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Ab initio and Monte Carlo modeling in Fe–Cr system: Magnetic origin of anomalous thermodynamic and kinetic properties
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Computational Materials Science 44 (2008) 1–8 www.elsevier.com/locate/commatsci
Ab initio and Monte Carlo modeling in Fe–Cr system: Magnetic origin of anomalous thermodynamic and kinetic properties D. Nguyen-Manh *, M.Yu. Lavrentiev, S.L. Dudarev EURATOM/UKAEA Fusion Association, Culham Science Centre, Abingdon OX14 3DB, United Kingdom Available online 7 March 2008
Abstract A multi-scale modeling approach is presented to investigate the phase stability and clustering in Fe–Cr alloys by combining density functional theory (DFT) calculations with statistical approaches involving cluster expansion (CE) and Monte Carlo (MC) simulations. This makes it possible to generate, in a systematic way, the low-energy configurations required for the subsequent DFT study of intrinsic defects (vacancies, interstitials) and impurity-defect interactions in the entire range of Fe–Cr alloy compositions under irradiation. The lowest mixing enthalpy configuration generated by MC simulation is found at Cr concentration of 6.25% that is consistent with the ab initio prediction of an intermetallic compound Fe15 Cr characterized by the negative heat of formation. The ordering structure Fe15 Cr is stabilized by lowest down-spin density of states value at the Fermi energy, showing Cr atom with a strong local magnetic moment aligned in one anti-ferromagnetic direction with the Fe atoms. Furthermore, it is shown that magnetism is responsible for anomalous nano-segregation of the a0 -Cr phase into various clustered configurations that are confirmed by a large scale kinetic Monte Carlo simulations. The impurity-interstitial defect interaction is investigated and we found that the binding energies of mixed dumbbell Fe–Cr in Fe15 Cr alloy are positive at variance with predictions made by elastic theory. Using the Stoner model within a tight-binding mean field approximation we are able to explain the origin of anomalous enthalpy of mixing as well as the complex correlation between magnetic moment distribution and phase stability in the Fe–Cr system. Ó 2008 Elsevier B.V. All rights reserved. PACS: 61.72.Ji; 61.82.Bg; 71.15.Mb; 71.10.Be; 75.10.Lp Keywords: Magnetism; Stoner model; Fe–Cr system; Ab initio; Monte Carlo
1. Introduction Low activated ferritic/martensitic steels based on Fe–Cr alloy system for future fusion power plant applications [1,2] represents a great challenge for computational material science because the magnetism in both constituting elements affects the properties of the alloy in a complex way [3]. In addition, since iron, chromium and Fe–Cr alloys inside the technological range of application are mainly investigated within the body-centered cubic (bcc) structure, there is a strong correlation between quantum mechanical
*
Corresponding author. Tel.: +44 1235 466284; fax: +44 1235 466435. E-mail address: [email protected] (D. Nguyen-Manh).
0927-0256/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2008.01.035
effect at the electronic level and mechanical properties at engineering level [4]. From theoretical point of view, an interplay between electronic and magnetic properties in a 3d transition metal system can only be treated correctly within many-electron approach [5]. Experimental results on the magnetic phase diagram for Fe–Cr alloys showed that the anti-ferromagnetic (AFM) phase boundary (Neel temperature) disappears at a concentration of 84%Cr whereas the ferromagnetic (FM) boundary (Curie temperature) has been observed up to 80%Cr [6]. Above the Curie temperature, where the alloy becomes paramagnetic, the conventional Fe–Cr phase diagram exhibits a large miscibility gap. Above 800 K, a region where the r phase becomes stable exists too [7]. In the Fe-rich region (where advances have been made in developing ferritic and
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martensitic steels with 2–12%Cr for power plants that are significant improvements over steels originally considered), the Fe–Cr system exhibits peculiar thermodynamic properties: the alloys are anomalously stable with negative enthalpy of mixing and then becoming positive above a critical concentration. The latter has been predicted to be at around 25%Cr by an earlier tight-binding calculation based on the generalized perturbation method (GPM) in ferromagnetic bcc Fe–Cr alloys [8] and also more recently by the Ising model with first and second nearest neighbor (NN) magnetic interactions [9]. Interestingly, according to a new ab initio (EMTO) study within coherent potential approximation (CPA) for a treatment of disordered ferromagnetic Fe–Cr alloys, the critical concentration is found to be 10%Cr and the minimum of enthalpy of mixing (8 meV/atom) is predicted to be around 5%Cr [10]. It is worth to mention here that by using the EMTO–CPA data for fitting inter-atomic potential, there are some doubts about the accuracy of these ab initio prediction [11] in reproducing experimental results carried out at 703 K on the short-range order the Fe-rich region [12]. The propose of this paper is to examine the origin of anomalous behavior in Fe–Cr system by going beyond EMTO–CPA formalism that is applicable only for random alloys and by using more rigorous approach of cluster expansion (CE) technique in combination with density functional theory (DFT) data base for enthalpy of mixing generated from ordered binaries. 2. Anomalous enthalpy of mixing and spin-polarized electronic structure of the ordered Fe15 Cr phase The exchange Monte Carlo (EMC) simulations based on CE approximation for the enthalpy of mixing in the binary Fe–Cr system have been recently investigated [13]. We show that in the region of small Cr concentration, the enthalpy of mixing remains negative up to fairly high temperatures whereas the clustering of Cr atoms begins at concentration exceeding approximately 10%Cr. Within a homogeneous Bragg–William approximation, for which many-body correlations are approximated by single-site clusters, we showed that the mixing enthalpy of the system is comparable with those obtained by the EMTO–CPA formalism and therefore justified that the results from [10] are considered as a random limit from the present CE–EMC study [14]. More importantly, it is found from CE–EMC simulations in the 4 4 4 supercell of bcc lattice that the configuration with minimum negative enthalpy of mixing at 6.25%Cr is exactly the one predicted by our independent DFT data-base calculations for Fe15 Cr ordered structure [14,15]. The spin-polarized density map calculated within the VASP code with the PBE exchange-correlation functional and the projector augmented wave (PAW) method [16] for the 4 4 4 supercell of the Fe15 Cr phase is shown in Fig. 1. We see a strong negative magnetic moment ð 1:7lB Þ on Cr atoms separated to each other by 6NN. It is aligned
Fig. 1. Spin density map calculated for the ordered Fe15 Cr structure.
