Parameters of self diffusion from non-empirical pair potential

Z. Phys. B 91, 225-228 (1993)

ZEITSCHRIFT FORPHYSIKB 9 Springer-Verlag 1993

Parameters of self diffusion from non-empirical pair potential S. Dorfman 1,., D. Fuks 2 1 Max-Planck-Institut fiir Festk6rperforschung, Hochfeld-Magnetlabor, BP 166X, F-38042 Grenoble Cedex, France 2 Materials Engineering Department, Ben-Gurion University of the Negev, Beer Sheva, Israel Received: 6 December 1992

Abstract. A scheme for construction of the pair potential from non-empirical calculations of electronic structure of solids is suggested. As an example, parameters of Lennard-Jones potential are obtained for fcc Cs, based on LMTO calculations of energy parameters. Vacancy formation and migration energies for fcc Cs are calculated from this first-principles pair potential. In addition, the frequency of vibration and the jump probability of an atom are calculated and it is shown that they are direction dependent.

The quantitative theory of diffusion is connected with the design of interatomic pair potentials, ~(r), in metals. It gives the possibility to construct the profiles of potenial barriers and to estimate thus the values of migration energies and jump probabilities in different directions. Diffusion in metals generally takes place by a series of jumps of individual atoms from site to site throughout the crystal. A number of mechanisms have been proposed for the elementary atomic jump [1]. The vacancy mechanism is the dominant mechanism for diffusion for most of the pure metals [2 5], and it is associated with the formation and the migration of vacancies. Here only simple vacancies will be considered. These vacancies represent holes on atomic sites in the lattice and the corresponding atoms have been removed to the surface of the crystal. In order to determine the activation energy of the diffusion process the evaluation of the energies of formation and migration is needed. Calculations of diffusion parameters for high-pressure phases are of great interest as there are difficulties in their experimental determination. In this paper we suggest a quantitative scheme for calculations of these parameters, and as a particular example we consider the high-pressure phase of fcc Cs, which has been already investigated experimentally [-6-8-1. * P e r m a n e n t address: Department of Physics, Technion-Israel Institute of Technology, 32000 Haifa, Israel

A review of calculation of diffusion parameters calculations with semiempirical and empirical pair potentials is given in [9]. The limitations of such a pair potential approach are obvious from the expression for the total energy, Etot, of the monoatomic crystal --1 ~, O(Ir~-rj[), E,o,-~

(1)

i,j;i,l: j

where the double summation over i and j is provided for all sites of atoms ri. This expression does not include three-, four- and higher-particle interactions and that is why it is restricted. On the other hand Eto t c a n be derived using more or less accurate solid state calculations [10, i 1]. The results of these calculations strongly depend on the number of electrons included in the core and may differ by several orders of magnitude. For example, the total energies of alkaline metals, calculated by means of pseudopotentials are of order of 0.4-4.0 Ry [12] and that, calculated by more exact band structure theory are about l0 s Ry [13]. So it is a very complicated task to compare these results even if one suggests that they are good enough. At the same time it is not a simple task to get good results for different properties of metals in the framework of a unified formalism. Here we present a straightforward procedure of evaluating different properties of metals and especially the pair potentials, that seem to include partly the manybody interatomic interactions. We shall base our calculations on the linear muffin-tin orbital (LMTO) approach for solids [14, 15]. Using self-consistent band calculations, one can evaluate the binding energy curves for a system of atoms on a given lattice [16, 17]. Analysis of such curves for elements and for simple compounds and alloys yields theoretical ground-state properties such as cohesive energies, equilibrium lattice separations, and bulk moduli that are in good agreement with experiment [17-20]. The LMTO procedure is known to be a very effective method for nonempirical calculations of band structures. Self-consistent band calculations can be carried out for different cell volumes O. They give the volume depen-

226 dence of the total energy,

Etot,

of the crystal:

Etot = E (12).

(2)

In order to perform self-consistent field (SCF) calculations in the atomic sphere approximation (ASA), the spherically averaged electronic density is used, and for this purpose the one-center expansions are sufficiently accurate. The total energy of the electrons in the ground state may, according to density functional theory [17] be estimated as oct

Etot = 2 ei - - g d o u b l e c o u n t i n g -[- gelectrostatic + Eexehang....... lation9

(3)

Here ei is the one-electrion terms, Edoubleco~nti.g the double counting energy, Eelectrostati e - the electrostatic energy which can be calculated from the Madelung constants and Ee~h,ng .... rrelation is the exchange-correlation energy. The electrostatic term can be calculated from the electronic density, p(r) and crystal potential, V(r), as

Eao~U. . . . . ti.g = 89 S p(r) V(r) d r,

(4)

where V(r) is a spherically symmetric potential. It is now fairly simple using the Born-Oppenheimer and the localdensity-functional (LDF) approximations [16] to carry out first-principles calculations of the full pressure [21, 22] at zero temperature, that is the change of the total energy with uniform compression, i.e.

