First-principles calculation of defect free energies: General aspects illustrated in the case of bcc-fe. D. Murali, M. Posselt *, and M.

Transcription

1 irst-principles calculation o deect ree energies: General aspects illustrated in the case o bcc-e D. Murali, M. Posselt *, and M. Schiwarth Helmholtz-Zentrum Dresden - Rossendor, Institute o Ion Beam Physics and Materials Research, Bautzner Landstraße 400, Dresden, Germany Abstract Modeling o nanostructure evolution in solids requires comprehensive data on the properties o deects such as the vacancy and oreign atoms. Since most processes occur at elevated temperatures not only the energetics o deects in the ground state but also their temperaturedependent ree energies must be known. The irst-principles calculation o contributions o phonon and electron excitations to ree ormation, binding, and migration energies o deects is illustrated in the case o bcc-e. irst o all, the ground state properties o the vacancy, the oreign atoms Cu, Y, Ti, Cr, Mn, Ni, V, Mo, Si, Al, Co, O, and the O-vacancy pair are determined under constant volume (CV) as well as zero pressure (ZP) conditions, and relations between the results o both kinds o calculations are discussed. Second, the phonon contribution to deect ree energies is calculated within the harmonic approximation using the equilibrium atomic positions determined in the ground state under CV and ZP conditions. In most cases the ZPbased ree ormation energy decreases monotonously with temperature, whereas or CV-based data both an increase and a decrease were ound. The application o a quasi-harmonic correction to the ZP-based data does not modiy this picture signiicantly. However, the corrected data are valid under zero-pressure conditions at higher temperatures than in the ramework o the purely harmonic approach. The dierence between CV- and ZP-based data D. Murali, M. Posselt, M. Schiwarth: irst-principles calculation o deect ree energies MS#BD

2 is mainly due to the volume change o the supercell since the relative arrangement o atoms in the environment o the deects is nearly identical in the two cases. A simple transormation similar to the quasi-harmonic approach is ound between the CV- and ZP-based requencies. Thereore, it is not necessary to calculate these quantities and the corresponding deect ree energies separately. In contrast to ground state energetics the CV- and ZP-based deect ree energies do not become equal with increasing supercell size. Third, it was ound that the contribution o electron excitations to the deect ree energy can lead to an additional deviation o the total ree energy rom the ground state value or can compensate the deviation caused by the phonon contribution. inally, sel-diusion via the vacancy mechanism is investigated. The ratio o the respective CV- and ZP-based results or the vacancy diusivity is nearly equal to the reciprocal o that or the equilibrium concentration. This behavior leads to almost identical CV- and ZP-based values or the sel-diusion coeicient. Obviously, this agreement is accidental. The consideration o the temperature dependence o the magnetization yields sel-diusion data in very good agreement with experiments. * Corresponding author, Address: Helmholtz-Zentrum Dresden - Rossendor, Bautzner Landstraße 400, Dresden, Germany Electronic address: M.Posselt@hzdr.de, Phone: , ax: Key words: irst-principles calculations; Deects; ree energy; bcc-e PACS numbers: y; Mb; Bb; J-; q D. Murali, M. Posselt, M. Schiwarth: irst-principles calculation o deect ree energies MS#BD

3 I. INTRODUCTION Processes at elevated temperatures can lead to nanostructure evolution and may cause the modiication o mechanical, electrical and magnetic properties o solids. Together with vacancies and sel-interstitials oreign atoms play an important role in the alteration o the nanostructure. Experimental investigations on deect or nanostructure thermodynamics and kinetics require a suiciently high temperature to observe the process o interest in a inite time. On the other hand, irst-principle calculations using Density unctional Theory (DT) have been widely used to determine the ground state properties o point deects, oreign atoms, and other nanoobjects. The data obtained or ormation, migration and binding energies are oten used in multiscale modeling o processes at elevated temperatures, by applying kinetic Monte Carlo simulations (c. e.g. Res. 1-4) or rate theory. This is clearly not a ully consistent approach. It is only recently that authors have pointed out the importance o considering ull DT-based ree energy data at nonzero temperatures instead o those obtained rom the energetics in the ground state Only ew studies on this topic were perormed earlier In general excitations o electrons, phonons, magnons and other quasi-particles can contribute to the ree energy o deects at elevated temperatures. 7,10,11,13 In the case o an extended spatial coniguration o a deect the contribution o its internal conigurational entropy must be considered as well in these calculations. An example is the dumbbell selinterstitial in bcc-e which exists in six equivalent orientations. In one o the irst ull DT-based study on the ree energy o oreign atoms in bcc-e Reith et al. 6 showed that the observed thermodynamic stability o Cu substitutional atoms cannot be properly explained using the DT ormation energy in the ground state, whereas a better agreement was obtained i the contribution o phonon excitations to the ree ormation energy is taken into account. The eect o the vibrational contribution to the nucleation ree energy o Cu clusters in bcc-e was clearly evidenced in the work o Yuge et al. 26 DT D. Murali, M. Posselt, M. Schiwarth: irst-principles calculation o deect ree energies MS#BD