anti-ferromagnetically in comparison with ferromagnetic moments at all the Fe sites. In order to understand correlation between magnetic properties and strongly negative enthalpy of formation in the ordered Fe15 Cr structure, we have performed spin-polarized electronic structure calculations for binary Fe15 Cr, Fe14 Cr2 and Fe13 Cr3 structures. Within our DFT data base [15], the enthalpy of mixing for these three structures changes the sign from negative (6.5 meV/atom for Fe15 Cr) to positive (22.2 meV/atom and 45.3 meV/ atom for Fe14 Cr2 and Fe13 Cr3 , respectively). The total density of states (DOS) of these three binaries are shown in Fig. 2 and compared with those calculated for pure bcc-Fe in 2 2 2 (16 atoms) supercell. We found that the total DOS at Fermi energy, nðEF Þ, decreases from 18.92 states/eV/cell for Fe to 13.11 states/eV/cell for Fe15 Cr, then to 8.8 states/eV/cell for Fe14 Cr2 and finally increases to 13.39 states/eV/cell for Fe13 Cr3 . Therefore, the minimum of total DOS at Fermi energy found at 12.5%Cr does not correspond to the minimum negative enthalpy of mixing found at 6.25%Cr in a contradiction with the argument discussed recently by Olsson et al. (see formula (2) in [17]). In fact, their Fig. 7 also shows clearly that the minimum of the total DOS (as well as the majority channel at Ef ) is found at 10%Cr whereas the minimum of enthalpy of mixing is located at 5%Cr within the EMTO–CPA calculations. This trend is somehow agreed with our calculations discussed above. However, if we look carefully at the spin-polarized DOS calculated again for these four structures (Fig. 3), we can see clearly that the minority channel DOS at Fermi energy is minimum for the ordered Fe15 Cr structure. The latter implies that the behavior of minority channel at Fermi energy where there is a pseudo-gap minimum due to spin-polarization, plays a crucial role in stabilizing an ordered structure in Fe-rich region. A strong negative moment on Cr site, as it is shown in Fig. 1, provides a coherent and physical picture of this correlation. It is important to emphasize here that our DFT/CE–EMC finding of stable structure demonstrates a long-ranged
D. Nguyen-Manh et al. / Computational Materials Science 44 (2008) 1–8
3
80 Fe16 (bcc) Fe15Cr Fe14Cr2 Fe13Cr3
Total density of states (states/eV)
70 60 50 40 30 20 10 0 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 Energy (eV)
3
4
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9
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Fig. 2. Total electronic density of states for ordered Fe15 Cr structure (thick solid line) compared with those calculated for bcc-Fe in the 2 2 2 supercell (thin solid line), for Fe14 Cr2 (dashed line) and for Fe13 Cr3 (point-dashed line) alloys. The Fermi level is located at zero energy.
DOS(states/spin/eV)
60 50
Fe_up Fe15Cr_up Fe14Cr2_up Fe13Cr3_up
40 30 20 10
DOS(states/spin/eV)
0 10 20
Fe_dn Fe15Cr_dn Fe14Cr2_dn Fe13Cr3_dn
30 40 50 60 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1
0
1
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4
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Energy (eV) Fig. 3. The same as in Fig. 2 but for decomposed spin-polarized density of states.
ordering presence of Fe15 Cr at low temperature region. Our result is at variance with previous theoretical works where Fe–Cr binary system is treated either within the CPA for a random alloy [10] or within the GPM approach with incorporation the short-range order effect [8]. These treatments are only valid, from the present formalism, for Fe–Cr alloys at high temperature limit [14].
3. Magnetic origin of nano-clustering in Fe–Cr system A combined DFT/CE–EMC modeling makes it possible to generate, in a systematic way, the low-energy configurations required for the subsequent DFT study of intrinsic defects (vacancies, interstitials) and impurity-defect interactions in the entire range of Fe–Cr alloy compositions
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under irradiation. These configurations were found by annealing simulations cooling down, from 2000 to 0 K, an initially random alloy configuration. Fig. 4 shows DFT spin density maps and corresponding atomic configurations calculated from the generated configurations in a 4 4 4 supercell for various Cr concentrations. The arrows indicate the values of enthalpy of mixing for each configuration obtained from CE–EMC calculations. At low Cr concentration where Cr atoms are isolated and far away from each other, the alloys are anomalously stable as discussed in previous section for the ordered Fe15 Cr phase. For alloy with 12.5%Cr, apart from isolated atoms embedded in bcc-Fe, there is clustering of Cr atoms as it can be seen from Fig. 4. This picture of nano-clustering is fully consistent with a larger scale Monte Carlo simulations showing that clustering of Cr atoms begins at concentrations exceeding 10%Cr at 800 K and 20%Cr at 1400 K [13]. At 37.5%Cr, the nano-clustering is clearly predicted by spin-density map shown in Fig. 4 where Cr atoms are arranged in a bcc-like cluster with small and anti-ferromagnetic moments of the a0 -Cr phase. Importantly, it is found that the Cr–Fe interfaces of these clusters are parallel to the [1 1 0] planes that is in turn in agreement again with larger scale MC simulations. Therefore, we learn from this section that thermodynamic clustering of Cr atoms has a strong correlation with nature of magnetic interactions in the Fe–Cr system. 4. Defect and kinetic properties in Fe–Cr alloys Recent DFT calculations have revealed a fundamental difference in point defect properties between the non-magnetic bcc transition metals [18,19] and ferromagnetic iron [20]. In bcc-Fe a self-interstitial atom (SIA) defect adopts
