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Real-space tight-binding approach to electronic structure and stability in substitutional alloys
Computational Materials Science 17 (2000) 217±223
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Real-space tight-binding approach to electronic structure and stability in substitutional alloys J.P. Julien a,c,*, P.E.A. Turchi b, D. Mayou c a
CUPF, Universite Francaise du Paci®que, BP 6570, Faa'a-Aeroport, Tahiti, French Polynesia, France b Lawrence Livermore National Laboratory, L-268, P.O. Box 808, Livermore, CA 94551, USA c LEPES-CNRS, 25 Avenue des Martyrs, BP 166, F-38042, Grenoble Cedex 9, France
Abstract A real-space approach based on the tight-binding approximation is proposed for studying electronic structure properties, stability and order in substitutional multi-component chemically random alloy based on periodic as well as topological disordered lattices. We show that the coherent potential approximation (CPA) equations including Shiba's o-diagonal disorder can be solved self-consistently in real-space with the same accuracy currently achieved in reciprocal space. An eective one-electron Green function is given by a continued fraction expansion, and the associated eective medium is used to determine the eective cluster interactions which enter the expression of the con®gurationl part of the total energy for describing order-disorder phenomena in alloys. Some applications will be presented. Ó 2000 Published by Elsevier Science B.V. All rights reserved.
1. Introduction The coherent potential approximation (CPA) [1,2] and its later Shiba's multiplicative o-diagonal disorder (ODD) [3] improvement is one of the most used mean ®eld theories of disordered substitutional alloys. Even in partially ordered systems, it is the basis for determining the interaction potentials that can lead to the prediction of phase stability [4]. This approximation leads to selfconsistent equations for the Green's function (GF), which are usually solved for each energy z. Here, we present a completely dierent way of solving the CPA equations within the Shiba's ODD. An auxilary GF associated to an eective Hamiltonian, which is energy-independent but acts
*
Corresponding author.
in greater space so that the recursion can be applied, is calculated and from this GF, it is possible to obtain all physical quantities, like componentprojected densities of states (DOS), the alloy average GF and also quantities used to compute the eective interactions that enter the con®gurational (or ordering) energy for a binary alloy, within the embedded cluster method (ECM) and the orbital peeling technique. 2. Real-space solution of CPA equation with Shiba's ODD The tight-binding (TB) Hamiltonian H for a given con®guration of the alloy is written in a form that exhibits both diagonal and ODD X X n jnihnj bnm jnihmj; 1 H n
n;m6n
0927-0256/00/$ - see front matter Ó 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 0 ) 0 0 0 2 7 - 6
218
J.P. Julien et al. / Computational Materials Science 17 (2000) 217±223
where the hopping integrals according to Shiba can be factorized as bnm an b0nm am and b0nm is a non-random o-diagonal quantity independent of species located at sites n and m, and common to all components of the alloy. an ; n are diagonal random quantities, depending on the occupation P species on site n : cn i pni cin (with c a or ). jn > is an atomic orbital centered on site n; pni is equal to one if site n is occupied by the atomic species i (having concentration ci ) and to zero otherwise, and in and ain are the on-site energy and a bandwidth parameter, respectively, associated with species i centered on site n and, for simplicity, independent on n in the following. The locator equation of motion for a site-diagonal element of the GF takes the form Gnn gn gnD^n Gnn ;
2
X c~i i
1 1 ~ G z: z ÿ i ÿ a2i D z z ÿ r z ÿ a2 D z 5
This self-consistent equation has to be solved to get the eective interactor, from which it is possible to obtain the physical i-component projected ÿ1 GF Gi z z ÿ i ÿ a2i D z i PA; B and the true alloy average GF, hG zi i ci Gi z. The expression for D associated with the average medium is given by an expression analogous to Eq. (3) but with each gm replaced by z ÿ rÿ1 , and am by a. The coherent potential r z is a complex energy-dependent on-site energy leading to the eective P Hamiltonian ofPthe average medium: Heff z n r zjnihnj n;m6n a2 b0nm jnihmj. Because of their analytical properties D z and r z can be expanded by CF
where gn z ÿ n ÿ1 is the locator associated with site n, and the interactor
r z A0 B21 =z ÿ A1 ÿ B22 =z ÿ A2 ÿ B23 =z ÿ
X X bnm gm bmn bnl gl blm gm bmn D^n
and
m6n
an
m6n l6m
X m6n
D z b21 =z ÿ a1 ÿ b22 =z ÿ a2 ÿ b23 =z ÿ
!