4 calculations o Lucas et al. 5 demonstrated that even at moderate temperatures the phonon contribution to the ree ormation energy o vacancies and sel-interstitials in bcc-e must not be neglected. Huang et al. 7 perormed DT studies on sel-diusion in bcc-e and showed that the consideration o phonon and electron contributions to the ree ormation and migration energy o the vacancy led to a very good agreement with experimental data. Using this methodology they could also reproduce the experimental diusion coeicients o W and Mo in bcc-e. A similar procedure to determine ree energies was also used by Mantina et al. 8 and Tucker et al. 10 to calculate sel- and impurity diusion in Al and Ni, respectively. Satta et al. 23,24 investigated the contribution o electronic excitations to the activation energy o seldiusion in W and Ta and ound that in the ormer case the eect is more pronounced than in the latter. The important role o vibrational contributions to the ree ormation energy and entropy o deects was also shown in DT calculations or non-metallic materials, namely, or the O vacancy and the Zn interstitial in ZnO (Re. 14), or the O vacancy in Sr-doped complex perovskites 15, as well as or native point deects in Cu 2 ZnSnS 4 (Re. 16), In 2 O 3 (Re. 20), and SiC (Re. 21). urthermore, several authors demonstrated that the eect o phonon excitations must be considered in irst-principles calculations o phase diagrams o binary and ternary metallic and non-metallic alloys ,22,27 While in most o the above-mentioned studies vibrational eects were treated within the harmonic or quasi-harmonic approximation anharmonic eects have been considered in a very recent paper. 11 These authors have investigated the temperature dependence o the ree ormation energy o the vacancy in Al and Cu and have ound that the ormation entropy is not constant, as oten assumed, but increases with temperature. This leads to a nonlinear temperature dependence o the ree ormation energy. The indings o Re. 11 have contributed to the explanation o apparent discrepancies between DT results and experimental data. D. Murali, M. Posselt, M. Schiwarth: irst-principles calculation o deect ree energies MS#BD

5 The ocus o the present paper is on the methodology o the calculation o deect ree energies within the ramework o the supercell approach and on the application o this procedure to bcc-e. Ater introduction o a oreign atom or an intrinsic deect into the supercell two alternative kinds o calculations may be perormed to determine the energetics in the ground state: (i) The positions o atoms are relaxed but the volume and shape o the supercell are held constant (constant volume calculations - CV). (ii) Both the positions o atoms and the volume and shape o the supercell are relaxed so that the total pressure or stress on the supercell is zero (zero pressure calculations - ZP). In the present work CV as well as ZP is perormed and the relation between the results obtained by the two methods is discussed, with a ocus on the dependence on the supercell size. Despite the wealth o literature on DT calculations on ground state energetics o deects such a discussion was perormed only in ew previous papers. The positions o atoms in the supercell obtained by CV and ZP are used to determine the contribution o phonon excitations to ree energies by relatively time-consuming, state-o-the-art calculations. A detailed discussion on the relation between the CV- and ZP-based deect ree energies and their size dependence is perormed because this was missing in the ew previous papers on corresponding DT calculations. In selected cases a quasi-harmonic correction is applied to ZP-based deect ree energies. Since experiments are usually perormed at zero external pressure these results are most interesting or a comparison. The electronic contributions to the ree energy are treated separately rom those o phonons within the meaning o the adiabatic approximation. 28 Contributions due to magnon excitations are not considered. In the present work the ground state energetics as well as the vibrational and electronic contributions to the ree ormation energy o the vacancy (v), o the oreign atoms Cu, Y, Ti, Cr, Mn, Ni, V, Mo, Si, Al, Co, O, and o the O-v pair in bcc-e are determined. urthermore, the ree binding energy o O-v, the ree migration energy o the vacancy, and the sel- D. Murali, M. Posselt, M. Schiwarth: irst-principles calculation o deect ree energies MS#BD

6 diusion coeicient in e are calculated. The results o present investigations shall contribute to a better understanding and an improved modeling o nanostructure evolution in erritic e and e-cr alloys which are important or various practical applications, in particular as basic structural materials or present and uture nuclear ission and usion reactors. or example, Cu, Ni, and Mn are characteristic solutes in conventional reactor pressure vessel steel, and Cr, Y, O, and Ti are basic alloying elements in oxide dispersion strengthened erritic e-cr steel that shows an excellent high-temperature creep strength and a high radiation resistance. 29 The solutes Mo, Al, Si, V, and Co can be present in both erritic e and e-cr alloys. The vacancy plays a undamental role in the diusion o most o the oreign atoms and in the ormation o nanoclusters containing these species. It should be noticed that in all these applications the relevant temperatures are between about 600 and 1000 K. The paper is organized as ollows. The ground state energetics o deects is considered in Sec. II. This includes the calculation o the ormation energy o the vacancy, oreign atoms and o the O-v pair, the vacancy migration energy as well as the binding energy o O-v. Secs. III and IV deal with the eect o excitations o phonons and electrons on the ree ormation, migration and binding energy. Both sections can be considered as the main part o the work. The ree ormation and migration energies o the vacancy determined by DT methods in the preceding sections are used in Sec. V to obtain the sel-diusion coeicient in bcc-e. The calculated data are compared with results o previous DT calculations and available experimental data. II. GROUND STATE ENERGETICS A. Calculation method The Vienna ab-initio simulation package (VASP) 30,31 was used to perorm DT calculations. Plane wave basis sets and pseudopotentials generated within the projected D. Murali, M. Posselt, M. Schiwarth: irst-principles calculation o deect ree energies MS#BD