the h1 1 0i configuration, while in vanadium, niobium, tantalum, molybdenum and tungsten the lowest-energy configuration of SIA defect has the h1 1 1i symmetry. This difference stimulated the development of a ‘magnetic’ semi-empirical potential for molecular dynamic simulations of defects in iron [21,22]. The understanding of defect behavior in Fe–Cr binaries is much less clear although we do know that for bcc-Cr the difference between energies of the h1 1 0i and h1 1 1i SIA configurations is very small [18]. According to classical elastic theory the binding energy of mixed Fe–Cr dumbbell h1 1 0i was predicted to be negative in the solid solution limit simply because the size of solute Cr atom is larger than those of the host Fe atom [23]. In order to understand defect behavior in Ferich region, we have carried out a systematic investigation of the mixed Fe–Cr dumbbell h1 1 0i defect in the ordered structure Fe15 Cr [14,15]. Fig. 5 shows the spin density map for this SIA defect calculated at the crystallographic site Fe3 where there are no nearest neighbors Cr atoms in its environment up to the 5NN. It is found that magnetic moment on Cr atom at the mixed dumbbell is 0.76 lB whereas the local magnetic moment on Fe atom at the defect is 0:35 lB . The calculated binding energy for this mixed dumbbell is positive (0.149 eV) that is in variance with the negative value (0.096 eV) calculated from the classical elastic theory for ferritic steel [23]. Obviously in the latter case, the magnetic interactions at the point defect were not taken into account and the finding of positive binding energy for mixed Fe–Cr SIA dumbbell again demonstrates that magnetism cannot be ignored in defect properties for a realistic model of Fe–Cr alloy. Our prediction of low mixed dumbbell formation energies is strongly supported by experimental evidence from electrical resistivity recovery measurements in Fe-rich region of Fe–Cr alloys
Fig. 4. Nano-clustering in Fe–Cr alloys across the Cr range of concentration larger than 10% Cr seen by DFT spin-polarized maps calculated from the 4 4 4 supercell atomic structures obtained by a combined cluster expansion with exchange Monte Carlo simulations.
D. Nguyen-Manh et al. / Computational Materials Science 44 (2008) 1–8
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Fig. 5. Spin-polarized density map calculated for mixing Fe–Cr h1 1 0i dumbbell for Fe3 site in the Fe15 Cr structure.
where such trapping defect configurations were observed and mobile at the temperature below the onset of long-rang SIA migration in bcc-Fe [24]. In order to investigate the kinetics of radiation induced precipitation in binary Fe–Cr alloys, a complete description of defect (vacancy and SIA) migration energy as a function of Cr concentration is needed. This would require a development of new statistical method to deal with the problem of environmentally magnetic dependence of potential barrier in calculating defect migration energy in a concentrated alloy. In the first approximation, we can use the broken bond approach in which the initial and final configuration energies in the presence of defect (monovacancy for example) are calculated from the set of CE energy interactions within a rigid lattice model. We also use the migration energy (0.57 eV) of Cr atom in solid solution limit and those of Fe mono-vacancy energy (0.64 eV) to study phase transformation kinetics during thermal ageing as the starting point of investigation [25]. Fig. 6 shows the result for 20%Cr alloy simulated by kinetic Monte Carlo (kMC) with a single vacancy inserted in the super-cell (20 20 20) of bcc lattice with 16,000 sites. This configuration is obtained from an initial random one after 108 kMC moves or 74872 s. Fig. 6 shows only clustering of Cr atoms obtained at T = 300 K from an initial random configuration. It is found that during the kMC ageing, the ratios of Cr atom in larger cluster size have increased in comparison with the equilibrium configuration obtained by EMC simulations. 5. The Stoner model for magnetic Fe–Cr alloys
Fig. 6. Cr cluster in 20% alloy obtained from kinetic Monte Carlo simulations with vacancy migration energies calculated within DFT scheme.
Fe-rich region, Cr nano-clustering and binding energy of defects in Fe–Cr system. In this section, a Stoner model investigated within the mean field tight-binding (TB) representation [4,26] for magnetic and binary alloy is developed to explain the origin of these peculiar properties. In the Stoner model, justified from spin density functional theory, in particular the local spin density approximation (LSDA), the moment formation is driven by the spin polarization energy, ELSDA spin . In a much used, and often quite good approximation [27,28] XX 0 ¼ ð1=4Þ J LSDA ð1Þ ELSDA spin il;il0 mil mil ; i
ll0
where mil is the partial magnetic moment at atom i with angular quantum number l and J LSDA is the LSDA elecil;il0 tron–electron exchange interaction between shells l and l0 from atom i. For transition metals, it is a good approximation to assume that the electronic states close to the Fermi level with essentially d character ðl ¼ 2Þ contribute to the spin polarization. Since the spin up and spin down energy bands are split more or less uniformly in LSDA, as –a in the older but more empirical Stoner theory – I Stoner id linear combination of the matrix elements J LSDA 0 , is conil;il ventionally referred to as the Stoner parameter [29]. The Stoner contribution to the magnetic energy is therefore given by X ¼ ð1=4Þ I Stoner m2id : ð2Þ EStoner spin id i
