Ab initio modelling of Y O cluster formation in g-fe lattice

Transcription

1 Phy. Statu Solidi B 253, No. 11, (2016) / DOI /pb Ab initio modelling of Y O cluter formation in g-fe lattice phyica tatu olidi baic olid tate phyic Alekej Gopejenko *,1, Yuri F. Zhukovkii 1, Eugene A. Kotomin 1, Yuri A. Matrikov 1, Pavel V. Vladimirov 2, Vladimir A. Borodin 3, and Anton M olang 2 1 Intitute of Solid State Phyic, Univerity of Latvia, Riga, Latvia 2 Intitut f ur Angewandte Materialien, Karlruhe Intitut f ur Technologie, Karlruhe, Germany 3 NRC Kurchatov Intitute, Mocow, Ruia Received 31 May 2016, revied 15 July 2016, accepted 25 July 2016 Publihed online 26 Augut 2016 Keyword defect, defect migration, denity functional theory calculation, iron, oxygen, vacancie * Correponding author: agopejen@inbox.lv, Phone: þ , Fax: þ Ab initio modelling of Y and O impurity atom a well a V Fe vacancie in the fcc-fe lattice i performed in order to calculate the interaction between thee defect, which are important for undertanding of nanoparticle formation within the oxide dipered trengthened teel. Large cale parallel calculation baed on plane-wave method realied in VASP computer code how that V Fe vacancie coniderably influence the binding between the impurity atom. In thi tudy, we preent the reult of performed calculation providing the detailed information about the binding energie between the defect, the change of their effective charge a well a diplacement of the ubtitute atom relatively the hot atom poition. The energy barrier for the migration trajectorie of impurity atom have been alo found by performing the large-cale calculation within the nudge elatic band method. 1 Introduction Development of oxide dipered trengthened teel (ODS), aimed at their mechanical trengthening and increaing their working temperature, ha been tarted in 1950 [1]. Since that time, their hightemperature creep and radiation reitance are being permanently improved through the variation of compoition of diperion particle and their ize ditribution by fine tuning the detail of fabrication proce and ubequent heat treatment. Thi proce i rather time-conuming a it implie trial and error approach. Modelling of diperion particle nucleation and growth baed on the reliable atomic cale mechanim and data would allow targeted variation of particle denity and ize ditribution thu improving mechanical and radiation reitance propertie of the bae material. Current generation of reduced activation ferriticmartenitic teel (RAFM) trengthened by oxide lead to increae of the operating temperature for future fuion and advanced fiion reactor by 100 8C, up to 650 8C or even higher [2]. The commonly ued metal oxide for the trengthening of RAFM teel i Y 2 O 3. It i one of the mot table oxide with melting temperature higher than that of the teel [3]. Thee material are produced by mechanical alloying for everal ten of hour, followed by a hot iotatic preing (hipping) under preure 100 MPa at temperature around C, which i above the phae tranition into fcc phae for low (<12 wt.%) chromium teel [4]. Experimental evidence ugget that after mechanical alloying a ignificant part of Y and O atom i located in teel matrix with concentration above their equilibrium olubility [5]. Thi might mean that nucleation and growth of Y 2 O 3 nanoparticle, according to numerou experimental and theoretical tudie [6], occur already during the hipping tage in fcc phae of low-cr teel. We purue a two-tep approach for the atomitic imulation of the oxide nanoparticle formation. The firt tep, decribed in thi tudy, include large-cale ab initio calculation of different defect complexe (vacancie, yttrium and oxygen atom) in variou patial configuration in order to extend interaction databae neceary for imulation of oxide particle nucleation and growth. Defect interaction energie and diffuion barrier of olute atom extracted from thee calculation will be ued in further atomitic imulation. The econd tep will include the lattice kinetic Monte Carlo (LKMC) imulation baed on the reult of ab initio calculation (or formalim of rate equation), in order to tudy the kinetic of the growth of the

