Full-Potential KKR calculations for Lattice Distortion around Impurities in Al-based dilute alloys, based on the Generalized-Gradient Approximation

Transcription

1 Trans. Mat. Res. Soc. Japan 40[2] (2015) Full-Potential KKR calculations for Lattice Distortion around Impurities in Al-based dilute alloys, based on the Generalized-Gradient Approximation C. Liu 1, M. Asato 2*, N. Fujima 3, and T. Hoshino 3 1 Graduate School of Science and Technology, Shizuoka University, Hamamatsu , Japan 2 National Institute of Technology, Niihama College, Niihama, Ehime , Japan 3 Graduate School of Engineering, Shizuoka University, Hamamatsu , Japan *asato@sci.niihama-nct.ac.jp We calculate systematically the atomic volume changes caused by Sc-Ge impurities in Al, using the formalism given by the Kanzaki model. All the parameters in the Kanzaki model, such as the Hellmann-Feynman (HF) forces and the lattice distortion, are calculated by the ab-initio calculations based on the generalized-gradient approximation in density functional formalism and the full-potential Korringa-Kohn-Rostoker (FPKKR) Green s function method for point defects. Most of the calculated results agree very well (within the error of 5%) with the available experimental results. We found that the calculated results for the HF forces on the 1st-nearest neighboring host atoms around impurities are strongly correlated with the volume changes per impurity in Al. The magnetism of 3d transition-metal impurities (Cr, Mn, Fe) is also discussed. Key words: Lattice Distortion, Al-based Dilute Alloys, Ab-initio Calculations, KKR, GGA in Density functional theory 1. INTRODUCTION The presence of a point defect in a crystal, such as a vacancy or an impurity atom, generally causes a displacement of the neighboring host atoms from their ideal lattice positions. For alloys, such a lattice distortion changes the lattice constant and this change can be measured by x-ray diffraction. King presented the volume size-factors of 469 substitutional solid solutions [1], in a convenient and compressive form, which were calculated from the experimental lattice parameter data of solid solutions, available in the literature [2]. It is noted that the size-factor concept has been applied to study the physical, chemical and mechanical properties of metallic solutions, although it treats the average change of the lattice parameter of metallic solid solutions. However, this information is not sufficient to estimate the interatomic distances because, generally, the distortion is different in magnitude for different atomic shells around the defects. More detailed information can be obtained by extended x-ray-absorption fine-structure (EXAFS) experiments. Using this method, Scheuer et al succeeded in determining the 1st-nearest neighboring distances around impurities in fcc (Al, Cu, Ni, Pd, Ag) and bcc (Fe, Nb, V) metals [3]. It is also noted that the interaction energies of two impurities in metals were determined by the perturbed-angular-correlation probe (PAC) experiments. Królas presented the accurate data for the interaction energies of impurities with PAC-probes (Rh, Pd, In, Sn, Sb) in fcc (Ni, Cu, Pd, Ag, Au), bcc (Fe) and hcp (Rh) metals [4]. From the theoretical point of view the treatment of structural relaxation due to defects in crystals is a difficult task. The problem of the lattice relaxation has been mostly dealt phenomenologically, e.g., by models of lattice statics or continuum theory. A reliable microscopic description of lattice relaxation effects based on first-principles electronic structure calculations requires very accurate total energies or forces and has mostly been attempted so far for simple metals and semiconductors on the basis of pseudo potential treatments. The difficulty arises mainly from the fact that energy differences due to local atomic displacements are quite small, of the order of 0.1 ev, compared with, e.g., the cohesive energy (5eV) of the solid. At present we can calculate accurately the atomic volume changes per impurity in transition metals, by using ab-initio all-electron calculations for the electronic structure of solids, based on the full-potential Korringa-Kohn-Rostoker Green s function method (FPKKR) and the density functional theory. Papanikolaou et al succeeded in calculating the atomic volume changes per Sc-Ge and Zr-Sn impurities in Cu [5]. In order to estimate the atomic volume changes per impurity, they used the Kanzaki model [6]. It is noted that the formalism given by the Kanzaki model is very suitable for the present work, because all the parameters in the Kanzaki model, such as the Hellman-Feynman forces and lattice