In the three previous Sections 2–4 we have demonstrated the influence of magnetism on anomalous phase stability in
In a metallic system when the local charge neutrality (LCN) condition is an excellent approximation [30,31],
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the Stoner energy (Eq. (2)) therefore represents the only correction coming from electron–electron interaction to total energy of the self-consistent one-electron system. Thus, the binding energy written within the TB bond model for a spin-polarized d-orbital system should read: X X X ¼ qrjd 0 ;id H id;jd 0 ð1=4Þ I Stoner ðm2id m2atom Þ Espin b id r¼";# id–jd 0
þ
ELCN prom
þ Erep :
i
ð3Þ
The first term in Eq. (3) represents the bond energy for the spin-polarized system with the density matrix elements qrjd 0 ;id and the tight-binding hopping integrals H id;jd 0 ; the second term is the so-called double counting energy coming from the magnetic (Stoner) exchange interaction; the third term is the energy of promotion that normally arises from the change in electronic configuration when atoms are brought together from infinite. Within a LCN condition, ELCN prom is simply represented by a constant energy. Finally, the last term of Eq. (3) contains the overlap and electrostatic repulsion between nuclei and is normally approximated by a simple pair-wise potential energy. The expression of forces within spin-polarized TBB model is given by simple formula [26]: X X qrjd 0 ;id ðoH id;jd 0 =ork Þ ðoErep =ork Þ; ð4Þ Fk ¼ r¼";# id–jd 0
where the first term in Eq. (4) is the usual Hellmann–Feynman force. The magnetic origin of anomalous enthalpy of mixing in the Fe–Cr alloy can now be investigated within the TB Stoner model (Eq. (3)) on the basis of d-band model of interatomic interactions as a function of band filling. Here, the magnetic contribution to binding energy difference as a function of Cr concentration is determined essentially by the first and second terms. By using the hopping integral parameters fitted for FM bcc-Fe [26] that in turn reproduce correctly the well-known structural stability trends across the 3d transition metal series [4], it is found that the spin-polarized AFM DOS of bcc-Cr calculated with the electron number N d ¼ 4:5 and the Stoner ðCrÞ ¼ 0:52 eV, is exchange interaction parameter I Stoner d comparable with those obtained from DFT calculations. The calculated magnetic moments, ±0.88 lB , and the corðCrÞhI Stoner ðFeÞ (=0.77 eV [26]) are responding value I Stoner d d in good agreement with various independent TB studies in [30,32]. The concentration dependent value of I Stoner d Fe–Cr system should be more complex than the approximately linear dependence in the ferromagnetic Fe–Co alloy [33]. Taking into account experimental evidence of the AFM states in Cr-rich region [6], we assume that the Stoner parameter on Cr atom is vanished when the Cr concentration is less than 80%. As we will discuss later, this assumption does not affect the presence of anti-ferromagnetic moment in Cr atom in the Fe-rich region, in particular for the Fe15 Cr phase shown in Fig. 1. The resulting calculated enthalpy of mixing using
Fig. 7. Enthalpy of mixing calculated from tight-binding Stoner model presented in this paper and compared with the corresponding DFT values.
TB Stoner model and compared with corresponding DFT data is shown in Fig. 7 for various binary structures. In addition to the three Fe-rich structures discussed in Section 2, we include in Fig. 7 the data of three segregated layer structures in the [001] direction: Fe6 Cr2 , Fe4 Cr4 , Fe2 Cr6 and also the B2 (FeCr) and the FeCr15 structures. The agreement between TB Stoner model and DFT data across the whole range of Cr concentration is excellent in particular for the change from small negative to positive enthalpy of mixing in the Fe-rich region. The difference between enthalpy of mixing between the B2 and the Fe4 Cr4 structure is found to be 132 meV/atom in the TB Stoner model that is very consistent with the corresponding DFT value of 139 meV/atom. For the Fe15 Cr structure, the TB Stoner model predicts a negative magnetic moment of 1:6 lB at Cr site that is again very consistent with the DFT result even though the Stoner parameter for Cr is assumed to be zero at this concentration. Therefore, we see that the anti-ferromagnetic moment at Cr atom in Fe15 Cr phase has been induced by the presence of ferromagnetic moments of surrounding Fe atoms and it is formed self-consistently within the present model via the LCN condition. In order to verify the validity of the present model beyond bcc-like structure, we have extended our investigation to stability of the complex r Fe–Cr phase [34]. In general, there are 5 different crystallographic sites in a r phase and we consider here the case where the site 2(a), 8(i) and 8(i) are occupied by Fe atoms and the sites 4(f) and 8(j) – by Cr atoms. The calculated enthalpy of mixing for this alloy is 81 meV/atom from the TB Stoner model that is in good agreement with the corresponding DFT value of 83 meV/ atom. The magnetic moment distribution at different sites is presented in Fig. 8 for the same scale between TB Stoner model and DFT cases. Importantly, we find that the magnetic moments in two Cr sites are again aligned anti-ferromagnetically in comparison with those at three Fe sites within both calculations for the Fe–Cr r phase.
D. Nguyen-Manh et al. / Computational Materials Science 44 (2008) 1–8
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Fig. 8. Magnetic moment distributions in Fe–Cr r phase (with 40%Cr) calculated within the tight-binding Stoner model (right) and compared with the corresponding DFT values (left).
