ELEMENTARY DIFFUSION PROCESSES IN AL-CU-ZN ALLOYS: AN AB INITIO STUDY

Transcription

1 ELEMENTARY DIFFUSION PROCESSES IN AL-CU-ZN ALLOYS: AN AB INITIO STUDY S. GRABOWSKI AND P. ENTEL Theoretische Tieftemperaturphysik, Gerhard-Mercator-Universität Duisburg, Duisburg, Germany The vacancy formation and migration energy of Al as well as the migration energies of Cu and Zn impurities were calculated at T = 0 K using the Local-Density Approximation based on ultrasoft pseudopotentials. Diffusion processes were studied in supercells with 32 and 108 lattice sites. Furthermore the effect of Cu and Zn atoms on the activation energy for diffusion jumps of Al atoms in an ordered Al-Cu-Zn system was investigated. In each case relaxation of atoms around the defect was taken into account by relaxing all atoms in the supercell. 1 Introduction Technologically important properties of alloys such as mechanical strength and toughness, creep and corrosion resistance are essentially controlled by the presence of precipitated particles of a second phase. This commonly results from decomposition of a solid after quenching. A fundamental understanding of the mechanism and kinetics of precipitation reactions and the arising microstructures in binary and ternary alloys is, therefore, of great interest. Phase separation in quenched systems is closely connected to diffusion processes via vacant or interstitial lattice sites. The aim of this work is to gain new microscopical insights into unusual fast diffusion processes in the early stages of spinodal ordering and decomposition in alloys. The Al-Cu-Zn system is particular interesting because, depending on composition, both spinodal ordering (Cu 3 Zn) on the Cu-rich side of the phase diagram as well as spinodal decomposition (Al-Zn) on the Zn-rich side 1 are observed. Vacancy migration is the most important diffusion mechanism in metals. Atoms diffuse by exchanging places with a vacancy at a nearest-neighbor lattice site. It is reasonable to assume that fast diffusion processes are a key to understand spinodal ordering and decomposition in Al-Cu-Zn alloys. Therefore, it is of interest to calculate the energy barriers for vacancy migration in Al-Cu-Zn systems from first principles. In our approach the interaction between valence electrons and the ionic core is described by ultrasoft pseudopotentials. Results of defect calculations in Al-Cu-Zn systems based on ultrasoft pseudopotentials have not yet been published. All calculations presented in this paper were performed using the Vienna ab initio simulation grabowski: submitted to World Scientific on November 24,

2 package. 2 In order to check the reliability of our calculations the results are compared with experimental results and theoretical calculations published by other groups. In order to gain experience in handling Al-Cu-Zn systems, we replaced some atoms in the fcc Al lattice by Cu or Zn atoms and studied their influence on the diffusion of Al atoms. In fcc metals atoms diffuse by jumping via vacant nearest-neighbor lattice sites. The diffusing atom has to push apart four nearest neighbors when approaching the position halfway between the original lattice site and the vacancy site (the so called saddle point configuration, see Fig. 1). The energy a migrating atom has to take up to perform a diffusion jump arises mainly from the repulsive interaction with its four nearest neighbors in the saddle point configuration and is commonly interpreted as the migration energy. We exchanged two of these atoms by Cu or Zn atoms which are smaller than Al atoms. This allows us to study if and to what extent the Cu or Zn atoms lower the activation energy barrier for Al atoms while passing through the saddle point configuration. Figure 1. Nature of the constriction which inhibits motion of an adjacent atom into a vacancy in the fcc lattice. The four atoms indicated by light grey circles are the four nearest neighbors of the jumping atom in the saddle point configuration and must give way before the indicated jump can occur. 2 Method In order to treat isolated defects and impurities in solids, one needs to consider large supercells because of the long range nature of the elastic interactions and because of loss of translational symmetry. A common approximation is to use supercells and artificially reimpose the translational symmetry by employing periodic boundary conditions. We start with the fcc lattice of Al and remove atoms in a periodic array so that each supercell of the lattice contains one grabowski: submitted to World Scientific on November 24,

