PHYSICAL REVIEW MATERIALS 3, 013605 (2019)
Editors’ Suggestion
Universality of point defect structure in body-centered cubic metals
Pui-Wai Ma* and S. L. Dudarev CCFE, Culham Science Centre, UK Atomic Energy Authority, Abingdon, Oxfordshire OX14 3DB, United Kingdom
(Received 20 September 2018; published 10 January 2019)
Formation and migration energies, elastic dipole, and relaxation volume tensors of nanodefects are the parameters determining the rates of evolution of microstructure under irradiation as well as macroscopic elastic stresses and strains resulting from the accumulation of defects in materials. To find the accurate values of these parameters, we have performed density functional theory simulations of self-interstitial and vacancy defects in all the body-centred cubic metals, including alkaline metals (Li, Na, K, Rb and Cs), alkaline-earth metal (Ba), nonmagnetic transition metals (V, Nb, Mo, Ta and W), and magnetic transition metals (Cr and Fe), correcting the computed values for the effect of finite cell size and periodic boundary conditions. The lowest energy structure of a self-interstitial atom defect is universal to all the nonmagnetic bcc metals, including metals of groups 1 and 2 of the periodic table, and has the 111 symmetry. The only exceptions are the 110 self-interstitial defect configuration in Fe, and a 11ξ configuration in Cr. We have also computed elastic dipole tensors and relaxation volumes of self-interstitial and vacancy defects in all the bcc metals and explored how elastic relaxation parameters vary along the defect migration pathways.
DOI: 10.1103/PhysRevMaterials.3.013605
I. INTRODUCTION a b where Pij and Pkl are the dipole tensors of defects a and b, r Dislocations and point defects in crystalline materials are is the relative position vector of the defects, and Gik is the created by mechanical deformation or by irradiation, followed elastic Green’s function. Gik can be evaluated numerically by the relaxation of locally distorted atomic configurations for an arbitrary elastically anisotropic material [10] from [1]. A Frenkel pair is a commonly occurring type of radi- the matrix elements of the stiffness tensor. Using Eq. (2), ation defect. In a Frenkel pair, an atom is removed from the energy of elastic interaction can be computed for any a lattice site and placed elsewhere in the lattice, forming a configuration of defects, which otherwise is difficult to treat pair of a vacancy and a self-interstitial atom (SIA) defect. using atomistic or electronic structure based methods due to In elasticity theory, the strain field associated with a defect the limitations imposed by the simulation cell size. can be described by its elastic dipole tensor [2–7]. Matrix Domain and Becquart [11] computed the dipole tensor of elements of elastic dipole tensor can be derived from atomic a vacancy (using an ab initio approach) and an SIA (using scale simulations, where interatomic forces are evaluated us- empirical potentials) in iron using the Kanzaki force formal- ing empirical potentials or ab initio methods. Using elastic ism. Varvenne and Clouet [6] showed how to evaluate the dipole tensors, discrete atomic configurations of defects can dipole tensor of a defect from average macrostresses in a be treated as objects of continuum elasticity, enabling simu- simulation cell and investigated the numerical convergence lations of evolution of large ensembles of defects on the time of calculations as a function of cell size. Treating point and spatial scales many orders of magnitude larger than those defects in zirconium, they compared the rate of convergence accessible to molecular dynamics or electronic structure based of the two methods and showed that the Kanzaki force and methods [8]. macrostress approaches produced similar results in the limit where simulation cells were sufficiently large. The dipole tensor of a defect Pij fully defines its properties within the theory of elasticity [9]. The energy of elastic Sivak et al. [12] investigated elastic interactions between interaction between a defect and an external strain field point defects and dislocations in iron, computing them using ext is molecular statics and linear elasticity. Matrix elements of ij dipole tensors of defects were derived from atomistic calcula- ext ext tions, where interaction between the atoms was described by E =−Pij , (1) int ij empirical interatomic potentials. They found that the elasticity whereas the energy of