distortion, caused by impurities in metals, are calculated by the present ab-initio calculation method. We also succeeded in calculating the atomic volume changes per Sc-Ge impurities in Fe [7], and the lattice relaxation energies of vacancies in Cu and Al [8]. It is also noted that the available experimental results for the interaction energies between the two impurities in Fe, such as I-I and I-Sn (I=Sc-Ge, Sn: perturbed-angularcorrelation probe), were reproduced very well by the FPKKR method [9, 10]. We are now studying the distortion effect for the defects in Al-based alloys, such as single and two impurities. In the present paper, we show the calculated result for the lattice relaxation due to Sc-Ge impurities 159

2 160 Full-Potential KKR calculations for Lattice Distortion around Impurities in Al-based dilute alloys, based on the GeneralizedGradient Approximation in Al-based dilute alloys. In Sect. 2, we describe the calculational method. In Sect. 3, we show the calculated results. The present calculations reproduce very well the available experimental results, such as the averaged volume changes per impurity in Al and the interatomic distances between the 1st-nearest neighboring (NN) Al atoms and impurities. In Sect.4, we summarize the main result of the present paper. The calculated results for the interaction energies between two impurities in Al will be published in the subsequent paper [11]. 2. CALCULATIONAL METHOD The calculations for the total energies and Hellmann-Feynman (HF) forces of the point defect systems are based on the full-potential Korringa-Kohn-Rostoker Green s function method and the generalized-gradient approximation in the density functional formalism [5,10,12]. The advantage of the Green s function method is that due to the introduction of the host Green s function, the embedding of point defects in an otherwise ideal crystal is described correctly, differently from the supercell and cluster calculations. Thus, the distance dependence of the HF forces around impurities and the impurity-impurity interaction energies, being usually long-ranged, is studied correctly by the present calculations. It is noted that although the potential perturbation due to the defects is localized in the vicinity of the defects, the change of the wave functions due to the defects is delocalized over the whole space. The practical advantage in using the Green s function method is to exploit the short-range nature of the defect potential. For example, in order to obtain the accurate and converged total energy of the defect system without the lattice distortion effect, it is enough to redetermine self-consistently only the potentials of the impurities and their neighboring potentials (up to the 1st (2nd)-NN atoms around impurities, in the case of fcc (bcc) structure), if the total energy change due to the perturbed wave functions over the infinite space is correctly evaluated by using the Lloyd s formula. The short-range nature of the defect potential is discussed in Ref. 12. On the other hand, it is noted that the study for the lattice distortion effect demands the self-consistent calculations for potentials in the larger region because the potentials at the host atoms connected to the displaced host atoms may be strongly perturbed [10]. Since we treat the lattice displacement of the 1st-NN Al atoms around the defect, as discussed in 3-1, the perturbation, at least, up to the 1st-NN atoms around the displaced Al atoms must be calculated self-consistently. In the present work, we calculate the HF forces induced on the host 201-atoms at the ideal atomic positions up to the 10th-NN atoms of an impurity, as shown in Fig.1. The geometry of the displaced atoms around the impurity keeps the Oh symmetry. Thus, using the Oh symmetry, we redetermined all the potentials in the 201-atom cluster. This cluster size may be enough to examine the lattice distortion up to the 2nd-NN atoms around the impurity, as discussed in 3-2, because the 201-atom cluster includes all the atoms up to the 4th-NN atoms around the 2nd-NN atoms. We also note that, in order determine the atomic displacement, the HF forces were found to be less than 1mRy/aB, as discussed in Ref.5. Fig atom cluster of O h symmetry, including one impurity with the surrounding 200 Al atoms up to the 10th-neighbors. There are two-kinds of 9th-neighbors, being noted as 9a and 9b. The representative atomic coordinates from the center to the 10th-neighbors are: Center(0,0,0), 1st (0,0.5,0.5), 2nd (0,0,1), 3rd (-0.5,0.5,1), 4th (1,1,0), 5th (-0.5,0,1.5), 6th (-1,1,1), 7th (1.0,0.5,1.5), 8th (0,0,2), 9th-a (0.5,-0.5,2), 9th-b (1.5,1.5,0), 10th (2,1,0).