dEt~

P(f2) =

d12 '

(5)

where 12 is the volume and we have neglected the zeropoint motion of the nuclei. Calculation of the pressurevolume relation enables one, in principle, to estimate the equilibrium atomic volume, f2o, as P(f2o)=0,

(6)

the bulk modulus, or inverse compressibility, B, as B = - [ dd~nO]-f2 o,

P-

123 + 125 -

(9)

Based on least-squares fitting by an optimization procedure the parameters e and fl were determined as e=2.47347 x 109 kBar x (a.u.) 3 and fl= 3.49432 x 1014 kBar x (a.u.) s. This enables us to carry out calculations of the ground state properties of our objects. In Fig. 2 the Lennard-Jones type pair potential that was constructed by this approximation is shown. Also by this fitting procedure the volume dependence of Er can be estimated as indicated in Fig. 1. This dependence is obtained by variating O o in Eq. (8). It is useful to note here, that the calculated values of pressure arc of the same order of accuracy as the calculated band structure in the framework of the ASALMTO scheme [22]. At the same time the results of the Eret calculations are much less accurate. That means

(7) P(kBar)

and the cohesive energy, Eooh, as Eooh= -- S e d f2.

We made first-principles calculations of the pressure at zero temperature, i.e., the change in the total energy, Etot, with uniform compression, was evaluated. The selfconsistent crystal potentials for fcc Cs were generated by ASA-LMTO band structure calculations. We included also the relativistic corrections in the scalar-relativistic scheme (without the spin-orbit coupling). Calculations have been performed for 9 different lattice parameters, on the mesh corresponding to 175 k-points in the irreducible wedge of the Brillouin zone. The integration has been done by the tetrahedron method [23, 24] and the Hedin-Lundquist exchange-correlation potential in the local density approximation (LDA) was used [25]. We found band structures that are practically identical with the results of the previous papers [26, 27]. Figure 1 shows the dependence of the pressure on the atomic volume 120. From this dependence the cohesive energy Ecoh at the equilibrium volume was estimated as Eooh=0.434 eV at the lattice parameter a = 11.43 a.u. This value is quite reasonable. It is clear that Ecoh, evaluated in this way, is a function of the volume of the system at equilibrium. We have selected on physical grounds [28-32] the following form for the fitting procedure:

E(eV)

(8)

Y~o

Using this approach it is possible to produce calculations of the full pressure with the same accuracy as calculations of the potential parameters in the L M T O scheme [14, 15, 2 1 ] . Performing these calculations on a large set of volumes we can receive the dependence of the pressure on the volume. From this dependence we can extract the equilibrium volume, 12o at P = 0 , the bulk modulus, B, and the cohesive energy, E~oh. These energies can serve as a measure of the stability of the investigated compounds.

..........................................................................

" .

289.0 '

v..o.!um.e.(~:u:} ................ o

.......... .........-'"'" Cohesion energy -0.44 843.0

Fig. 1. The dependence of pressure and cohesion energy vs an atomic volume for fcc Cs

227 nate sphere. For simplicity we can omitte for Cs the Friedel-type part, this term is the most important for transition metals and for Cs it is reasonably small (8.1 x 10 -a eV) [12, 34]:

Potential(eV)

0.

[[~:.u.,). -........1.........................................................................................

,

-0.03 20

Fig. 2. The Lennard-Jones type pair potential for fcc Cs

that the fitting procedure for pair potentials that is possible in principle directly from (1) will not be accurate enough for the description of such properties as cohesion energy, shear modulus, etc. The potentials obtained by this approach may be made more accurate by fitting with experimental data of shear and bulk modulus, but sometimes these values are unmeasurable. For phases under critical conditions these problems are the most complicated and tedious ones. We choose the form of potential as the Lennard-Jones (6-12) form and add the oscillatory part of the Friedel type [-33]. The last term is typical for metals and takes into account the averaged density of electron gas. It is the result of including in that term the Fermi wave vector, k e. Thus, we will have an electron density-dependent potential that will be useful for describing volume-dependent properties of solids. Therefore, we suggest to fit our results of SCF-calculations with the following form of potential, ~b(r): 0 a {b(r)=z-~+ r~+q

cos(2kfr ) r 3

(10) ,

where Z is a constant. Substituting Off) into Etot, Eq. (1), and making use of (8) we can express the pressure in terms of volume derivatives of potential, and take into account controlled number of coordinate spheres for the lattice under investigation. Now we have several possibilities of estimation of the parameters of the pair potential:

Unfourtunately, there is no experimental data for this value. However, specific heat measurements by adiabatic calorimetry yielded a value of 0.28 eV for the vacancy energy of formation in bcc Cs [2J. Calculations of Fumi [33 provides for Cs a formation energy of 0.26 eV, and pseudopotential calculations [4] gives the value 0.33 eV. From Eq. (14) we have estimated parameters of LennardJones pair potential. From the estimated values of 0 and o- the potential barrier, i.e. (bu, can be obtained for different crystallographic directions. Figure 3 shows a profile constructed by this method which indicates the potential barrier between nearest neighbors in two directions. The height of the barrier is ~u=0.0185eV, for the (110> direction, and the value ~,=0.024 eV for the (100> direction was obtained. The lower potential barrier for the (110> direction shows that the probability of migration of a vacancy in this direction is higher. The activation energy for diffusion in fcc Cs for the (110> direction is E~+Eu, where E, is the migration energy. This energy can be obtained as the difference between two energies of stages of the diffusion jump: the first - t h e crystal with one vacancy

~,V

Diffusion Barriers

t,/,/

i

(11)

Here the summation is produced over a number of coordinate spheres, i, with radius, R~, up to the N-th coordi-

~

2 iiiii i",',,,/

--12 Rr 5

-3 cos(2kf Ri) -R~-2ky.sin(2kyR~) tl -~i ] .

(12)

E~ = 0.217 eV.

/ /

,-., 6

.

Such procedure gives the opportunity of design of the nonempirical pair potential, but this potential is in fact takes into account all three-, four- and highest-order interactions that can be reduced to pair ones. Now the energy of formation of vacancies, E,, for fcc Cs can be evaluated by using the Eooh at equilibrium. We assume that during the formation process of a single vacancy in the bulk all the atomic bonds are broken, but half of them are restored when the atom is on the surface. This means that E~ = Econ/2 which results in

1) by solving the system of equations in number equal to the number, n, of potential parameters for given values of P(O,), P(f2:) ..... P(f2,); 2) by fitting the potential parameters in the overfilled system of equations for a lot of values P(f2~). It can be easily received from (13) the following equation for a pressure P(Ri)=

~i9 - 2

\ \

" \ '-k

,,~,

:7 "1

kX},

1

-0.03 0 F i g . 3. P r o f i l e s o f p o t e n t i a l

12 barriers for fcc Cs constructed

for differ-

ent crystallographic directions, l, 2, 3 - interatomic potentials; 4, 5 -potential barriers for (100) and (110> directions, respectively

228

and the second - the energy of crystal with two nearest neighbour vacancies and an atom in the saddle point between these vacancies. Using the obtained effective pair-potential for fcc Cs we carried out the calculations of these energies and obtained the value of E, = 0.070 eV. Thus the activation energy for the self-diffusion in fcc Cs is E,ct=0.287 eV. All the calculated migration energies of bcc alkaline metals with pseudopotentials [-5] are low (no data are available for Cs), i.e. about 17-22% of the energy of formation. An exceptional low value for bcc Na was evaluated [-35]. No direct self diffusion data are available for comparison, but we can expect a close value if we take the experimental activation energies for bcc alkaline metals. Finally we speculate additional possibilities of suggested approach: diffusion theory [36, 37] usually considers the jump probabilities co of ions from their equilibrium position in the lattice to the vacancy in the form co= v. exp (-- A ~/k T),

(13)

where A 9 is the potential of the barrier (and is equivalent to E, for the self diffusion indicated earlier) and v is the frequency of jump. As we showed above the value of A ~b is anisotropic and differs for different directions. This is the reason of the fact that co is direction dependent. The frequency, v, can be evaluated by applying the Taylor's expansion for the potential ~b(r) near the equilibrium position ro as (r) = ~ (to) q- 2 (r -- ro) 2 .

(14)

After fitting of this parabolic curve to the Lennard-Jones potential (in our case for fcc Cs) the elastic constant ~c can be evaluated and the value of v can be obtained as i

v=~m

K

K~.

(15)

Here m is the mass of the atom. As indicated in Fig. 3, the profile of the curve for (100} direction differs somewhat from that for (110) direction. This leads to another tr and hence to a different v for this direction. The obtained values are V=1.96 x 1011 s-1 and v ( 1 1 o ) = 3 . 0 2 x 1011 S - 1 By using these values of the frequences the j u m p p r o b abilities of the a t o m s in fcc Cs are: ~O=7.07 x 10 l~ s -1 a n d co=1.09 x 1011 s -1. We are grateful to J. Pelleg, R. Brener, J. Felsteiner, T. Maniv and P. Wyder for stimulating discussions. S.D.* would like to thank the Hochfeld-Magnetlabor of Max-Planck-Institut ffir Festk6rperforschung in Grenoble for the kind hospitality during his stay. This work was supported in the part by Israel Ministry of Science, Grant 3616-2-92 and by Germany Academy of Science DAAD.

