS and Charge density plots In this sub-section the partial d-dos and the charge density plots providing information on the trend of edos surrounding the transition metal impurity in perfect and initial states are presented. We mainly compare the d-dos of the transition elements V to Ni, where marked localization effect in transition metals is found. First we look at the d-dos of the transition elements in their perfect state. Values of the d-dos at the Fermi level of the system, marked in FIG. 5. 4, is the data plotted in FIG

175 FIG Figure illustrating the partial density of states of the d-orbital of the transition metals: V, Cr, Mn, Fe, Co and Ni in Al in perfect configuration after full relaxation from non-spin polarized calculation. The d-dos at the fermi energy of the Al-TM system is marked. (Note: The Fermi energy (E f ) of the Al-TM system is deducted from the energy of different states of the d-orbital for the plot, to indicate E f at 0) 156

176 From FIG we can see that the trend of the d-dos at E f is Al-V > Al-Cr > Al- Mn > Al-Fe > Al-Co > Al-Ni. Higher the d-dos at E f, greater is the scattering of the host conduction electrons. This would result in a higher charge density around the transition element. To support the trend observed above from the d-dos plots we compare the charge densities of these systems by looking at the difference of charge density between their perfect structures. (a) Al-V to Al-Cr (b) Al-Cr to Al-Mn (c) Al-Mn to Al-Fe FIG Figure illustrating the positive charge density (red colored lines indicate positive values as shown in the colorbar) difference between Al-TM systems as captioned in the plots, from charge density of perfect states. Now we look at the d-dos of the transition elements in their initial states. The trend of d-dos for the initial state is Al-Cr > Al-V > Al-Mn > Al-Fe. Looking at the charge density plots for the initial states we see a similar trend as that for the perfect state. 157

177 FIG Figure illustrating the partial density of states of the d-orbital of the transition metals: V, Cr, Mn and Fe in Al in initial configuration after full relaxation from non-spin polarized calculation. (a) Al-V to Al-Cr (b) Al-Cr to Al-Mn (c) Al-Mn to Al-Fe FIG Figure illustrating the positive charge density difference between adjacent Al- TM systems as captioned in the plots. Charge density of non-spin polarized relaxation of initial states is used. 158