3 C. Liu et al. Trans. Mat. Res. Soc. Japan 40[2] (2015) 161 Fig.2 Calculated results without the spin-polarization effect (solid lines), for the HF forces induced on Al atoms at ideal atomic positions in the vicinity of impurities (Sc-Ge) in Al. For Cr and Mn impurities, the calculated results with the spin-polarization effect (broken lines) are also shown. 3. CALCULATED RESULTS AND DISCUSSIONS If a defect (impurity or vacancy) is inserted in a metal, the HF forces are induced on the host atoms around the defect and the lattice relaxation occurs. The equilibrium positions of the neighboring host atoms are determined by the zero-force condition. The atomic volume change per impurity may be calculated by using the formalism of the Kanzaki model [5-8]. In the present work, using our ab-initio calculation method, we can calculate accurately all the parameters in the Kanzaki model, such as the HF forces and the lattice distortion around impurities. In 3-1, we show the calculated results for the distance dependence of the HF forces on the host atoms around impurities. In 3-2, we show the calculated results for the lattice distortion of the 1st-NN host Al atoms around impurities. In 3-3, we show the calculated results for the relative volume changes per impurity in Al-based dilute alloys. In the present paper we treat Sc-Ge impurities. 3.1 Distance dependence of HF forces on the host atoms around impurities in Al Fig.2 shows the calculated results for the Hellman-Feynman (HF) forces on the host atoms at ideal atomic positions (up to the 10th-NN atoms) in the vicinity of an impurity, shown in Fig.1. The positive forces mean the outward relaxation around a defect, negative the inward one. The fundamental features of the calculated results are summarized as follows, (1) The HF force on the 1st-NN atoms is dominant. (2) The HF force becomes weak with the distance from the impurity, although it remains up to the distant neighbors, showing the Friedel-type oscillating behavior. (3) The HF forces are almost zero over the 10th-NN atoms. These fundamental features are almost the same as the calculated results for Sc-Ge impurities in Fe, although the oscillating behavior due to the screening effect of the 4sp electrons in Fe is very complicated, differently from the simple oscillating behavior for 3sp electrons in Al, as discussed in Ref Lattice distortion of 1st-NN and 2nd-NN host atoms around the impurities in Al In the present calculations, the 3p electrons of impurities I are treated as the valence states for I=Sc, Ti, and V, while the core states for I=Cr-Ge. The treatment for 3p semi-core states of impurities has been discussed in Ref. 5. First we consider only the relaxation of 1st-NN atoms around the impurity because the HF forces are dominant on the 1st-NN atoms, as shown in the preceding section. Figure 3 shows the change of the total energy with the different radial shifts from the ideal positions of the 1st-NN atoms, together with the change of the HF forces exerted on the 1st-NN atoms. The positions of the force-zero agree well with those of the minima in total energies. This result demonstrates the accuracy of the present calculations. It is also noted that the calculated results for impurities I (=Mn, Cu, Zn, Ga, Ge) agree well with the available experimental results [3], as listed in Table I (a). It is noted that the experimental results are distributed in some range, depending on the models

4 162 Full-Potential KKR calculations for Lattice Distortion around Impurities in Al-based dilute alloys, based on the Generalized- Gradient Approximation Fig3. Distortion energies, radial forces on the 1st-nearest neighboring host atoms around impurities, as a function of lattice distortion (the 1st-nearest neighboring interatomic distance). Fig4. Local magnetic moments (solid lines) of impurities, as a function of the 1st-nearest neighboring interatomic distance. The total magnetic moments (broken lines) including the magnetic moments of the surrounding Al, induced by an insertion of magnetic impurities, are also shown. used for the EXAFS analysis and the experimental accuracy [13]. For I (=Cr, Mn, and Fe), we showed the calculated results with the spin-polarization effect, because the spin-polarized states are stable. The local magnetic moments of I (=Cr, Mn, Fe), as a function of the change of the 1st-NN distance, are shown in Fig. 4. The magnetism is very strong for the almost isolated impurity atoms at the ideal atomic positions, because the 1st-NN distance is very large compared with the atomic sizes of these impurity atoms, as shown in Fig.6. It is noted that the magnetism becomes weak together with the increase of the Al-I interactions of the surrounding Al, caused by the decease of the 1st-NN distance. The equilibrium positions of the surrounding Al atoms are determined by the minimization of the total energies including the magnetic energies of 3d transition-metal (Cr, Mn, and Fe) impurities and the bonding energies between the 3d transition-metal impurities and the surrounding Al