7 augmented wave (PAW) approach were employed. or transition metals and magnetic systems the PAW method is generally preerred compared to the ultra-sot pseudopotential (USPP) approach In all calculations the spin polarized ormalism was applied and the generalized gradient approximation GGA-PBE 35 was used to describe the exchange and correlation eects. The Brillouin zone sampling was perormed using the Monkhorst-Pack scheme. 36 In all cases a plane wave cuto o 500 ev was used. Calculations on the energetics o oreign atoms, the vacancy (v), and the oxygen-vacancy pair (O-v) were perormed or a bcc-e supercell with 54 lattice sites and Brillouin zone sampling o k points as well as or a cell with 128 sites and k points. In selected cases supercells with 250 sites and k points were considered. Ater introduction o the deect into the supercell two types o calculations were carried out in order to determine the energetics in the ground state: (i) The positions o atoms were relaxed at constant volume and shape o the supercell (constant volume calculations - CV). (ii) Ater step (i) the positions o atoms as well as the volume and shape o the supercell were relaxed so that the total stress/pressure on the supercell became zero (zero pressure calculations - ZP). Special care was taken to ensure a high precision o the results. This was achieved by perorming several calculations to obtain the most suitable parameters or the iterations carried out in VASP. There are two important criteria to be considered: (i) irst, CV and ZP are carried out until the residual orce acting on any atom alls below a given threshold, and (ii) at each step o CV and ZP the relaxation o the electronic degrees o reedom is perormed until the total energy change alls below another threshold. It was ound that the optimum threshold values depend on the example considered. They vary between 10-2 and 10-5 ev/å in the irst case and between 10-5 and 10-7 ev in the second case. In the ground state the ormation energy o a substitutional deect X is deined by E ( X) = E ( X) E + µ µ (1) D 0 0 X D. Murali, M. Posselt, M. Schiwarth: irst-principles calculation o deect ree energies MS#BD

8 (c. Res. 13, and 37-39) where ED ( X ) and E 0 denote the total energy o the supercell with and without the deect, respectively. The quantities µ 0 and µ X are the chemical potentials o the e and the X atoms, respectively, in the corresponding reservoirs or reerence systems. In the case o a very dilute alloy µ 0 can be set equal to the energy per atom E 0,A in perect bcc-e. According to the common practice in DT calculations o deect energetics in bcc-e 2,5,6,7,32,34,40-49 in the present work the perect crystal o the element X (existing under standard conditions) is chosen as reservoir or reerence system. Then, the ormation energy o a oreign atom X on a lattice site is calculated by E ( X) = E ( X) ( N 1) E E (2) D 0,A X,A where N and E X,A are the number o lattice sites in the supercell and the energy per atom in in the bulk reerence crystal o the element X, respectively. In the case o the vacancy the ormation energy is deined by a similar relation but without the term E X,A. In the present work the ollowing reerence systems are used to determine the ormation energy o the corresponding oreign atom in bcc-e: (i) hcp-y, (ii) hcp-ti, (iii) bcc-cr, (iv) cc-cu, (v) Mn (cubic), (vi) cc-ni, (vii) bcc-v, (viii) bcc-mo, (ix) Si (diamond structure), (x) cc-al, (xi) bcc-co, and (xii) eo (NaCl structure). The ground states determined or Ni and Co are erromagnetic while the ground states o Cr, and eo are antierromagnetic. The other reerence systems have nonmagnetic ground states. Table I shows the lattice parameters o bcc-e and the reerence materials obtained by present DT calculations in comparison with experimental data. A good agreement is ound. The number o atoms in the supercell and the number o k points used in the calculation are also given in Table I. A relatively large size o the supercell is required to obtain a high accuracy o the phonon requencies (c. Sec. III). The table contains as well the notations o the PAW-PBE pseudopotentials employed in this work. D. Murali, M. Posselt, M. Schiwarth: irst-principles calculation o deect ree energies MS#BD

9 The ormation energy o an O interstitial atom in bcc-e is determined using the relation E ( O) = E ( O) ( N 1) E E (3) D e,a eo,u where EeO, U is the energy o eo per ormula unit. The ormation and binding energy o the O-v pair are calculated using E (O-v) = E (O-v) -( N -2) E - E (4) D e,a eo,u E (O-v) = E (O-v) -E ( O) - E (v) (5) bind By deinition the value o E bind is negative i attraction between O and v dominates. The energy barrier or vacancy migration in bcc-e was evaluated using the Nudged Elastic Band method 62,63 by considering the exchange o a e atom with the nearest neighbor vacancy. The states with the initial and inal positions o the vacancy were connected by a number o system images that were constructed along the reaction path. These images were determined by (restricted) energy minimization under both CV and ZP conditions. In order to ind an exact saddle point with a minimum number o images the climbing image method was employed as implemented in the vtsttools. 64 B. ormation energy o the vacancy and oreign atoms Table II shows the calculated ormation energies E o the vacancy, Cu, Y, Ti, Cr, Mn, Ni, V, Mo, Si, Al, Co, O, and the O-v pair in bcc-e. The most avorable position o the O atom is the octahedral interstitial site while the other species preer regular lattice sites. or all these examples the data obtained by CV and ZP are presented or supercells with 54 and 128 bcc sites. The ormation energy o the vacancy and the Cu atom was also determined or the supercell with 250 sites. A clear quantitative comparison o the E data obtained or the dierent deects is not simple since this depends on the choice o the reerence system. In the D. Murali, M. Posselt, M. Schiwarth: irst-principles calculation o deect ree energies MS#BD