interaction between two defects sepa- approximation described the energy of interaction fairly well rated by a distance many times their size is if a defect and a dislocation line were separated by a distance just a little over three lattice constants, in agreement with ∂2 ab = a b the analysis of dislocation core effects by Boleininger et al. Eint Pij Pkl Gik(r), (2) ∂xj ∂xl [13]. Sivak et al. [12] also evaluated elastic dipole tensors of vacancies and SIA defects in iron at equilibrium and at saddle points. Since the elastic energy of interaction between a defect and *Corresponding author: [email protected] external strain field can be readily evaluated using the dipole
2475-9953/2019/3(1)/013605(16) 013605-1 Published by the American Physical Society PUI-WAI MA AND S. L. DUDAREV PHYSICAL REVIEW MATERIALS 3, 013605 (2019) tensor formalism, elastic interaction effects can be included in The only ab initio study of SIA defects in alkaline metals coarse-grained models, for example in object kinetic Monte carried out so far was performed by Breier et al. [32]. They Carlo (kMC). By performing kMC simulations, Sivak et al. evaluated the formation energies of 111 , 110 , and 100 estimated the effect of elastic interactions on the diffusion SIA dumbbells in sodium, computing them using DFT in the of defects in bcc iron and vanadium [14], and also on the local density approximation (LDA). The 110 dumbbell was diffusion of hydrogen in bcc iron [15,16]. They concluded, found to have the lowest energy, although it was only 0.01eV in agreement with the earlier works by Margvelashvili and lower than that of the 111 dumbbell configuration. Calcula- Saralidze [17], Wolfer and Ashkin [18], and Brailsford and tions in Ref. [32] were performed using a relatively small 55- Bullough [19] that dislocations were more efficient sinks for atom cell. Below, we compare these results with calculations SIA defects than for vacancies. The sink strength of an edge performed using more accurate density functionals and larger dislocation was several times larger than that of a screw dislo- simulation cells. We also investigate the structure of defects in cation for both SIA and vacancy defects in Fe and V. Assess- all the other bcc metals in the periodic table, including alkaline ing the effect of elastic field of a dislocation on the diffusion metals. In iron, the 110 dumbbell is the most stable SIA of a hydrogen interstitial, they showed that the predicted sink configuration [11,30,31]. The three-dimensional translation- strength efficiency was relatively high for an edge dislocation, rotation migration pathway of the defect [29] explains the rel- but was significantly lower for a screw dislocation. atively high activation temperature for the onset of diffusion This brings into focus the question about the structure of of SIAs in iron [24,25,28], which is close to 120 K. a defect in a metal. Paneth [20] suggested that in an alkaline The structure of an SIA defect in chromium remains metal, an SIA defect would adopt a 111 configuration. He somewhat uncertain. Calculations assuming a NM electronic evaluated the formation energy of a 111 crowdion in sodium ground state [30,31] or an antiferromagnetic (AFM) state using a model where a positive point charge was embedded [39,40] showed that the difference between the formation in a uniform field of negative charge density, and found the energies of 111 and 110 dumbbells was within the margin formation energy of the defect to be close to 0.3 eV. Below, of error of ab initio calculations. Recently, we found [41] we show that a more reliable value can now be derived from that the most stable configuration of an SIA defect in Cr has density functional theory (DFT) calculations. the 11ξ orientation, where ξ varies from 0.355 to 0.405, Other researchers later argued that it was the 110 dumb- regardless of magnetic order. bell that represented the most stable SIA defect configuration In this paper, in addition to determining the structure and in Fe and Mo [3,21]. This assertion was based on the data energies of various defect configurations, we have also eval- derived from the analysis of diffuse x-ray scattering experi- uated their elastic properties needed for large-scale dynamic ments. Interpreting diffuse x-ray diffraction data in definitive simulations of microstructure [42]. We start by computing terms is difficult since even at cryogenic temperatures radia- the formation and migration energies of