6. Conclusion In this paper, we have demonstrated the important role of magnetism in understanding of anomalous phase stability, nano-clustering, defect and kinetic properties for the Fe–Cr system by using a combined DFT and Monte Carlo studies. The crossover from negative to positive enthalpy of mixing at the Fe-rich region is dominated by the presence of ordered Fe15 Cr phase which is in turn characterized by the minimum of minority-spin density of states at Fermi level. We have proposed a simple tightbinding Stoner model for the Fe–Cr system that allows us to explain not only the origin of this crossover in enthalpy of mixing in the bcc-like phase but also provides a new possibility to extend our study to more complex feature of Fe–Cr alloy including the role of r phase. The development of a more rigorous Stoner model in Fe–Cr system for a large-scale atomistic modeling will be undertaken in our future works. It would also be good to have the experimental confirmation of the Fe15 Cr, for example, via the 3D atom probe analysis. Due to a small negative enthalpy of formation predicted by DFT calculations, the esistence of this ordering phase has not yet been observed up to now. Acknowledgements The authors would like to thank I.A. Abrikosov, G.J. Ackland, J.L. Boutard, A. Caro, P.M. Derlet, R. Drautz, P.A. Kozhavyi, P. Olsson, A.V. Ruban, A. Van der Ven, A. Zunger and F. Williame for stimulating discussions. This work, supported by the United Kingdom Engineering and Sciences Research Council and European Communities under the contract of association between EURATOM and UKAEA, was carried out within the framework of the European Fusion Development Agreement (Task No. D051-TW6-TTMS-007). The views and opinions expressed
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Computational Materials Science 44 (2008) 1–8 www.elsevier.com/locate/commatsci
Ab initio and Monte Carlo modeling in Fe–Cr system: Magnetic origin of anomalous thermodynamic and kinetic properties D. Nguyen-Manh *, M.Yu. Lavrentiev, S.L. Dudarev EURATOM/UKAEA Fusion Association, Culham Science Centre, Abingdon OX14 3DB, United Kingdom Available online 7 March 2008
Abstract A multi-scale modeling approach is presented to investigate the phase stability and clustering in Fe–Cr alloys by combining density functional theory (DFT) calculations with statistical approaches involving cluster expansion (CE) and Monte Carlo (MC) simulations. This makes it possible to generate, in a systematic way, the low-energy configurations required for the subsequent DFT study of intrinsic defects (vacancies, interstitials) and impurity-defect interactions in the entire range of Fe–Cr alloy compositions under irradiation. The lowest mixing enthalpy configuration generated by MC simulation is found at Cr concentration of 6.25% that is consistent with the ab initio prediction of an intermetallic compound Fe15 Cr characterized by the negative heat of formation. The ordering structure Fe15 Cr is stabilized by lowest down-spin density of states value at the Fermi energy, showing Cr atom with a strong local magnetic moment aligned in one anti-ferromagnetic direction with the Fe atoms. Furthermore, it is shown that magnetism is responsible for anomalous nano-segregation of the a0 -Cr phase into various clustered configurations that are confirmed by a large scale kinetic Monte Carlo simulations. The impurity-interstitial defect interaction is investigated and we found that the binding energies of mixed dumbbell Fe–Cr in Fe15 Cr alloy are positive at variance with predictions made by elastic theory. Using the Stoner model within a tight-binding mean field approximation we are able to explain the origin of anomalous enthalpy of mixing as well as the complex correlation between magnetic moment distribution and phase stability in the Fe–Cr system. Ó 2008 Elsevier B.V. All rights reserved. PACS: 61.72.Ji; 61.82.Bg; 71.15.Mb; 71.10.Be; 75.10.Lp Keywords: Magnetism; Stoner model; Fe–Cr system; Ab initio; Monte Carlo
1. Introduction Low activated ferritic/martensitic steels based on Fe–Cr alloy system for future fusion power plant applications [1,2] represents a great challenge for computational material science because the magnetism in both constituting elements affects the properties of the alloy in a complex way [3]. In addition, since iron, chromium and Fe–Cr alloys inside the technological range of application are mainly investigated within the body-centered cubic (bcc) structure, there is a strong correlation between quantum mechanical
*
Corresponding author. Tel.: +44 1235 466284; fax: +44 1235 466435. E-mail address: [email protected] (D. Nguyen-Manh).
0927-0256/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2008.01.035
effect at the electronic level and mechanical properties at engineering level [4]. From theoretical point of view, an interplay between electronic and magnetic properties in a 3d transition metal system can only be treated correctly within many-electron approach [5]. Experimental results on the magnetic phase diagram for Fe–Cr alloys showed that the anti-ferromagnetic (AFM) phase boundary (Neel temperature) disappears at a concentration of 84%Cr whereas the ferromagnetic (FM) boundary (Curie temperature) has been observed up to 80%Cr [6]. Above the Curie temperature, where the alloy becomes paramagnetic, the conventional Fe–Cr phase diagram exhibits a large miscibility gap. Above 800 K, a region where the r phase becomes stable exists too [7]. In the Fe-rich region (where advances have been made in developing ferritic and
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martensitic steels with 2–12%Cr for power plants that are significant improvements over steels originally considered), the Fe–Cr system exhibits peculiar thermodynamic properties: the alloys are anomalously stable with negative enthalpy of mixing and then becoming positive above a critical concentration. The latter has been predicted to be at around 25%Cr by an earlier tight-binding calculation based on the generalized perturbation method (GPM) in ferromagnetic bcc Fe–Cr alloys [8] and also more recently by the Ising model with first and second nearest neighbor (NN) magnetic interactions [9]. Interestingly, according to a new ab initio (EMTO) study within coherent potential approximation (CPA) for a treatment of disordered ferromagnetic Fe–Cr alloys, the critical concentration is found to be 10%Cr and the minimum of enthalpy of mixing (8 meV/atom) is predicted to be around 5%Cr [10]. It is worth to mention here that by using the EMTO–CPA data for fitting inter-atomic potential, there are some doubts about the accuracy of these ab initio prediction [11] in reproducing experimental results carried out at 703 K on the short-range order the Fe-rich region [12]. The propose of this paper is to examine the origin of anomalous behavior in Fe–Cr system by going beyond EMTO–CPA formalism that is applicable only for random alloys and by using more rigorous approach of cluster expansion (CE) technique in combination with density functional theory (DFT) data base for enthalpy of mixing generated from ordered binaries. 2. Anomalous enthalpy of mixing and spin-polarized electronic structure of the ordered Fe15 Cr phase The exchange Monte Carlo (EMC) simulations based on CE approximation for the enthalpy of mixing in the binary Fe–Cr system have been recently investigated [13]. We show that in the region of small Cr concentration, the enthalpy of mixing remains negative up to fairly high temperatures whereas the clustering of Cr atoms begins at concentration exceeding approximately 10%Cr. Within a homogeneous Bragg–William approximation, for which many-body correlations are approximated by single-site clusters, we showed that the mixing enthalpy of the system is comparable with those obtained by the EMTO–CPA formalism and therefore justified that the results from [10] are considered as a random limit from the present CE–EMC study [14]. More importantly, it is found from CE–EMC simulations in the 4 4 4 supercell of bcc lattice that the configuration with minimum negative enthalpy of mixing at 6.25%Cr is exactly the one predicted by our independent DFT data-base calculations for Fe15 Cr ordered structure [14,15]. The spin-polarized density map calculated within the VASP code with the PBE exchange-correlation functional and the projector augmented wave (PAW) method [16] for the 4 4 4 supercell of the Fe15 Cr phase is shown in Fig. 1. We see a strong negative magnetic moment ð 1:7lB Þ on Cr atoms separated to each other by 6NN. It is aligned
Fig. 1. Spin density map calculated for the ordered Fe15 Cr structure.