atoms. Secondly we consider both the relaxation of 1st-NN and 2nd-NN atoms. Since the HF forces on the 2nd-NN atoms are weak, we can easily obtain the force-zero on the 1st- and 2nd-NN atoms by starting from the atomic positions of force-zero on the 1st-NN atoms. The calculated results for the relaxation are shown in Table I (b). It is noted that the relaxation of the 1st-NN atoms doesn t change very much by the inclusion of the relaxation of the 2nd-NN atoms. Before closing this section, we discuss the convergence of the present calculations. We have already found that the present results for the displacement of the 1st-NN atoms agree with the calculation results obtained by redetermining all the potentials in the 405-atom cluster, including up to the 16th-NN atoms [11]. 3.3 Relative volumes changes per impurity in Al-based dilute alloys A lattice expansion or compression due to single impurities results in a change of the lattice constant. In the case of cubic metals the volume change due to an impurity is given by the first moment of the Kanzaki forces [6] 1 3 n V V-V0 FK Rn (1) K n where V and V0 are the atomic volumes of the impurity system and ideal host, respectively, and K is the bulk modulus of pure Al (fcc). We use the values of V0 and K [12] obtained by the present ab-initio calculation method. The Kanzaki forces n n nn F K F - sn (2) n are the forces that would induce the same local distortions s n in the host crystal as those the direct n forces F causes in the defect systems, as discussed in n n Ref.5. ΔΦ is a difference of the force-constant matrices of the systems with and without a defect ( Φ, Φ 0 ), given by the difference between the slopes of the force curves.

5 C. Liu et al. Trans. Mat. Res. Soc. Japan 40[2] (2015) 163 Table I. Calculated results for the lattice relaxation (in % of interatomic distance) around impurities in Al. The lattice relaxation is confined only to 1st-neighboring host atoms (a) and to 1st- and 2nd-neighboring host atoms (b). The experimental results in Ref. 3 are also listed in (a). For Cr, Mn, and Fe, the spin-polarization effect is included in the calculations. (a) Imp. Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge 1st Expt (b) Imp. Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge 1st nd In Fig.5 (a), we present the calculated relative volume changes ΔV/V0 for all the systems considered in this work. Most of the calculated results (I=Ti, Cr, Mn, Cu, Zn, Ga) agree very well (within the error of 5%) with the available experimental results. For I=V, there is a large discrepancy (20%) between the calculated and experimental results, which is puzzling, especially in view of the good agreement obtained for the other impurities. At present we cannot explain the reason of the disagreement. Here we discuss the micro mechanism of the expansion or compression of the atomic volumes per impurities in Al. The chemical trend (parabolic behavior) of the calculated results is similar to that of the atomic sizes of the bulk states in the fcc structure, being obtained by the present ab-initio calculations and shown in Fig.6. The lattice parameters in the ferromagnetic states of Fe and Co ( in Fig.6), and the aniti-ferromagnetic state of Mn ( in Fig.6), are also shown because those states are the calculated ground states in fcc state. It is noted that the expansion due to Sc, Ga, Ge impurities and the compression due to the other impurities (Ti Zn) are very well understood by comparison of the atomic sizes of impurities with that of Al. The large compression due to Mn, Fe, Co, Ni, and Cu is due to the very small (> 10%) atomic size of 3d transition metals, compared with that of Al. It is also noted that the chemical trend of the volume change difference agrees very well with the HF forces induced on the 1st-NN Al atoms, shown in Fig. 5(b). It is obvious that there is a strong correlation between the atomic volume changes and the HF forces induced on the 1st-NN Al atoms. We also found that the experimental results for Cr and Mn are reproduced very well by the calculations with spin-polarization effect. It is also noted that the magnetic effect is very large for Cr (30%) and Mn (50%) impurities in Al. The large compression obtained by the calculations without the spin-polarization effect is reduced very much by the calculations with spin-polarization effect. This magnetic effect may be easily understood by considering that the magnetic energy is increased by the increase of the interatomic distance between the impurity and the 1st-NN Al. Fig.5 (a) Calculated results for the relative volume changes per impurity, together with the available experimental results. (b) Calculated results for the HF forces induced on the host atoms at the ideal 1st-nearest neighboring atomic positions around impurities. Fig.6 Calculated equilibrium lattice parameters of metals (Sc-Ge) of non-magnetic states, in fcc structure. The lattice parameters of the ferromagnetic (for Fe and Co) and antiferromagnetic (for Mn) states are also shown. The lattice parameter of Al is 7.65 a.u..