10 present work the reerence systems were chosen in the same or similar manner as in related papers dealing with such deects e. 2,5,6,7,32,34,40-49 In most cases the reerence crystal was selected to model the condition o a very dilute e alloy where the solubility or equilibrium concentration o the oreign atom C ( X ) is directly related to the total ree ormation energy ( XT, ) (c. e.g. Res. 13, 38, and 39) tot eq C eq = exp tot ( XT, ) kt B (6) with tot vib el ( XT, ) = E( X) + ( XT, ) + ( XT, ) (7) el ( XT, ) and ( XT, ) denote contributions o phonon and electron excitations to vib vib el ( XT, ) (c. Secs. III and IV). I one ignores the quantities ( XT, ) and ( XT, ) the tot equilibrium concentration is approximately determined by the value o E ( X ). The comparison with available phase diagrams 65 shows that ormation energy data given in Table II ollow a similar trend as the solubility o the oreign atoms in bcc-e. A low ormation energy correspond to a much higher solubility than a high ormation energy. Amongst the oreign atoms considered in Table II, Y and O have the highest ormation energies. According to the phase diagrams these atoms have also the lowest solubility in bcc-e. Regarding the act that that in Table II certain values o E ( X ) are negative we reer to the corresponding results and their discussion in the literature. 32,40,41,44,45,47,49,66,67 In Table II DT data rom the literature are shown as well. Most o them were obtained by CV. Apart rom Cr a good agreement with results o the present work is obtained i PAW pseudopotentials were employed in these calculations 5-7,32,34,40-42,46,49,68 while a larger dierence is ound i the less accurate USPP pseudopotentials were used Similar dierences were also ound by other authors. 6,33,34 The dierence to Olsson s PAW-based D. Murali, M. Posselt, M. Schiwarth: irst-principles calculation o deect ree energies MS#BD

11 value or the substitutional energy o Cr (Res. 32 and 68) could be explained by the act that in the present work another pseudopotential was used or Cr (c. Table I and Re. 68). In many cases results o this work obtained or supercells with 54, 128, and 250 bcc sites are nearly equal, both or CV and ZP. This is an indication that the choice o a supercell with 54 bcc sites already leads to reasonable results. Moreover, or a given supercell size results o CV and ZP are oten very similar. The total internal pressure in the supercell obtained in the case o CV is given in Table III. I the ZP procedure is applied the volume o the supercell expands or contracts depending on the type o the point deect. The corresponding data are also given in Table III. While the introduction o v and Si leads to a contraction the other oreign atoms cause an expansion. As expected, the internal pressure p and the relative change o the cell volume ( V V0)/ V0 are considerably reduced i the supercell size increases. On the other hand the absolute volume dierence ( V V0 ) does not change signiicantly with supercell size, which is another indication that a cell with 54 lattice sites is oten an acceptable choice. In most cases an isotropic expansion or contraction is observed so that the cubic shape o the supercell is conserved. or the O octahedral interstitial a larger expansion in the direction o the two nearest neighbor e atoms than in the other two directions is obtained by ZP. Such a tetragonal distortion was also observed or other oreign atoms on octahedral sites, in particular or carbon. 69 In general it is ound that the additional relaxation o volume and shape o the supercell leads to a slight decrease o the ormation energy. In the case o Y and O a supercell with 128 bcc sites is obviously not large enough to yield completely converged data or the ormation energy. The atomic size o Y is very large compared to the e host atoms and O is an extra atom in the bcc lattice on the octahedral interstitial position. Thereore, or Y and O the total internal pressure observed or CV and the expansion o the supercell ound by ZP are highest compared to the other oreign atoms (c. D. Murali, M. Posselt, M. Schiwarth: irst-principles calculation o deect ree energies MS#BD

12 Table III). The dierence between the ormation energy obtained by CV and ZP can be determined by 70 CV ZP V E E = Cijkl εij εkl (8) 2 where ε mn is the homogeneous strain on the supercell in the case o ZP and Cijkl is the tensor o the elastic constants. I ZP leads to an isotropic expansion or contraction this dierence is given by 2 CV ZP 1 V V 1 V 1 V E E = BV= p = pv ( V ) 2 V 2 B 2 V (9) where B 0 is the bulk modulus o bcc-e. In the case o a tetragonal distortion o the supercell a slightly dierent relation is valid. Taking the values o CV E and ZP E rom Table II and the pressure and volume data rom Table III this kind o relation was employed to conirm that the results obtained by CV and ZP are ully consistent with each other. Details can be ound in the Supplemental Material. 71 Eq. (9) clearly illustrates that E CV E tends towards zero with increasing supercell size, ZP CV ZP E E 1/ N, since ( V V0 / V0) 1/ N and V N, V0 N. That means that results o CV and ZP become equal or a suiciently large supercell, i.e. in the case o a macroscopic crystal (c. also Re. 70). C. Energetics o the O-v pair The ormation energy o the O-v pair at a distance equal to about a hal o the lattice parameter is also given in Table II. The dierence between the values o E (O-v) determined by CV and ZP is smaller than or E (O). The binding energy o the O-v pair determined by Eq. (5) is and ev i ormation energies calculated by CV and ZP D. Murali, M. Posselt, M. Schiwarth: irst-principles calculation o deect ree energies MS#BD

13 are used, respectively. The latter number agrees well with the value o ev obtained by Barouh et al. 72 using the SIESTA code and zero pressure conditions while the ormer is in the range o the literature data obtained under constant volume conditions (Re. 42: ev, Re. 40: ev, Re. 49: ev, Re. 41: -1.7 ev). Obviously, the O-v pair is energetically highly avored compared to the coexistence o a single vacancy and a single O interstitial. More distant O-v pairs show a signiicantly lower binding. These results are ully consistent with indings o the other researchers ,49,72 The energy gain by the pair ormation is due to the reconiguration o the electronic structure and the reduction o the elastic strain. The latter can be clearly seen by comparing the corresponding lines in Table III. Similarly to the case o the O interstitial a tetragonal distortion o the supercell is ound by ZP. The energy barrier D. Migration energy o the vacancy E mig or vacancy migration in bcc e was ound to be 0.68 ev both or CV and ZP, or a supercell containing 54 bcc sites. This is similar to the case o vacancy ormation energy where the dierence between results o CV and ZP is very small. The above value or the migration energy agrees well with data obtained by previous DT calculations (Re. 7: 0.64 ev, Re. 43: 0.65 ev, Re. 46: 0.67 ev, Re. 47: 0.68 ev, Re. 48: 0.66 ev, Re. 73: 0.67 ev). III. EECT O PHONON EXCITATIONS A. Computational procedure In order to determine the contribution o phonon excitations to the ree energy, the vibrational requencies o the corresponding supercells were calculated using the method implemented in the VASP code. This procedure employs the harmonic approximation and the rozen phonon approach (c. Res. 7, 74, and 75). In the present work two kinds o D. Murali, M. Posselt, M. Schiwarth: irst-principles calculation o deect ree energies MS#BD