anti-ferromagnetically in comparison with ferromagnetic moments at all the Fe sites. In order to understand correlation between magnetic properties and strongly negative enthalpy of formation in the ordered Fe15 Cr structure, we have performed spin-polarized electronic structure calculations for binary Fe15 Cr, Fe14 Cr2 and Fe13 Cr3 structures. Within our DFT data base [15], the enthalpy of mixing for these three structures changes the sign from negative (6.5 meV/atom for Fe15 Cr) to positive (22.2 meV/atom and 45.3 meV/ atom for Fe14 Cr2 and Fe13 Cr3 , respectively). The total density of states (DOS) of these three binaries are shown in Fig. 2 and compared with those calculated for pure bcc-Fe in 2 2 2 (16 atoms) supercell. We found that the total DOS at Fermi energy, nðEF Þ, decreases from 18.92 states/eV/cell for Fe to 13.11 states/eV/cell for Fe15 Cr, then to 8.8 states/eV/cell for Fe14 Cr2 and finally increases to 13.39 states/eV/cell for Fe13 Cr3 . Therefore, the minimum of total DOS at Fermi energy found at 12.5%Cr does not correspond to the minimum negative enthalpy of mixing found at 6.25%Cr in a contradiction with the argument discussed recently by Olsson et al. (see formula (2) in [17]). In fact, their Fig. 7 also shows clearly that the minimum of the total DOS (as well as the majority channel at Ef ) is found at 10%Cr whereas the minimum of enthalpy of mixing is located at 5%Cr within the EMTO–CPA calculations. This trend is somehow agreed with our calculations discussed above. However, if we look carefully at the spin-polarized DOS calculated again for these four structures (Fig. 3), we can see clearly that the minority channel DOS at Fermi energy is minimum for the ordered Fe15 Cr structure. The latter implies that the behavior of minority channel at Fermi energy where there is a pseudo-gap minimum due to spin-polarization, plays a crucial role in stabilizing an ordered structure in Fe-rich region. A strong negative moment on Cr site, as it is shown in Fig. 1, provides a coherent and physical picture of this correlation. It is important to emphasize here that our DFT/CE–EMC finding of stable structure demonstrates a long-ranged
D. Nguyen-Manh et al. / Computational Materials Science 44 (2008) 1–8
3
80 Fe16 (bcc) Fe15Cr Fe14Cr2 Fe13Cr3
Total density of states (states/eV)
70 60 50 40 30 20 10 0 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 Energy (eV)
3
4
5
6
7
8
9
10
Fig. 2. Total electronic density of states for ordered Fe15 Cr structure (thick solid line) compared with those calculated for bcc-Fe in the 2 2 2 supercell (thin solid line), for Fe14 Cr2 (dashed line) and for Fe13 Cr3 (point-dashed line) alloys. The Fermi level is located at zero energy.
DOS(states/spin/eV)
60 50
Fe_up Fe15Cr_up Fe14Cr2_up Fe13Cr3_up
40 30 20 10
DOS(states/spin/eV)
0 10 20
Fe_dn Fe15Cr_dn Fe14Cr2_dn Fe13Cr3_dn
30 40 50 60 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1
0
1
2
3
4
5
6
7
8
9 10
Energy (eV) Fig. 3. The same as in Fig. 2 but for decomposed spin-polarized density of states.
ordering presence of Fe15 Cr at low temperature region. Our result is at variance with previous theoretical works where Fe–Cr binary system is treated either within the CPA for a random alloy [10] or within the GPM approach with incorporation the short-range order effect [8]. These treatments are only valid, from the present formalism, for Fe–Cr alloys at high temperature limit [14].
3. Magnetic origin of nano-clustering in Fe–Cr system A combined DFT/CE–EMC modeling makes it possible to generate, in a systematic way, the low-energy configurations required for the subsequent DFT study of intrinsic defects (vacancies, interstitials) and impurity-defect interactions in the entire range of Fe–Cr alloy compositions
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under irradiation. These configurations were found by annealing simulations cooling down, from 2000 to 0 K, an initially random alloy configuration. Fig. 4 shows DFT spin density maps and corresponding atomic configurations calculated from the generated configurations in a 4 4 4 supercell for various Cr concentrations. The arrows indicate the values of enthalpy of mixing for each configuration obtained from CE–EMC calculations. At low Cr concentration where Cr atoms are isolated and far away from each other, the alloys are anomalously stable as discussed in previous section for the ordered Fe15 Cr phase. For alloy with 12.5%Cr, apart from isolated atoms embedded in bcc-Fe, there is clustering of Cr atoms as it can be seen from Fig. 4. This picture of nano-clustering is fully consistent with a larger scale Monte Carlo simulations showing that clustering of Cr atoms begins at concentrations exceeding 10%Cr at 800 K and 20%Cr at 1400 K [13]. At 37.5%Cr, the nano-clustering is clearly predicted by spin-density map shown in Fig. 4 where Cr atoms are arranged in a bcc-like cluster with small and anti-ferromagnetic moments of the a0 -Cr phase. Importantly, it is found that the Cr–Fe interfaces of these clusters are parallel to the [1 1 0] planes that is in turn in agreement again with larger scale MC simulations. Therefore, we learn from this section that thermodynamic clustering of Cr atoms has a strong correlation with nature of magnetic interactions in the Fe–Cr system. 4. Defect and kinetic properties in Fe–Cr alloys Recent DFT calculations have revealed a fundamental difference in point defect properties between the non-magnetic bcc transition metals [18,19] and ferromagnetic iron [20]. In bcc-Fe a self-interstitial atom (SIA) defect adopts