b0nm am gm am b0mn an a2n Dn :
3
By multiplying on right- and left-hand side the equation of motion by an , using the notation an An an A^n A g; G or D and further inversion, one gets ^nn ^ gnÿ1 ÿ Dn ÿ1 G
z ÿ n ÿ Dn a2n
6
ÿ1 :
4
The idea of the CPA is to build an eective medium which is determined self-consistently in such a way that the averaged occupancy of an impurity on the substituted site, embedded in the eective medium, reproduces the eective medium itself. Coming back to the original quantities and in agreement with this single-site assumption, the average medium surrounding site n 0 has to be build according to the following equation (using P c~i ci a2i =a2 with a2 i ci a2i )
7
Thus, the basic principle of the method is to determine the CF coecients fAp ; Bp g for r and faq ; bq g for D, rather than calculating the value of these functions at each complex energy z. The CF expansion of r z (or D z) is equivalent to the schematic representation of the associated TB Hamiltonian by a semi-linear chain whose onsite energies are the Aq (or ap ) and the nearestneighbor hopping integrals are the Bq (or bp ). This equivalence immediately allows us to replace the eective Hamiltonian Heff z for the disordered alloy by a Hamiltonian H~ represented by a semilinear chain Aq ; Bq attached to each orbital of the lattice on which the alloy is based. These chains exactly represent the eect of the self-energy r z as presented in Fig. 1 in the case of an alloy based on an in®nite linear chain lattice. The advantage of this formulation resides in the fact that methods developed for solving TB Hamiltonians, such as the recursion technique [9], can now be directly used with this energy-independent eective Hamiltonian H~ . If one calculates D z by the recursion
J.P. Julien et al. / Computational Materials Science 17 (2000) 217±223
219
~ Fig. 2. Schematic representation of Gi z and G z associated ~ respectively. with Hi and H,
Fig. 1. Equivalent representations of the eective Hamiltonian describing chemical disorder, within the CPA, here for an alloy based on an in®nite linear chain (thick solid lines).
method with an initial vector j0g which is taken to be the atomic orbital j0i centered on site 0, then it is well known that the successive vectors jpg of the recursion extend away from site 0. In particular, jpg will have non-zero components on sites q of the chain attached to the sites of the real crystal up to a ®nite value of q. Thus, in order to calculate jpg and also ap ; bp , we do not need to know all the Aq ; Bq . By a precise analysis of the recusion scheme, one can obtain an expression for the ap and bp in terms of the Aq ; Bq of the following type: ap ap Apÿ1 ; Bpÿ1 ; . . . ; B1 ; A0 ; bp bp Bpÿ1 ; Apÿ2 ; . . . ; B1 ; A0 :
Fig. 3. Upper panel: Schematic representation of the Hamiltonian H~ expressed in the new basis fjpgs; jpgAS g (b), and compared with its representation in the original basis fjpgA ; jpgB g (a) for a binary alloy. Lower panel: Equivalent representations of the self-energy r z.