14 calculations were perormed or the supercells with deects: The dynamical matrix with the orce derivatives was determined using ground-state atomic positions obtained either by the CV or ZP. In the VASP code the calculation o phonon requencies is done only or the gamma point in the space o the phonon wave vectors. Thereore, a reasonably large supercell size must be chosen both or the cell with the deect and with the reerence material (c. Table I). Most o the vibrational calculations or supercells with a deect were perormed using 54 bcc sites and a Brillouin zone sampling o k points. In order to check the convergence o the vibrational contribution to the ree energy with respect to supercell size, some calculations were also carried out or supercells with 128 sites and k points as well as with 250 sites and k points. The diagonalization o the dynamical matrix yields all vibrational requencies ω i o the supercell. Within the harmonic approximation the vibrational ree energy vib ( T ) o a system o N atoms is given by ω i 1 kt B ( ) = + ln 1 e (10) 3N 3 vib T ωi kt B i= 1 2 ( ) = ( ) ( ) (11) vib vib vib T U T T S T where U vib ( T ) and S vib ( T ) denote the vibrational contribution to internal energy and entropy, respectively, and k B is the Boltzmann constant. S vib ( T ) can be obtained by S vib ( T) = T vib (12) The vibrational contribution to the ree ormation energy o a oreign atom X on a lattice site is determined similarly to Eq. (2) ( T) = ( T) ( N 1) ( T) ( T) (13) vib vib vib vib D 0,A X,A (c. Res. 5 and 6) where vib ( T ) denotes the vibrational ree energy o the bcc-e supercell D D. Murali, M. Posselt, M. Schiwarth: irst-principles calculation o deect ree energies MS#BD

15 with the oreign atom X and N lattice sites, whereas vib 0, A ( ) T and vib X, A( T) are the vibrational ree energy per atom in perect bcc-e and in the bulk reerence crystal o the element X, respectively. Modiied relations must be used or the vacancy and the O atom. The vibrational contribution to the ree ormation and binding energy o the O-v pair is calculated in a manner similar to Eqs. (4,5). The vibrational contribution to the ormation entropy is determined by a relation similar to Eq. (12). The vibrational contribution to the ree migration energy o the vacancy is calculated by the dierence between the vibrational ree energy at the saddle point and at the equilibrium state o the vacancy. ( T) = ( T) ( T) (14) vib vib vib mig SP 0 The quantities on the right-hand side o Eq. (14) are determined using Eq. (10) with the exception that in the irst term the sum is over 3N 4 vibrational degrees o reedom. In calculations o vib ( T) atomic positions determined by both CV and ZP were used. Within mig the harmonic transition state theory (c. Res. 10, 76, and 77) the vacancy diusivity in the bcc lattice is obtained by vib ( ) 2 kt Dmig T E B mig DT ( ) = a exp exp h kt B kt B (15) where h is Planck s constant. At this point the urther use o the acronyms CV and ZP must be explained. In this work the contributions o phonon and electron excitations to deect ree energies are determined or the two dierent systems (supercells with deects) obtained by energy minimization in the ground state under CV and ZP conditions. The corresponding results are thereore also denoted by the terms CV and ZP. However, the ZP-based data or the vibrational requencies and the corresponding contributions to the ree energy can only describe approximately the zero pressure case, namely at low temperatures. Additionally, the D. Murali, M. Posselt, M. Schiwarth: irst-principles calculation o deect ree energies MS#BD

16 ZP-based vibrational requencies are corrected according to the quasi-harmonic approach (c. e.g. Re. 78 and 79) ZP, qh ZP V ωi = ω i 1 γ V (16) This relation allows the determination o requencies or zero-pressure conditions at higher temperatures than the purely harmonic approach but it is limited to temperatures, where anharmonic eects do not prevail. The calculation o the volume increase V due to thermal expansion, and also due to the small eect o zero point vibrations, is explained in the Supplemental Material. 80 ZP The quantity ωi denotes ZP-based requencies determined within the harmonic approximation, i.e. using the dynamical matrix, γ is the thermodynamic or phonon Grüneisen parameter o bcc-e, and V is the volume o the supercell with the deect determined by ZP in the ground state (c. Table III). In general Eq. (16) holds i V / V is suiciently small. In the ollowing the quasi-harmonic approach (16) is applied to correct the ZP-based vibrational contributions to deect ree energies in selected cases. In order to obtain these data, because o Eq. (13) the quasi-harmonic approach must be also employed in the calculation o the requencies and the vibrational ree energy o the supercell containing the reerence material. Since the majority o experiments are perormed at zero external pressure and elevated temperatures the deect ree energy data resulting rom the quasi-harmonic correction are most interesting or a comparison. In this work the quasi-harmonic approach [Eq. (16)] is used under the assumption that the volume change V o a supercell with a deect can be determined in a similar manner as or a supercell containing bulk bcc-e and that the values o γ are identical or both types o supercells. This procedure can be considered as a irst approximation. In the ollowing a more accurate (quasi-harmonic) approach is described briely. However, since it requires a very high computational eort it is not employed here. or given temperatures the vibrational ree energy may be calculated or D. Murali, M. Posselt, M. Schiwarth: irst-principles calculation o deect ree energies MS#BD