the h1 1 0i configuration, while in vanadium, niobium, tantalum, molybdenum and tungsten the lowest-energy configuration of SIA defect has the h1 1 1i symmetry. This difference stimulated the development of a ‘magnetic’ semi-empirical potential for molecular dynamic simulations of defects in iron [21,22]. The understanding of defect behavior in Fe–Cr binaries is much less clear although we do know that for bcc-Cr the difference between energies of the h1 1 0i and h1 1 1i SIA configurations is very small [18]. According to classical elastic theory the binding energy of mixed Fe–Cr dumbbell h1 1 0i was predicted to be negative in the solid solution limit simply because the size of solute Cr atom is larger than those of the host Fe atom [23]. In order to understand defect behavior in Ferich region, we have carried out a systematic investigation of the mixed Fe–Cr dumbbell h1 1 0i defect in the ordered structure Fe15 Cr [14,15]. Fig. 5 shows the spin density map for this SIA defect calculated at the crystallographic site Fe3 where there are no nearest neighbors Cr atoms in its environment up to the 5NN. It is found that magnetic moment on Cr atom at the mixed dumbbell is 0.76 lB whereas the local magnetic moment on Fe atom at the defect is 0:35 lB . The calculated binding energy for this mixed dumbbell is positive (0.149 eV) that is in variance with the negative value (0.096 eV) calculated from the classical elastic theory for ferritic steel [23]. Obviously in the latter case, the magnetic interactions at the point defect were not taken into account and the finding of positive binding energy for mixed Fe–Cr SIA dumbbell again demonstrates that magnetism cannot be ignored in defect properties for a realistic model of Fe–Cr alloy. Our prediction of low mixed dumbbell formation energies is strongly supported by experimental evidence from electrical resistivity recovery measurements in Fe-rich region of Fe–Cr alloys
Fig. 4. Nano-clustering in Fe–Cr alloys across the Cr range of concentration larger than 10% Cr seen by DFT spin-polarized maps calculated from the 4 4 4 supercell atomic structures obtained by a combined cluster expansion with exchange Monte Carlo simulations.
D. Nguyen-Manh et al. / Computational Materials Science 44 (2008) 1–8
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Fig. 5. Spin-polarized density map calculated for mixing Fe–Cr h1 1 0i dumbbell for Fe3 site in the Fe15 Cr structure.
where such trapping defect configurations were observed and mobile at the temperature below the onset of long-rang SIA migration in bcc-Fe [24]. In order to investigate the kinetics of radiation induced precipitation in binary Fe–Cr alloys, a complete description of defect (vacancy and SIA) migration energy as a function of Cr concentration is needed. This would require a development of new statistical method to deal with the problem of environmentally magnetic dependence of potential barrier in calculating defect migration energy in a concentrated alloy. In the first approximation, we can use the broken bond approach in which the initial and final configuration energies in the presence of defect (monovacancy for example) are calculated from the set of CE energy interactions within a rigid lattice model. We also use the migration energy (0.57 eV) of Cr atom in solid solution limit and those of Fe mono-vacancy energy (0.64 eV) to study phase transformation kinetics during thermal ageing as the starting point of investigation [25]. Fig. 6 shows the result for 20%Cr alloy simulated by kinetic Monte Carlo (kMC) with a single vacancy inserted in the super-cell (20 20 20) of bcc lattice with 16,000 sites. This configuration is obtained from an initial random one after 108 kMC moves or 74872 s. Fig. 6 shows only clustering of Cr atoms obtained at T = 300 K from an initial random configuration. It is found that during the kMC ageing, the ratios of Cr atom in larger cluster size have increased in comparison with the equilibrium configuration obtained by EMC simulations. 5. The Stoner model for magnetic Fe–Cr alloys
Fig. 6. Cr cluster in 20% alloy obtained from kinetic Monte Carlo simulations with vacancy migration energies calculated within DFT scheme.
Fe-rich region, Cr nano-clustering and binding energy of defects in Fe–Cr system. In this section, a Stoner model investigated within the mean field tight-binding (TB) representation [4,26] for magnetic and binary alloy is developed to explain the origin of these peculiar properties. In the Stoner model, justified from spin density functional theory, in particular the local spin density approximation (LSDA), the moment formation is driven by the spin polarization energy, ELSDA spin . In a much used, and often quite good approximation [27,28] XX 0 ¼ ð1=4Þ J LSDA ð1Þ ELSDA spin il;il0 mil mil ; i
ll0
where mil is the partial magnetic moment at atom i with angular quantum number l and J LSDA is the LSDA elecil;il0 tron–electron exchange interaction between shells l and l0 from atom i. For transition metals, it is a good approximation to assume that the electronic states close to the Fermi level with essentially d character ðl ¼ 2Þ contribute to the spin polarization. Since the spin up and spin down energy bands are split more or less uniformly in LSDA, as –a in the older but more empirical Stoner theory – I Stoner id linear combination of the matrix elements J LSDA 0 , is conil;il ventionally referred to as the Stoner parameter [29]. The Stoner contribution to the magnetic energy is therefore given by X ¼ ð1=4Þ I Stoner m2id : ð2Þ EStoner spin id i
In the three previous Sections 2–4 we have demonstrated the influence of magnetism on anomalous phase stability in
In a metallic system when the local charge neutrality (LCN) condition is an excellent approximation [30,31],
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the Stoner energy (Eq. (2)) therefore represents the only correction coming from electron–electron interaction to total energy of the self-consistent one-electron system. Thus, the binding energy written within the TB bond model for a spin-polarized d-orbital system should read: X X X ¼ qrjd 0 ;id H id;jd 0 ð1=4Þ I Stoner ðm2id m2atom Þ Espin b id r¼";# id–jd 0
þ
ELCN prom
þ Erep :
i
ð3Þ
The first term in Eq. (3) represents the bond energy for the spin-polarized system with the density matrix elements qrjd 0 ;id and the tight-binding hopping integrals H id;jd 0 ; the second term is the so-called double counting energy coming from the magnetic (Stoner) exchange interaction; the third term is the energy of promotion that normally arises from the change in electronic configuration when atoms are brought together from infinite. Within a LCN condition, ELCN prom is simply represented by a constant energy. Finally, the last term of Eq. (3) contains the overlap and electrostatic repulsion between nuclei and is normally approximated by a simple pair-wise potential energy. The expression of forces within spin-polarized TBB model is given by simple formula [26]: X X qrjd 0 ;id ðoH id;jd 0 =ork Þ ðoErep =ork Þ; ð4Þ Fk ¼ r¼";# id–jd 0