defects, treating tion induced defects diffuse [22], interact and can form fairly interactions with periodic images in the linear elasticity ap- complex clusters. proximation. Then, we evaluate dipole tensors and relaxation Depending on the structure of an SIA configuration, SIA volume tensors of all the defect structures in all the bcc metals, defects have different mobilities. A 111 defect can easily including alkaline metals Li, Na, K, Rb, and Cs, alkaline-earth translate itself through the lattice in the direction parallel to metal Ba, NM transition metals V, Nb, Mo, Ta, and W, and its axis, suggesting that a 111 crowdion should be highly magnetic transition metals Cr and Fe. Thermal migration of mobile even at fairly low temperatures [22,23]. This explains a 111 SIA defect proceeds through a sequence of 111 the high diffusivity of SIAs observed in resistivity recov- crowdion ↔ 111 dumbbell transformations. This involves ery experiments performed on electron irradiated bcc metals almost no variation of the elastic dipole tensor. On the other [24–26]. Experimental observations, confirmed by theoretical hand, when investigating the migration of a 110 defect analysis [22], show that in many bcc metals SIA defects configuration in Fe, we find that it involves some significant still move at temperatures below 6 K. High-voltage electron variation of the dipole tensor along the migration pathway. microscope experiments [27] also show that the activation energy characterizing long-range migration of SIA defects II. THEORY in tungsten is low. On the other hand, a 110 dumbbell in Fe migrates through a sequence of translation and rotational We start by briefly reviewing the methodology for comput- jumps [3,28], and its three-dimensional diffusion is strongly ing the formation and migration energies of defects, dipole, thermally activated. Ab initio DFT calculations have been and relaxation volume tensors, and the variation of these recently applied to the evaluation of formation and migration quantities along defect migration pathways. F energies of defects in bcc metals and to the calculation of The formation energy of a defect Edef is relaxation volumes of defects [29–38]. In all the nonmag- N F = − def netic (NM) bcc transition metals, the most stable SIA defect Edef Edef(Ndef) Ebulk(Nbulk ), (3) structure was found to have the 111 symmetry [30,31]. Nbulk Swinburne et al. [22] showed that the migration energy of a where Edef is the energy of a simulation cell containing a 111 dumbbell in tungsten is close to 2 meV, and that the defect, Ebulk is the energy of a reference defect-free cell, and saddle point configuration on a defect migration pathway is Ndef and Nbulk are the numbers of atoms in the respective cells. a 111 crowdion. Quantum fluctuations of atomic positions Equation (3) leaves open the question about whether the make the 111 SIA defects extremely mobile at temperatures simulation cell should be fully relaxed or the relaxation of as low as 1 K [22]. ion positions should be performed in a constrained manner,
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corr leaving the boundaries of the cell fixed. The subtlety of Eq. (3) approximation. Estrain is the self-strain correction energy as- D is also associated with the fact that it implicitly assumes the sociated with the regularization of strain ij (r) produced by limit Ndef,Nbulk →∞. Equation (3) defines the formation the defect in the simulation cell itself. This regularization is D energy of a defect, assuming that the defect is embedded required to make ij (r) compatible with the fact that under in an infinite medium, an approximation that is not satisfied the condition of no overall relaxation of the simulation cell, in a practical simulation. To circumvent the need to treat the total macroscopic strain must vanish [7]. the surface termination effect, simulations of defects are per- The above consideration also applies to any configuration formed using periodic boundary conditions. This is equivalent related to the lowest energy defect structure through an adia- to evaluating the energy associated with embedding an infinite batic transformation. Equation (6) remains valid at any point number of defects in an infinite medium. Naturally, to relate on a migration pathway of a defect found using an NEB the formation energy defined by Eq. (3) to a calculation calculation. performed using periodic boundary conditions, it is necessary Numerical values of Eint can