3 vacancy. In the impurity calculations another Al atom in the supercell is substituted by an impurity atom so that each supercell contains one vacancy and one impurity. For a given atomic configuration we calculate the total energy of the supercell using a first-principles quantum mechanical method. Static relaxation of the atomic geometry was performed via a conjugate-gradient minimization of the total energy with respect to the atomic positions. After each update of the atomic coordinates the groundstate of the electronic system is self-consistently recalculated. The total energy calculations presented in this paper are based on the selfconsistent iterative solution of the Kohn-Sham equations 3 of Density Functional Theory. 4 We use the Local-Density Approximation (LDA) with the exchange correlation functional of Ceperley and Alder 5 which was parameterized by Perdew and Zunger. 6 The solutions of the Kohn-Sham equations are expanded in a plane-wave basis set and nonlocal ultrasoft pseudopotentials are used to describe the electron-ion interaction. 7,8 Table 1. Valence configurations of the ultrasoft pseudopotentials used in our calculations. Al Cu Zn Valence Configuration 3s 2 3p 1 3d 10 4s 1 3d 10 4s 2 Brillouin zone integrations necessary to calculate the band-structure energy were performed on Monkhorst-Pack 9 sets of special k-points. Partial occupancies of the Kohn-Sham one-electron states corresponding to a first order Methfessel-Paxton 10 function with σ =1.0eV, are introduced as a tool for reducing the number of k-points necessary to calculate an accurate bandstructure energy. All these features of the computational technique are implemented in the Vienna ab initio simulation package which we used for our calculations. A more detailed description of the program can be found in the publications of its authors Vacancy formation energy in Al Denoting by E(N,M,Ω) the energy of the system of N atoms and M vacancies, which occupy N + M lattice sites in a volume Ω, the vacancy formation energy at constant volume Ω is E VF = E[N 1, 1, Ω] N 1 E(N,0, Ω). (1) N grabowski: submitted to World Scientific on November 24,

4 We took Ω = NΩ 0 with Ω 0 being the LDA equilibrium volume per atom at T = 0 K. The LDA lattice constant at T =0Kisa 0 =3.985 Å. The quantity of physical interest is obtained for N,withΩ/N = constant. In oder to minimize the elastic interaction between vacancies, the supercell should be large. We present results calculated for supercells containing 32 and 108 lattice sites. Due to computational limitations it was not possible to perform calculations with the same supercell geometry for supercell sizes larger than 108 lattice sites. For both supercell sizes structural relaxation around the vacancy was taken into account by relaxing all atoms in the supercell. Figure 2. Geometry of the supercells used in the calculations. The small (large) supercell with 32 (108) lattice sites consists of 2 3 (3 3 ) cubic fcc elementary cells. Each supercell contains one vacancy so that the vacancies are distributed on a simple cubic lattice with lattice constant 2a (3a) with a=3.985 Å being the LDA lattice constant of the Al fcc lattice at T =0K. 2.2 Migration energy for self-diffusion in Al In the framework of transition state theory 11,12 the mobility of vacancies is proportional to the Boltzmann factor exp( E VM /k B T ). The vacancy migragrabowski: submitted to World Scientific on November 24,

5 tion energy E VM is given by the difference in potential energy for two relaxed static configurations, the first one with the considered atom in the saddlepoint configuration between the initial and final state of the jump, and the second configuration with the atom at the initial potential minimum. For the calculation of the migration energy, the unrelaxed saddle-point configuration is obtained when a nearest-neighbor atom of the vacancy is displaced halfway in the [110] direction towards the vacancy. By fixing the displaced atom in the saddle point and relaxing the remaining atoms in the supercell the relaxed saddle-point configuration is reached. The vacancy migration energy E VM is defined as the difference between the total energy of the saddle-point configuration and the energy E[N 1, 1, Ω] in Eq. (1) which corresponds to the total energy of the initial minimum configuration. 2.3 Migration energy for impurity diffusion in Al For the impurity calculations a nearest neighbor of the vacancy in the unrelaxed supercell was replaced by a Cu or Zn atom. After relaxation of all atoms in the supercell we obtained the total energy of the initial minimum configuration for the impurity jump. Starting with the unrelaxed initial minimum configuration the impurity was displaced halfway in the [110] direction towards the vacancy. By fixing the displaced impurity and relaxing the positions of all Al atoms in the supercell, we obtained the total energy of the relaxed saddle-point configuration. Just as in the case of self-diffusion the migration energy is defined as the difference of these two energies. 3 Results 3.1 Vacancy formation and migration energy of Al A detailed investigation of the convergence with respect to the supercell size, Brillouin zone sampling and plane-wave cutoff is one of the major problems in order to obtain meaningful results for defect data from electronic structure calculations in metals. Vacancy formation and migration energies for Al were calculated for supercells with 32 and 108 lattice sites and various n k n k n k Monkhorst-Pack sets of special k-points. The results are presented in Fig. 3. From the data in Fig. 3 we conclude that the results for E VF and E VM are converged within 10 mev with respect to the total number of k-points N k = n 3 k for the small (large) supercell with 6 n k (4 n k ). According to these calculations the vacancy formation and migration energy in Al is E VF =0.7eV and E VM =0.6eV, respectively. The difference between the results for the small and the large supercell is smaller than 5 mev. Relaxation grabowski: submitted to World Scientific on November 24,