where the first term in Eq. (4) is the usual Hellmann–Feynman force. The magnetic origin of anomalous enthalpy of mixing in the Fe–Cr alloy can now be investigated within the TB Stoner model (Eq. (3)) on the basis of d-band model of interatomic interactions as a function of band filling. Here, the magnetic contribution to binding energy difference as a function of Cr concentration is determined essentially by the first and second terms. By using the hopping integral parameters fitted for FM bcc-Fe [26] that in turn reproduce correctly the well-known structural stability trends across the 3d transition metal series [4], it is found that the spin-polarized AFM DOS of bcc-Cr calculated with the electron number N d ¼ 4:5 and the Stoner ðCrÞ ¼ 0:52 eV, is exchange interaction parameter I Stoner d comparable with those obtained from DFT calculations. The calculated magnetic moments, ±0.88 lB , and the corðCrÞhI Stoner ðFeÞ (=0.77 eV [26]) are responding value I Stoner d d in good agreement with various independent TB studies in [30,32]. The concentration dependent value of I Stoner d Fe–Cr system should be more complex than the approximately linear dependence in the ferromagnetic Fe–Co alloy [33]. Taking into account experimental evidence of the AFM states in Cr-rich region [6], we assume that the Stoner parameter on Cr atom is vanished when the Cr concentration is less than 80%. As we will discuss later, this assumption does not affect the presence of anti-ferromagnetic moment in Cr atom in the Fe-rich region, in particular for the Fe15 Cr phase shown in Fig. 1. The resulting calculated enthalpy of mixing using
Fig. 7. Enthalpy of mixing calculated from tight-binding Stoner model presented in this paper and compared with the corresponding DFT values.
TB Stoner model and compared with corresponding DFT data is shown in Fig. 7 for various binary structures. In addition to the three Fe-rich structures discussed in Section 2, we include in Fig. 7 the data of three segregated layer structures in the [001] direction: Fe6 Cr2 , Fe4 Cr4 , Fe2 Cr6 and also the B2 (FeCr) and the FeCr15 structures. The agreement between TB Stoner model and DFT data across the whole range of Cr concentration is excellent in particular for the change from small negative to positive enthalpy of mixing in the Fe-rich region. The difference between enthalpy of mixing between the B2 and the Fe4 Cr4 structure is found to be 132 meV/atom in the TB Stoner model that is very consistent with the corresponding DFT value of 139 meV/atom. For the Fe15 Cr structure, the TB Stoner model predicts a negative magnetic moment of 1:6 lB at Cr site that is again very consistent with the DFT result even though the Stoner parameter for Cr is assumed to be zero at this concentration. Therefore, we see that the anti-ferromagnetic moment at Cr atom in Fe15 Cr phase has been induced by the presence of ferromagnetic moments of surrounding Fe atoms and it is formed self-consistently within the present model via the LCN condition. In order to verify the validity of the present model beyond bcc-like structure, we have extended our investigation to stability of the complex r Fe–Cr phase [34]. In general, there are 5 different crystallographic sites in a r phase and we consider here the case where the site 2(a), 8(i) and 8(i) are occupied by Fe atoms and the sites 4(f) and 8(j) – by Cr atoms. The calculated enthalpy of mixing for this alloy is 81 meV/atom from the TB Stoner model that is in good agreement with the corresponding DFT value of 83 meV/ atom. The magnetic moment distribution at different sites is presented in Fig. 8 for the same scale between TB Stoner model and DFT cases. Importantly, we find that the magnetic moments in two Cr sites are again aligned anti-ferromagnetically in comparison with those at three Fe sites within both calculations for the Fe–Cr r phase.
D. Nguyen-Manh et al. / Computational Materials Science 44 (2008) 1–8
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Fig. 8. Magnetic moment distributions in Fe–Cr r phase (with 40%Cr) calculated within the tight-binding Stoner model (right) and compared with the corresponding DFT values (left).
6. Conclusion In this paper, we have demonstrated the important role of magnetism in understanding of anomalous phase stability, nano-clustering, defect and kinetic properties for the Fe–Cr system by using a combined DFT and Monte Carlo studies. The crossover from negative to positive enthalpy of mixing at the Fe-rich region is dominated by the presence of ordered Fe15 Cr phase which is in turn characterized by the minimum of minority-spin density of states at Fermi level. We have proposed a simple tightbinding Stoner model for the Fe–Cr system that allows us to explain not only the origin of this crossover in enthalpy of mixing in the bcc-like phase but also provides a new possibility to extend our study to more complex feature of Fe–Cr alloy including the role of r phase. The development of a more rigorous Stoner model in Fe–Cr system for a large-scale atomistic modeling will be undertaken in our future works. It would also be good to have the experimental confirmation of the Fe15 Cr, for example, via the 3D atom probe analysis. Due to a small negative enthalpy of formation predicted by DFT calculations, the esistence of this ordering phase has not yet been observed up to now. Acknowledgements The authors would like to thank I.A. Abrikosov, G.J. Ackland, J.L. Boutard, A. Caro, P.M. Derlet, R. Drautz, P.A. Kozhavyi, P. Olsson, A.V. Ruban, A. Van der Ven, A. Zunger and F. Williame for stimulating discussions. This work, supported by the United Kingdom Engineering and Sciences Research Council and European Communities under the contract of association between EURATOM and UKAEA, was carried out within the framework of the European Fusion Development Agreement (Task No. D051-TW6-TTMS-007). The views and opinions expressed
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