6 164 Full-Potential KKR calculations for Lattice Distortion around Impurities in Al-based dilute alloys, based on the Generalized- Gradient Approximation 3. SUMMARY We calculated systematically the atomic volume changes per impurity (Sc-Ge) in Al, by using the formalism given by the Kanzaki model [5-8]. All the parameters in the Kanzaki model, such as the Hellmann-Feynman(HF) forces and the lattice distortion around impurities in Al, were calculated by the present ab-initio calculations. The obtained atomic volume changes per impurity agree very well (within the error of 5%) with the available experimental results, except V impurity. We also found that the chemical trend of the atomic volume changes per impurity in Al is understood very well by considering the sign and magnitude of the calculated HF forces on the 1st-NN Al atoms around impurities. The present data are very useful for understanding the atomic and electronic structures of Al-based alloys and the construction of the reliable interaction parameter model such as the embedded-atom-method potentials [14]. Acknowledgement The authors are grateful for the financial support from the Ministry of Education, Culture, Science and Technology (JSPS KAKENHI Grand Nos and ). References [1] H. W. King, J. Mater. Sci. 1 (1966), 79. [2] W. B. Pearson, A Handbook of Lattice Spacings and Structures of Metals and Alloys (pergamon, London, 1958), Vol.1 and Vol.2. [3] U. Scheuer and B. Lengler, Phys. Rev. B 44 (1991), [4] K. Królas: Hyperfine Interact. 60 (1990), 581. [5] N. Papanikolaou, R. Zeller, P. H. Dederichs, and N. Stefanou, Phys. Rev. B 55 (1997), [6] G. Leibfried and N. Breuer, Point Defects Metal I (Springer, Verlag, Berlin, 1978). [7] C. Liu, M. Asato, N. Fujima, and T. Hoshino, The Proceeding of 8th Pacific Rim International Congress on Advanced Materials and Processing (2013), [8] T. Hoshino, N. Papanikolaou, R. Zeller, P. H. Dederichs, M.Asato, T. Asada, and N. Stefanou, Comput. Mater. Sci. 14 (1999), 56. [9] C. Liu, M. Asato, N. Fujima, and T. Hoshino, Mater. Trans. 54 (2013), [10] M.Asato, C.Liu, K.Kawakami, N.Fujima and T.Hoshino, Mater.Trans. 55 (2014), [11] C. Liu, M. Asato, N. Fujima, and T. Hoshino, to be submitted to Mater.Trans. [12] T. Hoshino, T. Mizuno, M. Asato, and H. Fukushima, Mater. Trans. 42 (2001), [13] For example, the experimental results for Cu in Al are estimated by using the experimental minimum and maximum values (-6.4= pm and -5.0= pm) of the shifts for the lowest concentration =0.5% and the lattice parameter of pure Al (284.9 pm), listed in Table I and II in Ref.3, because the present paper treats the dilute alloys. The minimum and maximum experimental results for the lattice relaxation become -2.2% (=-6.4/ ) and -1.8% (=-5.0/ ), respectively. [14] M. Asato, R. Tamura, N. Fujima, and T. Hoshino, Mater. Sci. Forum (2007), (Received October 7, 2014; Accepted March 13, 2015)