References 1. Varotsos, PA., Alexopoulos, K.D.: Thermodynamics of point defects and their relation with bulk properties. Amsterdam: North-Holland 1986 2. Kraftmakher, Ya.A., Strelkov, PG.: In: Seeger, A., Schilling, W., Diehl, J. (eds.) Vacancies and interstitials in metals. Vol. 63. Proceedings of the International Conference. Jfilich, Germany, 1968. Amsterdam: North-Holland 1968 3. Fumi, F.G.: Philos. Mag. 46, 1007 (1955) 4. Ho, ES.: Phys. Rev. B3, 4035 (1971) 5. Torrens, I.M., Gerl, M.: Phys. Rev. 187, 912 (1969) 6. Takemura, K., Shimomura, O., Fujihisa, H.: Phys. Rev. Lett. 66, 2014 (1991) 7. Kennedy, G.C., Jayaraman, A., Newton, R.C.: Phys. Rev. 126, 1363 (1962) 8. Anderson, M.S., Gutman, B.J., Packard, J.R., Swenson, S.A.: J. Phys. Chem. Solids 30, 1587 (1969) 9. Johnson, R.A.: Papers printed at a seminar of the American Society for Metals. Metals Park, Ohio, 25. Oct. 14-15, 1972 10. Harrison, W.A.: Electronic structure and the properties of solids. The physics of the chemical bond. San Francisco: W.H. Freeman and Company 1982 11. Theory of the inhomogeneous electron gas. Lundquist, S., March, iN.H. (eds.) New York: Plenum Press 1983 12. Portnoy, K.I., Bogdanov, V.I., Fuks, D.L.: The calculation of interaction and stability of phases. Moscow: Metallurgia 1981 13. Moruzzi, V.L., Janak, J.F., Williams, A.R.: Calculated electronic properties of metals. New York: Pergamon Press 1978 14. Andersen, O.K.: Phys. Rev. BI2, 3060 (1975) 15. Skriver, H.L.: The LMTO method. Berlin, Heidelberg, New York: Springer 1984 16. Kohn, W.: Density functional theory: fundamental and applications. In: Bassani, F., Fumi, F., Tosi, M.R (eds.). Highlights of condensed-matter theory. Vol. 1. Amsterdam: North-Holland 1985 17. Skriver, H i . : Phys. Rev. B31, 1909 (1985) 18. Moruzzi, V.L., Janak, J.F., Schwarz, K.: Phys. Rev. B37, 790 (1988) 19. Sanchez, J.M., Stark, J.P, Moruzzi, V.L.: Phys. Rev. B44, 5411 (1991) 20. Pettifor, D.G.: Commun. Phys. 1, 141 (1976) 21. Andersen, O.K., Jepsen, O., G16tzel, D.: Canonical description of the band structures of metals. In: Bassani, F., Fumi, F., Tosi, M.P. (eds.) Highlights of condensed-matter theory. Vol. 59. Amsterdam: North-Holland 1985 22. Pettifor, D.G.: J. Phys. F: Metal. Phys. 8, 219 (1978) 23. Jepsen, O., Andersen, O.K.: Solid State Commun. 9, 1763 (1971) 24. Lehmann, G., Taut, M.:Phys. Status Solidi B54, 469 (1972) 25. Hedin, I., Lundquist, B.I.: J. Phys. C4, 2064 (1971) 26. Sigalas, M., Bacalis, N.C., Papaconstantopoulos, D.A., Mehl, M.J., Switendick, A.C.: Phys. Rev. B42, 11637 (1990) 27. Mazin, I.L, Maksimov, E.G., Rashkeev, S.N., Uspenskii, Yu.A.: JETP Lett. 44, 690 (1986) 28. Atomistic simulation of materials: beyond pair potentials. Vit e l V., Srolovitz, D.J. (eds.). New York: Plenum Press 1989 29. Hafner, J.: From Hamiltonian to phase diagrams. Berlin, Heidelberg, New York: Springer 1987 30. Johnson, R.A.: Phys. Rev. B37, 3924 (1988) 31. Carlsson, A.E.: Solid State Phys. 44, 1 (1990) 32. Moriarty, J.A.: Phys. Rev. B38, 3924 (1988) 33. Friedel, J.: Chapt. 8. Transition metals. Electronic structure of the d-Band. It's role in the crystalline and magnetic structures. In: Ziman, J.M. (ed.) The Physics of metals. Vol. 340. Cambridge: University Press 1969 34. Animalu, A.O.E.: Intermediate quantum theory of crystalline solids. Englewood Cliffs: Prentice-Hall 1979 35. Feder, R., Charbnau, H.P.: Phys. Rev. 149, 464 (1966) 36. Yamamoto, R., Doyama, M.: Phys. Rev. B8, 2586 (1973) 37. Johnson, R.A.: J. Phys. F: Metal. Phys. 3, 295 (1973)