6 of the atomic coordinates reduces the calculated values of E VF and E VM by 0.07 ev and 0.3 ev, respectively, which indicates the importance of relaxation especially for E VM. Figure 4 shows the dependence of E VF and E VM on the plane-wave cutoff, E cut, calculated for the small supercell. If E cut is increased from ev to ev, the calculated value of E VF and E VM increases by 30 mev and 5 mev, respectively. Both E VF and E VM can be considered as converged values with respect to E cut if E cut 200 ev. From the data in Fig. 3 and 4 we conclude that the vacancy formation and migration energy for Al is 0.73 ev and 0.6 ev, respectively. This is in good agreement with the experimental results E VF =0.67 ± 0.03 ev and E VM =0.61 ± 0.03 ev. 13 Previous results of ab initio calculations published by other groups range from 0.6 ev to 0.84 ev for E VF and from 0.5 ev to 0.9 ev for E VM (see Table 2). Table 2. Comparison of experimental and theoretical results for E VF and E VM calculated with different ab initio techniques. Defect Energy Expt. (ev) Calc. (ev) Method E VF 0.67 ± US-PP (this paper) 0.84 FLAPW NC-PP KKR 19 E VM 0.61 ± US-PP (this paper) 0.57 NC-PP ± 0.2 NC-PP Migration energies for Cu and Zn impurities in Al Our calculations of migration energies for Cu and Zn impurities in Al are restricted to a supercell with 32 lattice sites because of the much higher cutoff energy of the Cu and Zn pseudopotentials compared with the Al pseudopotential. For the Brillouin zone sampling we used n k n k n k Monkhorst-Pack sets of special k-points with 3 n k 8. The results are presented in Fig. grabowski: submitted to World Scientific on November 24,

7 E VF (ev) E VM (ev) N l N k N l N k Figure 3. Vacancy formation and migration energy versus N l N k. N k = n 3 k is the total number of k-points sampled in the Brillouin zone and N l is the number of lattice sites in the supercell. Total energy calculations were performed using n k n k n k Monkhorst-Pack sets of special k-points with 3 n k 10 (3 n k 6) for the small (large) supercell. Circles (squares) represent data for the small (large) supercell. Open (filled) symbols represent data with (without) relaxation. For these calculations the plane-wave cutoff was chosen to be ev E (ev) E cut (ev) Figure 4. Vacancy formation energy (circles) and vacancy migration energy (squares) versus plane-wave cutoff E cut calculated for the small supercell (32 sites) using an k-point mesh. 5. According to our calculations the relaxed value of the impurity migration energy of Cu (Zn) impurities is 0.56 ev (0.37 ev). Lattice relaxation reduces E VM for Zn by 0.06 ev. For the Cu impurity E VM increases by 0.1 ev if lattice relaxation is taken into account. With a Cu atom in the minimum configugrabowski: submitted to World Scientific on November 24,