8
In practice, the Aq and Bq are obtained from the ap ; bp once again by recursion in the eective medium. Using the representation of r and D in terms of semi-in®nite linear chains, we are interested in the evaluation of the diagonal elements of the Green ~ associated operators si f0jGi j0gsi and f0jG zj0g with Hi and H~ , as in Fig. 2. The Hamiltonian H~ is given by the sum of the two Hamiltonians HA and HB which are totally uncoupled. Using the properties of the sum-space [12], H~ can be conveniently expressed in a new basis, dierent from the original basis fjpgSA ; jpgSB ; p 0; 1; 2; . . .g. For each p, we de®ne two orthonormal states
q c~A d0p cA 1 ÿ d0p jpgSA q c~B d0p cB 1 ÿ d0p jpgSB ; q c~B d0p cB 1 ÿ d0p jpgSA q ÿ c~A d0p cA 1 ÿ d0p jpgSB ;
jpgS
jpgAS
9
which generates the sum-space S associated with H~ . After analytically computing the matrix elements of H~ in the new basis and by simple identi®cation of the two representations (in the old and new basis) of the Hamiltonian H~ given in Fig. 3 (upper picture), one sees that r can be represented
220
J.P. Julien et al. / Computational Materials Science 17 (2000) 217±223
by the object on the lower part of Fig. 3 (righthand side representation). An additional recursion on that object, starting from the state j0gs, will provide the CF coecients of r leading to its semiin®nite chain (left-hand side representation). Using a similar procedure, the Aq and Bq can be easily obtained from ap ; bp for higher order multicomponent alloys and/or multi-orbital and multisite cases [8]. 3. Embedded cluster method and eective interaction When dealing with the atomistic description of chemical order in alloys, one has to combine electronic structure calculations with a statistical treatment, both requiring a high degree of accuracy. One way to meet this challenge is to cast the quantum mechanical description of the energetics of an alloy in the form of an Ising model which is most appropriate for a subsequent statistical mechanics treatment of order-disorder phenomena in alloys as functions of temperature and concentration [4]. The mapping of the energetics resulting from the solution of the Hamiltonian (1) (in its multi-orbital generalization), onto an Ising form has been originally achieved within the so-called generalized perturbation method (GPM) [5], which is a perturbation treatment applied to the CPA reference medium, and was generalized in the ECM [6] to account for the correlations inside ®nite clusters embedded in the CPA medium. In the ECM, the con®gurational (or ordering) energy for a binary alloy is expressed by an expansion in terms of eective pair and multi-site (concentration-dependent) interactions X X 2 Vn 1 dcn Vnm dcn dcm ; DEord fpn g n
n;m6n
10 where dcn refers to the ¯uctuation of concentration on site n; dcn pn ÿ c, where c is the concentration in B-species, and pn is an occupation number associated with site n, equal to 1 or 0 depending on whether or not site n is occupied by a B-species. The eective single interaction Vn 1 , which plays
only a role if there exist inequivalent sites, associated with the interchange of a B with an A species at site n is given by Vn 1
Z
EF
ÿ1
ÿ EF nBn ÿ nA n d;
11
where EF is the Fermi energy of the CPA medium, and, with i referring to the species and ÿ1 The nin ÿI=p limg!0 z ÿ in ÿ ai2 n Dn z . 2 eective pair interaction (EPI), Vnm , given by the dierence in single site interaction at site n when site m 6 n is occupied by an A or a B species, can be computed with standard recursions within the so-called orbital peeling method (OPM) [7] according to i; j A; B pÿ1 p n0 X X X 2 X 1ÿd 2 ÿ I ÿ1 ij zk;ij ÿ pak;ij Vnm a p ij a1 a1 k1 !
nPk;ij ÿ nZk;ij EF : 12 and pak;ij (with their respective numbers, nk;ij zk;ij Z a and nk;ij P ) are the zeros and the poses, up to EF , of k;ij Gk;ij nm , the resolvent of a Hamiltonian Hnm which de®nes a system where the two species i and j located at sites n and m, respectively, are embedded in the CPA medium, and for which the orbitals from 1 to k ÿ 1 are omitted at site n. In practice, since the self-energy rn z is known by its continued fraction expansion, it is a simple matter to determine the continued fraction expansion of Gk;ij nm with a recursion scheme by (i) branching on each site of the real lattice, except at sites n and m, a semi-linear chain representing rn z or n0 chains representing rnl z, in the multiorbital case; (ii) locating the species i and j at sites n and m, respectively; and (iii) setting the projection of the recursion vector on site n, at each step of the recursion, to zero for the orbitals 1 to k ÿ 1, in accordance with the OPM. Once these interactions are known, the ground-state properties of the alloy at zero-temperature, i.e., the possible ordered state which is stable at each concentration, can be predicted. Finally, combined with a
J.P. Julien et al. / Computational Materials Science 17 (2000) 217±223
statistical model such as the cluster variation method or with Monte Carlo (MC) simulations, the con®gurational part of the free energy can be computed, and hence, the phase diagram of an alloy which summarizes the phase stability properties as functions of temperature and concentration [4].