be computed from the dipole to subtract from the result of an ab initio calculation the tensor of the defect and the anisotropic elastic Green’s func- energy of interaction between a defect and all its periodic tion of the crystal [4–7]. The regularized energy of elastic in- images, and estimate the contribution associated with the teraction between a defect and its periodic images, compatible lattice deformation generated by the infinite number of images with the condition of zero average elastic strain, is [7] of the defect. E = Etotal + Ecorr, (8) An alternative approach is to relax the orientation and int int int position of boundaries of the cell to arrive at a stress-free state where of the material. Varvenne et al. [5,6] noted that the stress- 1 ∂ ∂ free approach exhibited better numerical convergence as a Etotal = P P G (R )(9) function of the simulation cell size, and produced lower values int 2 ij kl ∂x ∂x ik n n =0 j l of formation energies. Relaxing positions and orientations 1 of boundaries of the simulation cell containing a defect is = Pij PklGik,jl(Rn) (10) app 2 equivalent to applying a homogeneous elastic strain .The n =0 elastic energy associated with this strain Eapp contributes to is a conditionally convergent sum of pairwise elastic interac- the formation energy of the defect. tions between a defect and its periodic images situated at R . The strain tensor app associated with the relaxation of n The term boundaries of the simulation cell satisfies the condition 1 ref + app = def Ecorr =− P P G (R − r)d3r (11) V (I ) V , (4) int 2V ij kl ik,jl n cell n =0 Vcell ref ={ ref ref ref} where I is the identity matrix, V L1 , L2 , L3 is the matrix of translation vectors of the reference cell and Vdef = regularizes the strain produced by the periodic images and {Ldef, Ldef, Ldef} is the matrix of translation vectors of the cell ensures the absolute convergence of the sum Eq. (10). The 1 2 3 self-strain correction energy is containing a defect. The elastic energy associated with the applied strain app is [5] 1 P corr =− − D = ij D 3 Estrain Pij ¯ij ij (r)d r. (12) ref 2 2V V cell Vcell Eapp = C appapp − P app, (5) 2 ij kl ij kl ij ij Since in a DFT calculation we do not explicitly compute D (r), and only need to correct the elastic part of the strain where V ref is the volume of the simulation cell and C is ij ij kl field of the defect, we again use the dipole tensor approxima- the elastic constant tensor. The quadratic term in Eq. (5)isthe tion and write elastic energy associated with the deformation of the simula- 1 tion box, whereas the second term is the energy of interaction corr =− 3 Estrain Pij PklGik,jl(r)d r. (13) between the defect and the applied strain. Since Eq. (5) is valid 2Vcell Vcell app in the limit 1, we neglect the difference between the This term corrects the total energy for the effect of elastic volume of the reference cell and the cell containing a defect. strain produced by the defect itself. The correction has the Equation (3) for the formation energy of a defect now has same form as Eq. (11), and corresponds to the first term of the the form series n = 0. N F = − app − def − corr Numerically, Eqs. (11) and (13) can be conveniently repre- Edef [Edef(Ndef) E ] Ebulk(Nbulk) Eel , Nbulk sented by surface integrals through the use of the divergence theorem [43], namely (6) corr 3 where Eel is a term taking into account the elastic field of PklGik,jl(r)d r = PkαGik,j (r)nαdS. (14) the defect and its interaction with its periodically translated Vcell Scell images. This elastic correction is a sum of two parts, Here n is the unit vector of external surface normal, and α de- corr = + corr notes a Cartesian component of this vector. Evaluating the first Eel Eint Estrain, (7) derivative of Green’s function Gik,j (r) is numerically more where Eint is the elastic energy of interaction with periodic expedient than the second derivative Gik,jl(r). Numerical tests images of the defect. Eint can be computed in the elastic dipole show that the results obtained using the above equations are in