2 Original Paper Phy. Statu Solidi B 253, No. 11 (2016) 2137 precipitate [7]. Our previou calculation [8 10] performed for g-fe phae have clearly hown that vacancie play a ignificant role in the formation of the ODS precipitate in the low-cr teel matrix. According to our previou reult, there i a trong repulion between overized Y olute atom and the atom of the iron lattice. Subtitutional yttrium willingly collect everal vacancie to reduce the elatic relaxation of the lattice in it neighbourhood. Thee V Fe vacancie could be healed later by oxygen atom, thu creating Y O bond. In the preent tudy, we have invetigated at atomic level interaction between V Fe vacancie, yttrium olute atom Y Fe a well a oxygen impurity atom (denoted a O Fe at regular lattice ite and a O i at intertitial poition) within g-fe matrix. Since a low migration of Y Fe atom, compared to much fater intertitial O i atom diffuion, might be conidered a the limiting factor of the of Y O precipitate growth inide iron matrix, the behaviour of Y Fe atom ha been conidered firt. In thi work, the reult obtained in our previou tudie (mainly formation energie of defect and binding energie) [8 10] are complemented with important data (effective charge and relaxation of defect inide lattice) recently obtained for the ame lattice configuration. Moreover, in thi article we conider new configuration of defective fcc-fe lattice decribed by the ame parameter and contruct the migration trajectorie of impurity atom uing the nudge elatic band method (NEB) [11]. 2 Model and method The firt principle calculation for g-fe have been performed uing the VASP 5.2 computer code. Thi i baed on the Denity Functional Theory (DFT) approach with a plane-wave (PW) bai et [11, 12] and Perdew-Wang-91 GGA [13] non-local exchange-correlation functional which operate with Fe core electron of (4 1 3d 7 outer hell), O (2 2 2p 4 ) and Y (4 2 4p d 2 ) atom with 8, 6 and 11 external electron, repectively. The core electron are decribed within the projector-augmented wave (PAW) method [14]. It i well known that fcc iron i paramagnetic [15], which mean that at temperature where g-fe i table in reality, the magnetic moment of iron atom are diordered. The magnetic moment of individual atom are not tatic, but change their direction dynamically giving, however, by averaging over ufficiently large volume a total magnetic moment equal to zero. An attempt to model uch behaviour uing DFT method would mean that for each defect configuration one i forced to calculate many configuration with variou random magnetic moment orientation. Intead, we aume that uch averaging would reult in obtaining zero magnetic moment on the mot of the atom and perform non-magnetic calculation. Nonmagnetic approach might be not completely correct for all cae, however, it give practical way in order to etimate the defect binding energie and other propertie neceary for modelling of yttrium oxide particle nucleation in g-iron. A erie of tet calculation have been performed, to define the parameter of calculation neceary to reproduce the baic experimental data for g-fe (lattice contant a 0, tructural relaxation around point defect dr i, coheive energy per atom E coh, etc.). Thi include an analyi how the convergence of the reult depend on the upercell ize, the cut-off energy and the k-point et in the correponding Brillouin zone. Our model upercell are cubic, with the dimenion of 4a 0 4a 0 4a 0 and 5a 0 5a 0 5a 0, extenion (containing 64 and 125 atom, repectively), the calculated optimied lattice contant of which have been found to be A both (cf A [8]). Core electron have been treated within the PAW-PW91 potential. Our tet calculation howed that the k - point et in the Brillouin zone hould be at leat k-meh for upercell (SC). We have choen to ample the Brillouin zone by the Monkhort Pack cheme [16]. The calculation on g-fe have hown that for achieving plauible reult for baic parameter mentioned above the cut-off kinetic energie hould increae at leat up to 800 ev [10]. Lattice parameter optimied for perfect fcc-fe have been ued for further calculation of defect configuration in g-iron. For arbitrary configuration of defective iron upercell, poition of all atom have been determined by a minimiation of the total energy. The effective atomic charge have been etimated uing the Bader topological analyi [17]. The choen computational model ha been verified for the key propertie of the perfect g-fe ingle crytal determined earlier both experimentally and theoretically [18 23]. The equilibrium lattice contant, bulk modulu and coheive energy for a cubic paramagnetic g-fe