8 ration the relaxation energy is approximately 0.15 ev larger than with a Zn or Al atom in the minimum configuration (see Table 3). Furthermore d/d 0 (see Table 3) for Cu is positive. This implies a repulsive interaction between the Cu atom and the vacancy at the nearest-neighbor lattice site. Table 3. Relaxation energies E sad rel, Emin rel or minimum configuration and the total relaxation energy E rel =E sad with a Cu, Zn or Al atom in the saddle point rel E min rel. d is the change of the distance between the Cu, Zn or Al atom in the minimum configuration and the vacancy if the lattice relaxation is taken into account. d 0 is the nearest-neighbor distance of the unrelaxed fcc lattice. E rel (ev) E sad rel (ev) E min rel (ev) d/d 0 Cu % Zn % Al % 0.6 E IM (ev) N l N k Figure 5. Migration energy of Cu and Zn impurities versus N l N k calculated for the small supercell containing N l = 32 lattice sites. Total energy calculations were performed using n k n k n k Monkhorst-Pack sets of special k-points with 3 n k 8. Circles (squares) represent data for the Cu (Zn) impurity. Open (filled) symbols represent data with (without) lattice relaxation. For these calculations the plane-wave cutoff was chosen to be ev. grabowski: submitted to World Scientific on November 24,

9 3.3 Comparison with experiment For a cubic supercell with 32 lattice sites the calculated value for the activation energy E VF +E VM of self-diffusion in Al is in good agreement with experimental values. 13 Our calculated values for E VF and E VM correspond to converged values with respect to the k-point density. Increasing the number of lattice sites from 32 to 108 does not change the result within 5 mev. The calculated activation energies for Cu and Zn impurities in Al are at least 0.1 ev lower than the experimental values. 14 One contribution to this error might be due to the pseudopotentials for Cu and Zn used in our calculations because it is difficult to describe perfectly the d-electrons within pseudopotential theory. Another source of error can be expected because of the use of the Local-Density Approximation. In general defect formation and migration energies in metals are temperature dependent so that systematic errors should be expected if zero-temperature theoretical results are compared with finite-temperature data obtained from experiments. Table 4. Experimental activation enthalpies of self- and impurity diffusion in Al compared with the sum E VF +E VM derived from the calculated values for E VF and E VM. In each case we took the calculated value E VF =0.73 ev for bulk Al as theoretical value for E VF. Experimental values for E VF and E VM for vacancies in Al are 0.67 ev ± 0.03 ev and 0.61 ev ± 0.03 ev, respectively. Al in Al Cu in Al Zn in Al Expt. (ev) Theo. (ev) Diffusion in Al-Cu-Zn alloys In order to study the influence of Cu and Zn atoms on the activation energy barriers for Al diffusion, we investigated the Al-Cu-Zn system shown in Fig. 6. During the jump into a vacant nearest-neighbor site at least four atoms have to be pushed apart by the jumping atom according to Fig. 1. We simply exchanged two of these atoms by Cu or Zn atoms. Total energies were calculated starting from five different initial configurations which are explained in Fig. 7 (b). After relaxing all 31 atoms in the supercell, except for the Al atom at the intermediate position in configuration 2 and 4, we obtained the total energy of each configuration. The differences in total energy between grabowski: submitted to World Scientific on November 24,

10 configuration i and i+1 (i =1...4) are presented in Fig. 7 (a). If all atoms in the supercell were of the same type, configuration 2 and 4 would correspond to saddle point configurations. Although configuration 2 and 4 do not exactly correspond to saddle point configurations of the systems studied here, it is worth comparing energy differences between the minimum configurations 1, 3, 5 and the configurations 2, 4 with the vacancy migration energy of pure Al. For pure Al the difference E 2 E 1 between the total energies E 1 and E 2 of configuration 1 and 2 is the vacancy migration energy. In the case of Al 6 Cu 2, Al 6 Zn 2 and pure Al, the configurations 1, 3, 5 as well as 2, 4 are equivalent because of symmetry reasons. Compared with pure Al the energy difference E 2 E 1 in Al 6 Cu 2 and Al 6 Zn 2 is reduced by at least 0.17 ev. For Al 6 Cu 2 the reduction in E 2 E 1 is larger than for Al 6 Zn 2. This can be explained if we assume that Cu is the smallest atom, Al the largest and the size of the Zn atom is in between the size of Cu and Al atoms. The results for Al 6 CuZn are more complicated. In this case neither the configurations 1, 3, 5 nor the configurations 2, 4 are equivalent. The Al 6 Cu 2 (Al 6 Zn 2 )systemhastwocu (Zn) atoms and two Al atoms as the four nearest neighbors of the jumping Al atom in the unrelaxed configuration 2 (4). In configuration 2 (4) of the Al 6 CuZn system also two Cu (Zn) atoms and two Al atoms are the nearest neighbors of the jumping Al atom. Surprisingly E 2 E 1 is larger than for Al 6 Cu 2 and E 4 E 3 is smaller than for Al 6 Zn 2. Al Cu, Zn Figure 6. The elementary cell of the Al-Cu-Zn system investigated here is shown on the left hand side. This cell consists of two cubic fcc cells. Six lattice sites are filled with Al atoms (light grey circles) and the remaining two lattice sites (dark grey and white circle) are filled with Cu or Zn atoms depending on the system studied. The Figure on the right shows the geometry of the supercell for which we investigated activation energy barriers for Al diffusion. The supercell consists of four elementary cells shown on the left hand side and contains 32 lattice sites. grabowski: submitted to World Scientific on November 24,