17 dierent values o the supercell volume. Since the pressure is the negative derivative o the ree energy with respect to the volume, the zero-pressure case corresponds to the minimum o the vibrational ree energy with respect to the supercell volume. In this manner the requencies and the corresponding vibrational ree energy may be determined at given temperature, or the supercells with and without the deect. rom these results deect ree energies may be determined using Eq. (13). In order to validate the computational method used in this work the calculated vibrational ree energy vib ( T ) and entropy S vib ( T ) o bulk bcc-e, cc-cu, cc-al, and hcp- Y were compared with data rom literature, taking into account recent developments and discussions related to the SGTE database. 28,79,81-84 In general a very good agreement is ound. or details the reader is reerred to the Supplemental Material. 80 B. Inluence o deects on vibrational requencies I a vacancy or a oreign atom is introduced into the bcc-e supercell under CV conditions the interatomic distances in the environment o the deect are increased or reduced and there is a buildup o a total internal pressure in the supercell (c. Table III). The changed equilibrium atomic positions lead to a modiication o the atomic orce constants. This eect and the act that the mass o the oreign atom is dierent to the e atomic mass cause the modiication o the vibrational requencies o the supercell. On the other hand, i the deect is introduced under ZP conditions not only the equilibrium positions o atoms in the deect environment are changed but there is also a position change o all the other atoms due to the increase or decrease o the supercell volume (c. Table III). Both eects cause a change o orce constants and lead, together with the mass eect, to an alteration o the vibrational requencies o the supercell. D. Murali, M. Posselt, M. Schiwarth: irst-principles calculation o deect ree energies MS#BD

18 igs. 1-3 depict the relative deviation o the vibrational requencies ω i o a supercell with a deect rom the phonon requencies 0 ω i o the bulk bcc-e supercell. The straight lines were obtained by a linear it to the data points and shall only show the trends. The representation o ( ω ω )/ ω as unction o 0 0 i i i 0 ω i was ound to be more suitable to illustrate requency modiications than depicting the phonon density o states. Note that in all examples 0 considered the numbering o ω i and ω i by the index i starts at the highest requency. In the case o the vacancy and the O interstitial where the total number o atoms in the supercell is not identical to that in the perect bcc-e supercell the 3 lowest requencies o the respective cell with the higher number o atoms were discarded. This does not aect the qualitative discussion on the modiication o the vibrational requencies due to the presence o a deect. ig. 1 (a) shows that the presence o the vacancy leads to a relative decrease o most o the phonon requencies. I the atomic positions were obtained by CV the relative decrease o the vibrational requencies is slightly higher than in the case o ZP. The eect ound or the vacancy in a supercell with 128 bcc sites is qualitatively similar to that or the smaller cell [ig. 1 (b)]. However, the relative decrease o the vibrational requencies is generally smaller since the eect o the vacancy on the whole phonon spectrum o the larger cell is weaker. igs. 2 (a) and (b) depict the results or Cu in bcc-e or two dierent supercell sizes. The presence o Cu leads predominantly to a relative decrease o the vibrational requencies. The relation between the data or the larger and smaller supercell is similar to the case o the vacancy. Present results or Cu are in accord with indings o Reith et al. 6 ig. 3 (a) illustrates the case o Y in a supercell with 54 bcc sites. Here a relative decrease o the vibrational requencies in the low requency range and an increase in the high requency range are observed i the equilibrium positions o atoms were obtained by CV. The vibrational requencies mainly decrease i the atomic positions were obtained by ZP. or the O octahedral interstitial the relative increase o the vibrational requencies dominates i the equilibrium D. Murali, M. Posselt, M. Schiwarth: irst-principles calculation o deect ree energies MS#BD

19 positions o atoms were determined by CV [ig. 3 (b)]. The ZP-based results show requency shits in both directions. In the case o the O interstitial there exist three high requencies with energies o about (0.089), (0.045), and (0.042) ev, or CV- (ZP)-based calculations [not completely shown in ig. 3 (b)]. ig. 3 (c) depicts the relative deviation o the vibrational requencies or the O-v pair. The behavior is qualitatively similar to that observed or the O interstitial. Three o the highest vibrational requencies increase essentially, by about 0.025, 0.012, and ev [not shown in ig. 3 (c)]. Ater showing these illustrative examples two aspects will be discussed: (i) The change o the vibrational requencies due to the introduction o the deect in the perect crystal under CV conditions, and (ii) the dierence between the phonon requencies in the supercell with the deect in the CV case and those in the ZP case. In order to understand the change o the vibrational requencies due to the presence o a deect the relaxation o the neighboring e atoms and the modiication o the orce constants is discussed in the CV case. or details the reader is reerred to the Supplemental Material. 85 The presence o the vacancy causes a signiicant relaxation o e atoms in the irst neighbor shell towards the vacancy position, whereas the opposite eect is observed or Cu. In both cases the relaxation in the second and every subsequent shell is decreasing but oscillates rom shell to shell. Interestingly, the relaxation leads to a decrease o orce constants o e atoms in the irst neighbor shell o both the vacancy and the Cu substitutional atom. urthermore, the orce constants o the Cu atom are smaller than those o perect bcc-e. On the average atoms in the second neighbor shell o v and Cu have orce constants close to values or perect e. rom these indings one can conclude that the relative decrease o most o the vibrational requencies observed or v and Cu in the CV case (c. igs. 1 and 2) is due to the decrease o orce constants (since the requency is proportional to the square root o the orce constant) which is caused by the change o interatomic distances around the deect. The act that the D. Murali, M. Posselt, M. Schiwarth: irst-principles calculation o deect ree energies MS#BD