4. Applications To compare and show the in¯uence of ODD in a simple case, we have calculated the DOS for the d-band of CuPd equi-concentration alloy based on a fcc lattice, using the Slater±Koster parametrization proposed by Harrison [10] and used in Ref. [13] for comparison: ®rst within the diagonal CPA neglecting the ODD, by simply averaging hopping integrals according to the so-called virtual crystal bandwidth approximation and applying our method by setting equal a's and second, within the full approach to Shiba's ODD (see DOS on Fig. 4). The DOS calculated with ODD has a slightly larger bandwidth and the high sharp peak, close to the top of the band in the diagonal CPA calculation has disappeared in this more elaborated computation, showing the in¯uence of the ODD. Our real space results agrees very well with the results of [13], obtained by a k-space solution for each energy. We now present some results for amorphous Zr±Ni alloys, for which, after determining the
221
CPA medium, the eective interactions have been calculated with the method described above. A full spd multi-orbital basis was used and the TB parameters were determined within the TB±LMTO approach [11] for the alloy composition Zr2 Ni. For a reliable amorphous structure [14] of 96 atoms, we determined the CPA±DOS of the chemically random alloy and compared it with the same composition chemically random alloy based on a bcc lattice (Fig. 5). As expected, the DOS of the amorphous phase is less structured than its bcc lattice counterpart, but in both cases, EF is located close to minimum of DOS where partial Ni-3d and Zr-4d bands cross. On these CPA results, the effective interactions were determined within the
Fig. 4. Total average DOS of the Cu50 Pd50 alloy calculated in the CPA recursion approach with no ODD (dashed line) and with Shiba's ODD (solid line).
Fig. 5. DOSs of bcc-based (a) and amorphous (b) Zr2 Ni (solid line: total; dashed line: Ni; dotted line: Zr).
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J.P. Julien et al. / Computational Materials Science 17 (2000) 217±223
Fig. 6. On-site interactions (a) and pair interactions (b) for a 96 atom-cell of amorphous Zr2 Ni as function of site number n and distance, respectively.
ECM with the OPM technique as described above. Fig. 6 shows results for the species-decomposed on-site terms Vn 1;Zr and Vn 1;Ni as a function of site number n and the EPIs which, here, are determined for all neighbors of each of the 96 sites up to a cut o distance of 4 A. Note that the on-site terms show a signi®cant drop beyond n 62, which is indicative of a dierent environment for Zr and Ni, since originally, when the structural model was constructed, the sites were sorted out, with the ®rst 64 being occupied by Zr and the last 22 by Ni. These terms will play a major role in MC simulations. Although the EPIs overall decrease with distance, large scatter in magnitude should be noticed, at a particular distance, their sign can even change. This must be attributed to the renormalized character of the interactions within the ECM, and the nature of the environment (topology) in which a given pair of atoms is embedded. However, as in the crystalline case for which the ®rst and second EPIs are equal to 14:1 and ÿ6:3 mRy/atom at distances of 2.88 and 3.33 A, respectively, the EPIs are indicative of a clear tendency towards order, that is, for this particular model of amorphous alloy, there is a tendency to have Ni±Zr pairs favoured over Ni±Ni and Zr± Zr pairs. With this energetics, MC simulations can be performed so that the atoms are positioned in agreement with the con®gurational part of the total energy . With this new atomic con®guration, the amorphous structure can be relaxed with TB molecular dynamics before a new cycle of
calculations, and so on until self-consistency is achieved.
5. Conclusion We have shown that the solution of CPA equations including Shiba's ODD, and calculation of the properties and energetics of alloys can be eciently performed in real-space using the recursion technique. The associated CF coecients are valid for the whole energy range, and avoid the search of solution for each energy and the CF expansion is truncated (using standard terminator techniques) at a desired accuracy, at a usually small number of coecients. This new method allows to study the stability and chemical order of alloys based on periodic or amorphous lattices, and opens applications to a wide class of systems with inequivalent sites, where traditional k-space CPA solutions are at least inecient because of numerical instabilities and computing time or even impossible if there are no symmetries on the underlying lattice.
References [1] P. Soven, Phys. Rev. 156 (1967) 809. [2] B. Velicky, S. Kirkpatrick, H. Ehrenreich, Phys. Rev. 175 (1968) 747. [3] H. Shiba, Prog. Theor. Phys. 46 (1971) 77.