lattice have been calculated to be (i) 3.44 A ; (ii) 162 GPa and (iii) 4.96 ev atom 1, repectively. Obtained value can be conidered to be in qualitative agreement both with reult of other ab initio calculation on g-fe iron: (i) A [18, 19, 21]; (ii) GPa [18, 20] and (iii) 4.42 ev atom 1 [20], repectively. The ame i true for experimental value: (i) 3.57 A [22] and (ii) GPa [23] for g-fe a well a (iii) 4.28 ev atom 1 [21] for a low-temperature ferromagnetic a-fe, which i believed to be cloe to coheive energy of g-fe. 3 Reult and dicuion Different configuration of defect cluter containing O and Y impurity atom a well a vacancie on fcc-fe lattice were conidered by u elewhere [8 10]. In thi article, we ytematie the main reult preented there (accompanied by the correponding reference) and complement them with additional propertie and the new reult obtained for additional configuration of defective upercell. The two Y Fe atom incorporated in vacancie have been placed at progreively increaing ditance (1NN, 2NN, 3NN and 4NN correponding to the nearet neighbour eparation of conequent atomic coordination phere in lattice). A no binding i found between them at any interatomic ditance (i.e., the binding energie have been found to be negative, varying from 0.45 ev down to

3 phyica p tatu olidi b 2138 A. Gopejenko et al.: Ab initio modelling of Y O cluter formation in g-fe lattice Table 1 Calculated parameter for Y Fe V Fe pair. configuration E bind,ev[8 10] Dq (Y Fe ),e a dr Y,A Table 2 Binding in V Fe V Fe pair. configuration E bind,ev[8 10] 1NN NN NN NN NN NN NN NN 0.12 a Etimated within the Bader topological analyi [17] ev), we have conidered influence of vacancie on the binding of Y Fe atom in fcc-fe lattice. According to our calculation, poitive binding energie E bind have been found for the configuration where Y Fe and V Fe are 1NN, 3NN and 4NN, while repulion ha been found at 2NN ditance (Table 1 and Fig. 1). The larget binding energy of 1.67 ev correpond to 1NN ditance between Y Fe and V Fe accompanied by ignificant diplacement of 1.25 A of yttrium atom toward vacancy (correponding approximately to a half of 1NN ditance ince during the relaxation, the cloet atom are moved too). No noticeable change of effective charge on impurity atom (Dq) i induced irrepectively of the length of Y Fe V Fe pair. The reult decribing the interaction between the two lattice vacancie preented in Table 2 and Fig. 2 how weak binding between two V Fe vacancie poitioned at 1NN and 3NN and a light repulion for 2NN and 4NN interdefect ditance. The reult preented in Table 3 and 4 clearly how a noticeable influence of vacancie, e.g., their number and arrangement (Fig. 3 and 4), on the binding energie of ubtituted Y Fe atom (three and four vacancie have been poitioned in the cloet ite). Rather mall binding energy ha been found between three vacancie (0.36 ev), however, a noticeable binding energy ha been obtained between four vacancie (1.36 ev). The initial configuration are hown in Fig. 3 and 4. Analogou qualitative concluion can be drawn when comparing the binding energy of one (Fig. 5 7) or two (Fig. 8 10) yttrium atom in cluter of different ize and configuration inide fcc-fe lattice (cf. Table 3 and 4, repectively). In the Y Fe 2V Fe (1NN) configuration (Fig. 5), yttrium impurity atom donate 1.52e to the cloet Fe atom while it Figure 2 Initial 1NN configuration of V Fe V Fe pair. Table 3 Calculated parameter for Y Fe ubtitute with multiple vacancie. configuration E bind,ev Dq, e a (Y Fe ) dr Y,A Y 2V Fe (1NN) Y Fe 2V Fe (2NN) Y Fe 3V Fe a Etimated within the Bader topological analyi [17]. diplacement from the initial ubtitute poition i 1.03 A (Table 3). When Y Fe and two vacancie are at the 2NN poition (Fig. 6) Y Fe donate maller amount of 1.24e to the cloet Fe atom while it hift from the initial poition i 0.61 A. The calculation of Y Fe urrounded by three V Fe vacancie (Fig. 7) reult in yttrium atom diplacement by 1.28 A accompanied by donation of electronic charge 1.4e to the cloet lattice atom. Thi configuration i decribed by Table 4 Calculated parameter for two Y Fe atom at different ditance from V Fe vacancy. configuration E bind, ev Dq, e a (Y Fe ) dr Y (1), A dr Y (2), A Y Fe V Fe Y Fe 2NN Y Fe V Fe Y Fe 3NN Y Fe V Fe Y Fe 4NN Figure 1 Initial 1NN configuration of V Fe Y Fe pair. a Etimated within the Bader topological analyi [17].