11 (a) Al Cu (b) 0.34 ev 0.34 ev 0.34 ev 0.34 ev Al Zn ev 0.43 ev 0.43 ev 0.43 ev Al CuZn ev 0.34 ev ev 0.51 ev 5 5 Figure 7. Fig. (a) shows the results of our calculations for the supercell in Fig. 6. Total energies were calculated for five configurations labeled by The Figure shows the difference in total energies between the configuration i and i + 1 (i = 1...4) for three different compositions. For each configuration a plane cut through the supercell is shown in Fig. (b). White circles represent vacant lattice sites and grey circles represent lattice sites occupied by Al atoms. The configurations only differ by the arrangement of atoms in this cut. This plane contains only Al atoms and a vacancy. In each case the supercell contains 31 atoms and one vacancy. Configurations 1, 3 and 5 are minimum configurations which are obtained by removing an Al atom from a regular lattice site and the remaining two configurations labeled by 2 and 4, are obtained by placing an Al atom in between the two vacant Al sites, see Fig. (b). grabowski: submitted to World Scientific on November 24,

12 4 Conclusions Using ultrasoft pseudopotentials we have calculated the vacancy formation and migration energy of Al and migration energies of Cu and Zn impurities in Al at T = 0 K. Our results are in good agreement with available experimental data even for the small supercell with 32 lattice sites. Furthermore we investigated the influence of Cu and Zn atoms on the migration energy of Al. Our calculations show that the migration energy of Al can be reduced by placing Cu or Zn atoms at the nearest-neighbor sites of a diffusing Al atom in the saddle point configuration. In order to have fast diffusion processes in Al-Cu-Zn alloys, the energy barrier for the elementary diffusion process via nearest-neighbor vacancy sites must be much lower. For diffusion in bcc lattices it has been proposed that phonons can considerably help the diffusing atom. 20,21 For fcc lattices phononassisted diffusion has not yet been studied. Preliminary investigations show that for the elementary diffusion path for an atom at (0,0,0) to the vacancy position at ( 1 2, 1 2, 0) in the fcc lattice will not entirely occur in the (0,1,0) plane but will first follow a path along the [1, 1, 1] direction and then going into the vacancy position. In this context it is of particular interest to note that all measured phonon spectra of Al-Cu-Zn systems alloys show very low lying TA [1, 1, 1] branches 22 which help to lower the potential barrier of the diffusing atom. Work along this lines is in progress. 5 Acknowledgements This work has been supported by the German Science Council (Deutsche Forschungsgemeinschaft) through the Graduate College Structure and Dynamics of Heterogeneous Systems. References 1. H. Löffler, Structure and Structure Development of Al-Zn alloys (Akademie Verlag, Berlin, 1995). 2. G. Kresse and J. Furthmüller, Phys. Rev. B 54, (1996). 3. W. Kohn and L.J. Sham, Phys. Rev. 140, A 1133 (1965). 4. P. Hohenberg and W. Kohn, Phys. Rev. 136, B 864 (1964). 5. D.M. Ceperley and B.J. Alder, Phys. Rev. Lett. 55, 566 (1980). 6. J.P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981). 7. D. Vanderbilt, Phys. Rev. B 41, 7892 (1990). 8. G. Kresse and J. Hafner, J. Phys. : Condens. Matter 6, 8245 (1994). grabowski: submitted to World Scientific on November 24,

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