20 mass o Cu is slightly higher than that o e may also contribute to this eect. or the explanation o the dierence between the vibrational requencies o the supercell with the deect in the CV case and those in the ZP case, it is examined whether the relation between both sets o data can be approximately described according to an expression similar to Eq. (16) ZP CV V V 0 ωi ωi 1 γ V0 (17) with data or V 0 and ( V V0 ) rom Table III. Indeed, i the dierence between ZP ωi and CV ω i is determined or each i, one obtains average values o γ between 1.65 and 1.88 (see Supplemental Material 86 ) which are close to the phonon Grüneisen parameter o bcc-e. 87,88 An average over all data gives the value γ = Thereore, the dierence between the CV- and ZP-based requencies is predominantly due to the volume change o the whole supercell in the ZP case, while the relative arrangement o atoms in the environment o the deects is nearly identical in the two cases. This is an important result which is used in the discussion o the dierence between CV- and ZP-based data or deect ree energies (Sec. III C 4). Note that it is not sel-evident that the relative arrangement o the e atoms around the deect is nearly the same or the CV and ZP results but this must be examined in each case considered. In the Supplemental Material 85 the relaxation o the e atoms in the irst ive neighbor shells o a deect is illustrated or the vacancy and the Cu atom. The trends obtained or the relative change o the orce constants in the ZP and CV case (c. Supplemental Material 85 ) are ZP CV consistent with the trend or the dierence between ωi and ω i. C. Vibrational contributions to ree ormation and binding energies 1. Vacancy and Cu D. Murali, M. Posselt, M. Schiwarth: irst-principles calculation o deect ree energies MS#BD

21 irst, the results or the single vacancy and the single Cu atom in bcc-e are presented since in these cases the only literature data are available or comparison. ig. 4 (a) shows the vibrational contribution to the ree ormation energy o the vacancy. or supercells with 54, 128, and 250 bcc sites the data are depicted by thin, dashed and thick lines, respectively. The dierence between the curves or the three cell sizes is very small or the ZP-based data ( ) whereas a size eect is ound or the CV-based data ( vib,zp vib,cv ). This indicates that only in the irst case calculations using a supercell with 54 lattice sites yield completely accurate results. The data o Lucas et al. 5 are also given in the igure. They were calculated using equilibrium positions o atoms obtained by CV, a supercell with 128 bcc sites, and k points. or supercells with 54 and 128 lattice sites the agreement with the CVbased data calculated in the present work is very good. On the other hand, ig. 4 shows that the results based on equilibrium positions o atoms obtained by ZP ( data based on CV ( vib,cv vib,zp ) dier rom the ). The dierence is considerably higher than that o the ormation energies E (Table II). The reason or this will be discussed later (Sec. III C 4). ig. 4 demonstrates that with increasing temperature the phonon contribution to the ree ormation energy leads to a considerable reduction o the total ree ormation energy o the vacancy, i.e. the thermodynamic stability o the vacancy increases, which also contributes to an increase o its equilibrium concentration [c. Eq. (6)]. Note that the nonzero value o vib at T = 0 is due to the zero point vibrations. The application o the quasi-harmonic correction [Eq. (16)] to ZP-based phonon requencies o the supercell with the vacancy and to that with perect bcc-e leads to values o the ree ormation energy, that are almost identical to the original ZP-based data shown in ig. 4. The slope o the curves corresponds to the vibrational contribution to the ormation entropy [c. Eq. (12)]. At relatively high temperatures vib S is a constant and about 4.8k B and 3.0k B in the CV and the ZP case, respectively. Above 500 K a constant value or D. Murali, M. Posselt, M. Schiwarth: irst-principles calculation o deect ree energies MS#BD

22 vib S is obtained or all examples depicted in igs This can be explained by considering Eq. (10) in the high-temperature limit ω i << kt B, in combination with Eqs. (12) and (13). Then vib S becomes independent o temperature and contains only the logarithmically averaged phonon requencies (c. e.g. Re. 78). Obviously, in the examples o igs. 4-7 such a relation is already valid at temperatures slightly above the Debye temperature o bcc-e. This is not surprising because or other physical properties a similar validity range is known or the high-temperature limit (c. e.g. Re. 78). The thin, dashed and thick lines in ig. 5 (a) depict the vibrational contribution to the ree ormation or substitutional energy o Cu, or supercells with 54, 128, and 250 bcc sites, respectively. There is only a small size eect suggesting that the choice o a cell with 54 lattice sites leads already to satisactory results. or comparison the data o Reith et al. 6 are also shown. These authors used a supercell with 64 bcc sites and equilibrium positions o atoms obtained by ZP. As in the case o the vacancy the thermodynamic stability o the Cu substitutional atom increases with temperature, which contributes to an increase o the Cu solubility in bcc-e [c. Eq. (6)]. There is a qualitative agreement between the data o Re. 6 and the ZP-based results ( vib,zp ) o the present work. The remaining deviations should be mainly related to the use o a non-cubic supercell in Re. 6. Similar to ig. 4 there is a nonnegligible dierence between the data calculated using equilibrium positions o atoms obtained by CV and ZP. On the other hand, the values o the ormation energy E determined using CV and ZP are nearly identical (Table II). Unlike the case o the vacancy, there is an outwards relaxation o the atoms around the Cu atom which leads to an increase o the volume o the supercell (Table III) in the ZP case. Obviously, this eect results in a stronger temperature dependence o the vibrational contribution to the ree ormation energy than in the CV-based case (see also Sec. III C 4). It must be mentioned that such a dierence was also D. Murali, M. Posselt, M. Schiwarth: irst-principles calculation o deect ree energies MS#BD