4 Original Paper Phy. Statu Solidi B 253, No. 11 (2016) 2139 Figure 3 Initial three-v Fe configuration. Figure 7 Y Fe atom and adjacent V Fe vacancie for Y 3V Fe configuration. Figure 4 Initial four-v Fe configuration. Figure 8 Two Y Fe atom at 2NN poition and V Fe. Figure 5 Y 2V Fe (1NN) configuration. Figure 9 Two Y Fe atom at 3NN poition and V Fe. Figure 6 Y 2V Fe (2NN) configuration. Figure 10 Two Y Fe atom at 4NN poition and V Fe.

5 phyica p tatu olidi b 2140 A. Gopejenko et al.: Ab initio modelling of Y O cluter formation in g-fe lattice the highet binding energy. For all thee configuration, the impurity atom move to the central poition between the vacancie. The calculation for two Y Fe atom and V Fe vacancy have been performed for Y Fe V Fe Y Fe configuration (Fig. 8 10) where two yttrium atom are located at the increaing relative ditance: 2NN, 3NN and 4NN (Table 4) while V Fe i poititioned at 1NN ditance to both Y Fe. The binding energy increae ignificantly when going from 2NN to 3NN and eem to be aturated at larger ditance. Both Y Fe atom are ymmetrically hifted inide 2NN configuration donating 1e to the nearet Fe atom each (Table 4). In the configuration when both Y Fe atom are located at 4NN ditance, firt atom i diplaced by 0.19 A, while the econd move toward V Fe vacancy by 1.31 A. Our previou calculation [8] performed for the ame fcc-fe upercell have hown that the oxygen impurity atom can reide on either octahedral (O) or tetrahedral (T) intertitial poition (Fig. 11). (The latter ha been found to be ev le energetically favourable than the former.) The exceive electronic charge attracted by ingle oxygen impurity in O poition i 1.4e. The preence of intertitial oxygen reult in a coniderable expanion of the two nearet coordination phere around an impurity atom (1.75% and 9% for octahedral and tetrahedral poition, repectively). A barrier for oxygen atom migration between the nearet O and T poition ha been found to be 1.1 ev [8] (cf ev, according to variou experimental tudie ytematied in the handbook [24]). On the other hand, a intertitial O i atom approache V Fe vacancy, it can be trapped by the latter which block further intertitial migration of oxygen impurity atom. During the relaxation O i atom move from octahedral intertitial poition toward the V Fe by 0.07 A, wherea O Fe atom poitioned at V Fe move toward the octahedral intertitial poition by 0.79 A. The latter configuration i by 0.5 ev more energetically favourable than the former one. The migration barrier of O Fe between the octahedral intertitial ite O and V Fe vacancy ha been found to be 1.5 ev. We have calculated in the current tudy the pair-wie interaction of the two O Fe atom eparated by different ditance (Table 5 and Fig. 11). Reult of performed calculation indicate no binding energy for all value of O Fe O Fe eparation. Both O Fe ubtitute accept lightly Figure 11 Initial 1NN configuration of O Fe O Fe pair. more than 1.0e from the nearet Fe atom. For both oxygen atom at 1NN poition, d O O ditance grow by 1.28 A during the relaxation. For the two O Fe atom poitioned at 2NN, d O O i reduced on 1.19 A during the relaxation, while both for 3NN and 4NN configuration, d O O i again increaed: by 1.28 and 1.04 A, repectively. It ha been alo important to tudy how the O impurity atom might affect the binding energie between the two Y atom [9, 10]. Configuration of Y 2 O cluter contain two Y Fe atom arranged ymmetrically around O atom, o that Y atom are either 1NN or 2NN, while oxygen occupie nearet ubitutional or intertitial poition (Fig. 12 and 13, repectively). The binding energie have been obtained for the configuration where O Fe atom i located at the ubtitution poition with the highet binding energy aeed for the 2NN configuration (Table 