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TABLE I. Elastic constants (in GPa units) calculated using the Le TABLE II. The total energy Edef of a simulation cell containing Page and Saxe [59] method for a two-atom cell, using a 30 × 30 × a defect, the applied strain energy Eapp, the elastic correction energy corr F 30 k-points mesh, and GGA-PBE exchange-correlation functional. Eel , and the formation energy Edef of a defect in W, where the 3 Atomic volumes (Å ) and lattice constants (Å) were taken from total energy of a perfect lattice cell is Ebulk =−1658.68112eV. relaxed 128 atoms perfect lattice simulations. Numbers in italic are The values are computed using 4×4×4 unit cells. The shape and the experimental values. Numbers in bold were computed using the volume of simulation cells containing defects are the same as those AM05 exchange-correlation functional. of the perfect lattice cell, with no relaxation of boundary conditions. All the values are given in eV units. 3 C11 (GPa) C12 (GPa) C44 (GPa) 0 (Å ) a0 (Å) app corr F W Edef E E E Li 18.14 11.85 11.43 20.24 3.434 el def 14.85a 12.53a 10.80a 21.27b 3.491b 111 d −1661.142709 0.0000 0.2101 10.2867 Na 9.34 7.44 5.96 36.94 4.196 111 c −1661.140788 0.0000 0.2100 10.2888 8.57c 7.11c 5.87c 37.71b 4.225b 110 d −1660.823317 0.0000 0.2401 10.5761 K 3.91 3.44 2.70 73.63 5.281 Tetra −1659.652937 0.0000 0.2697 11.7169 4.17d 3.41d 2.86d 71.32b 5.225b 100 d −1659.149546 0.0000 0.2941 12.1959 Rb 3.07 2.65 1.99 90.92 5.665 Octa −1659.074298 0.0000 0.3001 12.2652 3.25e 2.73e 1.98e 87.10b 5.585b Vac −1642.491107 0.0000 0.0083 3.2233 Cs 2.16 1.85 1.38 116.74 6.158 2.47f 2.06f 1.48f 110.45b 6.045b Ba 12.06 7.31 10.39 63.55 5.028 using the equation [4–7] 13.0g 7.6g 11.8g 63.25b 5.02b = app − V 279.59 142.02 26.72 13.40 2.993 Pij Vcell Cij kl kl σ¯ij , (15) 308.53 147.96 31.31 12.91 2.955 227.9h 118.7h 42.6h 13.91b 3.03b where Nb 248.76 135.24 19.46 18.33 3.322 1 273.54 143.06 23.62 17.68 3.282 σ¯ij = σij dV (16) h h h b b V 246.6 133.2 28.1 17.97 3.30 cell Vcell Mo 469.07 157.72 99.71 15.77 3.160 is the average macroscopic stress in the simulation box. The 505.43 175.26 108.04 15.24 3.124 relaxation volume tensor of a defect, proportional to the so- 464.7h 161.5h 108.9h 15.63h 3.15b called λ-tensor [2], is defined as Ta 266.28 161.36 76.75 18.29 3.320 293.44 168.18 82.08 17.62 3.278 = S P , (17) 266.0h 161.2h 82.4h 17.97b 3.30b ij ij kl kl W 518.26 199.77 142.09 16.14 3.184 where S = C−1 is the elastic compliance tensor, satisfying 569.73 211.52 157.16 15.61 3.149 = 1 + h h h b b the condition Cij kl Sklmn 2 (δimδjn δinδjm). The relaxation 522.4 204.4 160.6 15.78 3.16 volume of the defect can be computed by taking the trace of Cr 448.12 62.03 102.13 11.72 2.862 tensor ij , namely 394.1i 88.5i 103.75i 11.94b 2.88b Fe 289.34 152.34 107.43 11.34 2.831 rel = Trij = 11 + 22 + 33. (18) 243.1j 138.1j 121.9j 11.82b 2.87b a In the next section, we summarize the results of ab initio Ref. [64]. calculations of energies and elastic properties of defects in bRef. [65]. cRef. [66]. d Ref. [67]. TABLE III. The total energy E of a simulation cell containing e def Ref. [68]. a defect, the applied strain energy Eapp, the elastic correction energy f Ref. [69]. Ecorr, and the formation energy EF of a defect in W, where the g el def Ref. [70]. total energy of a perfect lattice cell is E =−1658.68112eV. h bulk Ref. [71]. The values are computed using 4×4×4 unit cells. The shape and i Ref. [72]. the volume of simulation cells containing defects are relaxed to the j Ref. [73]. stress-free condition. All the values are given in eV units.
app corr F W Edef E Eel Edef agreement with calculations performed using code ANETO 111 d −1661.631853 −0.4825 0.2063 10.2839 developed by Varvenne et al. [5]. 111 c −1661.630294 −0.4832 0.2060 10.2864 The dipole tensor of a localized defect can be calculated 110 d −1661.226956 −0.3991 0.2319 10.5798 from macrostresses developing in a simulation box due to Tetra −1660.049231 −0.3938 0.2652 11.7190 the presence of a defect [5,7]. The calculation is exact if 100 d −1659.579432 −0.4271 0.2861 12.2012 all the nonlinear deformations associated with the defect Octa −1659.509293 −0.4320 0.2930 12.2693 structure are contained entirely within the cell [7], even if Vac −1642.502871 −0.0130 0.0085 3.2242 the cell size is relatively small. The dipole tensor is computed
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TABLE IV. Formation energies EF (in eV units) of various self-interstitial atom (SIA) configurations and a vacancy in bcc metals, computed using the GGA-PBE functional. An SIA defect may adopt a 111 dumbbell, 111 crowdion, 110 dumbbell, tetrahedral site interstitial, 100 dumbbell, and octahedral site interstitial configuration. In antiferromagnetic chromium, a 11ξ (ξ ≈ 0.355) dumbbell is the most stable SIA configuration.
PBE Li Na K Rb Cs Ba V Nb Mo Ta W Cr Fe