23 discussed qualitatively by Reith et al. 6 but these authors showed only their ZP-based data explicitly. Quasi-harmonic corrections were applied to the ZP-based ree ormation energy and slight modiications at temperatures above 700 K were ound [ig. 5 (b)]. It must be noticed that in igs. 4, 5, and in the ollowing igures the temperature ranges up to 1200 K which is above the temperature o the erromagnetic-to-paramagnetic transition (1043 K) and also slightly above the temperature o the α -to-γ -phase transition (1183 K) o iron. rom the viewpoint o magnetism the presented data are only valid below the Curie temperature. On the other hand the temperature dependence o the spontaneous magnetization in the erromagnetic state is not considered here. I not stated otherwise it is always assumed that the magnetization o bulk iron corresponds to its value at T = Y, Ti, Cr, Mn, Ni, V, Mo, Si, Al, Co, O The vibrational contribution to the ree ormation energy o several solutes is shown in igs. 6 (a)-(k) by the thick lines. The results were calculated or a e supercell with 54 bcc sites using the equilibrium positions o atoms determined by CV and ZP. In general a signiicant dierence between the CV-based ( ) and the ZP-based ( vib,cv vib,zp ) data is ound. With the exception o Si the temperature dependence is qualitatively similar to that shown in ig. 5 or Cu, i.e. the data or vib,cv are higher than those or vib,zp. This should be also due to act that in these cases ZP yields an increase o the supercell volume (see also Sec. III C 4). or Si the behavior is similar to that o the vacancy (ig. 4) because o the lower volume in the case o ZP. In contrast to igs. 4 and 5 in many cases the CV- and ZP-based curves show opposite trends: While the latter decrease mostly with temperature, the ormer show oten an increase. In this case, the thermodynamic stability o the oreign atom in bcc-e decreases with temperature. Such a behavior was also ound in the ew recent papers on deect ree energies, i.e. in the work o Reith et al. 6 or the pair o e atoms in cc-cu, in the work o D. Murali, M. Posselt, M. Schiwarth: irst-principles calculation o deect ree energies MS#BD

24 Bjørheim et al. 14 or the oxygen vacancy in ZnO, in the paper o Gryaznov et al. 15 or the oxygen vacancy in Sr-doped complex pervoskites, and in the work o Kosyak et al. 16 or some point deects in Cu 2 ZnSnS 4. The dierence between vib,cv and vib,zp is highest or Y and O, e.g. about ev at 1000 K. In both cases the corresponding ormation energies and ZP E dier only by about 0.2 ev (Table II). On the other hand, the dierence between the values o the ormation energy o Si determined by CV and ZP is only about ev which CV E might be the reason or the small dierence o the corresponding curves or vib,cv and vib,zp in ig. 6 (h) (about ev at 1000 K). In the other cases the dierence E CV E is ZP between 0.04 and ev, whereas 0.1 and 0.2 ev at 1000 K. vib,cv and vib,zp show larger dierences, e.g. between The application o quasi-harmonic corrections to the ZP-based ree ormation energy was investigated or Y, Ti, Ni, V, Mo, Al, and Co. or Y, Ni, and Co the modiications obtained are negligible. In the other cases the corresponding curves are shown in the igures. In general the change o the corresponding values o the ree ormation energy is small and only visible above about 700 K. A stronger modiication is ound or Al. The dierent results can be explained by the interplay o the three terms in Eq. (13) which depends on the particular oreign atom considered. The shit o the vibrational requencies according to Eq. (16) was determined using values o the Grüneisen parameter which were obtained rom literature data or the volume expansion coeicient, the bulk modulus, the density, and the speciic heat capacity Details are given in the Supplemental Material. 80,91,92 3. O-v pair The temperature dependence o the vibrational contribution to the ree ormation energy o the O-v pair depicted in ig 7 (a) looks somewhat dierent to that o single oreign D. Murali, M. Posselt, M. Schiwarth: irst-principles calculation o deect ree energies MS#BD

25 atoms. The dierence between the curves or vib,cv and vib,zp is about 0.1 ev at 1000 K which is substantially smaller than in the case o the O interstitial. This trend is similar to that ound or the dierence between the corresponding ormation energies (Sec. II C). ig 7 (b) illustrates the vibrational contribution to the ree binding energy o the O-v pair. The CV- and ZP-based data dier considerably, e.g. 0.3 ev at 1000 K. Such dierences should be generally expected or the ree binding energy o nanoclusters consisting o point deects and oreign atoms. At 1000 K vib,zp bind is about 0.34 ev while the ground state value ZP E bind is ev (Sec. II D). Neglecting vib,zp bind would thereore lead to an error o the total ree binding energy o about 20%. 4. Relation between vib,cv and vib,zp Summarizing the results depicted in igs. 4-7 one can conclude that the vibrational contribution to the ree ormation and binding energy is generally not negligible. In the examples considered, the ratio vib ( T = 1000K) / E ranges between 0.01 and 8. The highest number is obtained or Mo because this solute has a very low ormation energy E. In most cases vib,zp decreases monotonously with temperature, which may lead to an increase o the equilibrium concentration or solubility o the deects [c. Eq. (6)], whereas or vib,cv both an increase and a decrease is ound. The consideration o the quasi-harmonic correction [Eq. (16)] leads only to a minor modiication o this picture. On the other hand, the corrected ZP-based data are valid under zero-pressure conditions at higher temperatures than the data obtained within the purely harmonic approach. These conditions are usually realized in experiments. Vibrational eects must be also considered in the determination o the total ree binding energy o deect clusters as illustrated in the case o the O-v pair. Another important inding concerns the origin o the dierence between CV- and ZP-based data. It is D. Murali, M. Posselt, M. Schiwarth: irst-principles calculation o deect ree energies MS#BD