6), while no Figure 12 Initial 1NN configuration between Y atom for the Y Fe O Fe Y Fe cluter. Table 5 Calculated parameter for O Fe O Fe pair in fcc-fe. configuration E bind,ev a Dq, e b (O Fe ) dr O (1), A dr O (2), A 1NN NN NN NN a The difference in the binding energie between the value obtained in the current tudy and that publihed in Ref. [10] i explained by the corrected value for formation energy of O Fe, which i found to be 1.1 ev lower. b Etimated within the Bader topological analyi [17]. Figure 13 Initial 1NN configuration between Y atom for the Y Fe O i Y Fe cluter.

6 Original Paper Phy. Statu Solidi B 253, No. 11 (2016) 2141 Table 6 Calculated parameter for the cluter Y Fe O Fe Y Fe and Y Fe O i Y Fe inide fcc-fe lattice configuration E bind,ev a Dq, e b (Y) Dq, e b (O) Y Fe O Fe Y Fe (1NN) Y Fe O Fe Y Fe (2NN) Y Fe O i Y Fe (1NN) Y Fe O i Y Fe (2NN) a Binding energie for triple-wie interaction in fcc-fe lattice have been defined and etimated in Ref. [10]. The difference in the binding energie for Y Fe O Fe Y Fe configuration 1NN and 2NN between the value obtained in the current tudy and thoe publihed in Ref. [10] can be explained by the corrected value for formation energy of O Fe found to be 1.1 ev lower. b Etimated within the Bader topological analyi [17]. binding ha been found when oxygen occupie intertitial poition. Thee reult together with relatively high binding energy of oxygen with vacancy indicate that formation of metal-oxygen bond with impurity or matrix atom i poible only when oxygen i at ubtitutional poition. It hould be noted that poition of Y and O atom in the mallet oxide cluter are different from thoe in Y 2 O 3 crytal. The latter i well-illutrated by eential tructural recontruction around impurity atom (Table 6 and 7). The only configuration where intertitial O i i not hifted i that where oxygen i located in the octahedral poition between Y Fe atom. Similarly to different configuration of Y or O impuritie hown in Fig. 8 13, Y Fe atom donate around e to the nearet O and Fe atom, while O both in intertitial and ubtitutional poition accept 1.0e from the nearet yttrium and iron atom. Thi can be wellillutrated by Fig. 14 and 15, which demontrate that during the relaxation O Fe ubtitutional atom i hifted toward the cloet Fe atom haring additional electron charge with them. On the other hand, appearance of negatively charged area between a pair of yttrium ubtitute and oxygen in Y Fe O Fe Y Fe (1NN) configuration can be oberved, which likely weaken the binding in that a compared to Y Fe O Fe Y Fe (2NN) configuration, where oxygen and yttrium atom are eparated enough in order to exclude overtrain of impurity atom. A to configuration Y Fe O i Y Fe, hown in Fig. 13, uch overtrain in it i eentially larger, which reult in configuration intability (Table 6 and 7). Figure 14 Difference electron charge denity map for 1NN Y Fe O Fe Y Fe configuration for Y Fe and O Fe impurity atom, wherea oxygen ubtitute atom i the firt nearet neighbour (1NN) for both Y atom. Solid (red), dahed (blue) and dah-dotted (black) ioline (with the increment of e A 3 ) repreent poitive, negative and neutral electronic charge, repectively. Table 7 Calculated parameter for the cluter Y Fe O Fe Y Fe and Y Fe O i Y Fe inide fcc-fe lattice configuration dr Y (1), A dr Y (2), A dr O,A Y Fe O Fe Y Fe (1NN) Y Fe O Fe Y Fe (2NN) Y Fe O i Y Fe (1NN) Y Fe O i Y Fe (2NN) Figure 15 Difference electron charge denity map for and 2NN Y Fe O Fe Y Fe configuration for Y Fe and O Fe impurity atom, wherea oxygen ubtitute atom i the firt nearet neighbour (1NN) for both Y atom. Solid (red), dahed (blue) and dah-dotted (black) ioline (with the increment of e A 3 ) repreent poitive, negative and neutral electronic charge, repectively.

7 phyica p tatu olidi b 2142 A. Gopejenko et al.: Ab initio modelling of Y O cluter formation in g-fe lattice Due to a relatively low migration barrier of oxygen impurity atom in fcc-fe lattice a mentioned above (a it exit in both O Fe and O i configuration), imulation of the diffuion path for ytrium impurity atom, which can occupy only V Fe and migrate via them, attract enhanced attention [8, 9]. NEB method implemented within VASP code [11] ha been ued in thi tudy for optimiation of migration trajectory for yttrium impurity (Fig. 16 and 17) and other defect a well a determination of the diffuion barrier (e.g., Table 8). It i clearly een that the energy barrier decreae with the growth of ditance between Y impurity atom and V Fe vacancy. Thi can be explained by the fact that the ditance between migrating and regular atom in the lattice along the migration path are maller when Y Fe and V Fe are cloer to each other. Thu, the repuling force grow and increae the energy barrier. On the contrary, with the increae of the ditance between Y Fe and V Fe the energy barrier decreae. Thee reult explain influence of vacancie on migration path of Y atom in fcc-fe lattice. In order to receive accurate reult for energy barrier, we have ued even intermediate configuration for the NEB calculation of the optimied migration path. The reult of performed large cale Figure 16 Initial upercell configuration for the migration of Y atom (in T poition) urrounded by two V Fe vacancie where red and light blue arrow how direction of hift for Fe and Y atom, repectively. Table 8 NEB imulation of Y atom migration. configuration energy barrier, ev Y Fe V Fe (1NN) 2.60 Y Fe V Fe (2NN) 2.37 Y Fe V Fe (3NN) 1.85 Y Fe V Fe (4NN) 1.81 ab initio calculation will be ued in the LKMC imulation [7] of the ODS cluter formation. 4 Concluion Uing ab initio DFT-GGA calculation the modelling of Y, O and V Fe defect ha been performed in fcc-fe lattice. The calculated binding energie between different defect, their effective charge and relaxation of lattice have been reported. Obviouly, formation of Y O nanocluter in fcc-fe matrix can be explained in pite of repulion between the arbitrary pair of Y impurity atom [8 10]. Indeed, thee cluter can be tabilied by the preence of V Fe vacancie or O Fe impuritie at the nearet coordination phere around Y Fe atom. Migration energie of the latter have been calculated for different Y V Fe configuration. A table binding ha been found alo for Y Fe O Fe Y Fe nanocluter configuration with oxygen located at ubtitutional ite (where yttrium impuritie are poitioned at 1 NN or 2NN ite relatively to oxygen), while Y Fe O i Y Fe nanocluter with oxygen at the nearet intertitial poition are found to be rather untable. The migration path of O Fe atom pae through octahedral O intertitial ite and V Fe vacancie. The NEB calculation, including optimization of trajectorie for migration of both yttrium and oxygen atom in iron lattice, are very important for further LKMC modelling of Y 2 O 3 nanoparticle formation in ODS teel. Acknowledgement Thi work ha been carried out within the framework of the EUROfuion Conortium and ha received funding from the Euratom reearch and training programme under grant agreement no The view and opinion expreed herein do not necearily reflect thoe of the European Commiion. Support from Latvian National Reearch Program IMIS2 i alo appreciated. Author thank J. Puran for timulating dicuion and the CSC Helio Supercomputer (Rokkaho, Japan) for the ue of their computational facilitie for our large-cale calculation. Reference Figure 17 Final upercell configuration for the migration of Y atom (hifting from regular lattice ite to O poition) potentially urrounded by two V Fe vacancie. [1] C. Cayron, E. Rath, I. Chu, and S. Launoi, J. Nucl. Mater. 335, (2004). [2] M. Klimiankou, R. Lindau, and A. M olang, J. Nucl. Mater , (2007). [3] F. Hanic, M. Hartmanova, G. G. Knab, A. A. Uruovkaya, and K. S. Bagdaarov, Acta Crytallogr. B 40, (1984). [4] Ternary Steel Sytem, Phae Diagram and Phae Tranition Data, Landolt-B orntein, New Serie IV, edited by E. Muntermann, P. Franke and H. J. Seifert, vol. 19C (Springer-Verlag, Berlin, Heidelberg, 2012), p. 401.

8 Original Paper Phy. Statu Solidi B 253, No. 11 (2016) 2143 [5] T. Okuda and M. Fujiwara, J. Mater. Sci. Lett. 14, (1995). [6] G. R. Odette, M. J. Alinger, and B. D. Wirth, Annu. Rev. Mater. Re. 38, (2008). [7] V. A. Borodin, P. V. Vladimirov, and A. M olang, J. Nucl. Mater , (2007). [8] A. Gopejenko, Yu. F. Zhukovkii, P. V. Vladimirov, E. A. Kotomin, and A. M olang, J. Nucl. Mater. 406, (2010). [9] A. Gopejenko, Yu. F. Zhukovkii, P. V. Vladimirov, E. A. Kotomin, and A. M olang, J. Nucl. Mater. 416, (2011). [10] A. Gopejenko, Yu. F. Zhukovkii, P. V. Vladimirov, E. A. Kotomin, and A. M olang, in: Proc. NATO ARW Nanodevice and Nanomaterial for Ecological Security, edited by Yu. N. Shunin and A. E. Kiv (Springer, Dordrecht, 2012), pp [11] G. Kree and J. Hafner, VASP the Guide (Univerity of Vienna, 2007), html. [12] G. Kree and J. Furthmueller, Phy. Rev. B 45, (1996). [13] J. P. Perdew and Y. Wang, Phy. Rev. B 45, (1992). [14] G. Kree and D. Joubert, Phy. Rev. B 59, (1999). [15] B. D. Cullity and C. D. Graham, Introduction to Magnetic Material, 2nd edn. (John Wiley & Son, IEEE Pre, Hoboken, New Jerey, 2009). [16] H. J. Monkhort and J. D. Pack, Phy. Rev. B 13, (1976). [17] G. Henkelman, A. Arnaldon, and H. Jonon, Comput. Mater. Sci. 36, (2006). [18] H. C. Herper, E. Hoffmann, and P. Entel, Phy. Rev. B 60, (1999). [19] M. Marman and J. Hafner, Phy. Rev. B 66, (2002). [20] W. Zhong, G. Overney, and D. Tomanek, Phy. Rev. B 47, (1993). [21] J. T. Wang, D. S. Wang, and Y. Kawazoe, Appl. Phy. Lett. 88, (2006). [22] V. L. Moruzzi, P. M. Marcu, K. Schwarz, and P. Mohn, Phy. Rev. B 34, (1986). [23] G. Ghoh and G. B. Olon, Acta Mater. 50, (2002). [24] Diffuion in Solid Metal and Alloy, Landolt B orntein, New Serie III, edited by H. Mehrer, vol. 26 (Springer- Verlag, Berlin, Heidelberg, 1990), p. 481.