J.P. Julien et al. / Computational Materials Science 17 (2000) 217±223 [4] P.E.A. Turchi, in: J.H. Westbrook, R.L. Fleischer (Eds.), Intermetallic Compounds: Principles and Practice, vol. 1, Wiley, New York, 1995, pp. 21±54 (Chapter 2). [5] F. Ducastelle, F. Gautier, J. Phys. F: Metals Phys. 6 (1976) 2039. [6] A. Gonis, X.-G. Zhang, A.J. Freeman, P.E.A. Turchi, G.M. Stocks, D.M. Nicholson, Phys. Rev. B 36 (1987) 4630. [7] N.R. Burke, Surf. Sci. 58 (1976) 349. [8] P.E.A. Turchi, D. Mayou, J.P. Julien, Phys. Rev. B 56 (1997) 1726.
223
[9] R. Haydock, in: H. Ehrenreich, F. Seitz, D. Turnbull (Eds.), Solid State Physics, vol. 35, Academic Press, New York, 1980, p. 215. [10] W.A. Harrison, Electronic Structure and the Properties of Solids, Freeman, San Francisco, 1980. [11] O.K. Andersen, O. Jepsen, Phys. Rev. Lett. 53 (1984) 2571. [12] D. Mayou, A. Pasturel, D. Nguyen Manh, J. Phys. C 19 (1986) 719. [13] J. Kudrnovsky, J. Masek, Phys. Rev. B 31 (1985) 6424. [14] W.Y. Ching, L.W. Song, S.S. Jaswal, Phys. Rev. B 30 (1984) 544.
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Real-space tight-binding approach to electronic structure and stability in substitutional alloys J.P. Julien a,c,*, P.E.A. Turchi b, D. Mayou c a
CUPF, Universite Francaise du Paci®que, BP 6570, Faa'a-Aeroport, Tahiti, French Polynesia, France b Lawrence Livermore National Laboratory, L-268, P.O. Box 808, Livermore, CA 94551, USA c LEPES-CNRS, 25 Avenue des Martyrs, BP 166, F-38042, Grenoble Cedex 9, France
Abstract A real-space approach based on the tight-binding approximation is proposed for studying electronic structure properties, stability and order in substitutional multi-component chemically random alloy based on periodic as well as topological disordered lattices. We show that the coherent potential approximation (CPA) equations including Shiba's o-diagonal disorder can be solved self-consistently in real-space with the same accuracy currently achieved in reciprocal space. An eective one-electron Green function is given by a continued fraction expansion, and the associated eective medium is used to determine the eective cluster interactions which enter the expression of the con®gurationl part of the total energy for describing order-disorder phenomena in alloys. Some applications will be presented. Ó 2000 Published by Elsevier Science B.V. All rights reserved.
1. Introduction The coherent potential approximation (CPA) [1,2] and its later Shiba's multiplicative o-diagonal disorder (ODD) [3] improvement is one of the most used mean ®eld theories of disordered substitutional alloys. Even in partially ordered systems, it is the basis for determining the interaction potentials that can lead to the prediction of phase stability [4]. This approximation leads to selfconsistent equations for the Green's function (GF), which are usually solved for each energy z. Here, we present a completely dierent way of solving the CPA equations within the Shiba's ODD. An auxilary GF associated to an eective Hamiltonian, which is energy-independent but acts
*
Corresponding author.
in greater space so that the recursion can be applied, is calculated and from this GF, it is possible to obtain all physical quantities, like componentprojected densities of states (DOS), the alloy average GF and also quantities used to compute the eective interactions that enter the con®gurational (or ordering) energy for a binary alloy, within the embedded cluster method (ECM) and the orbital peeling technique. 2. Real-space solution of CPA equation with Shiba's ODD The tight-binding (TB) Hamiltonian H for a given con®guration of the alloy is written in a form that exhibits both diagonal and ODD X X n jnihnj bnm jnihmj; 1 H n
n;m6n
0927-0256/00/$ - see front matter Ó 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 0 ) 0 0 0 2 7 - 6
218
J.P. Julien et al. / Computational Materials Science 17 (2000) 217±223
where the hopping integrals according to Shiba can be factorized as bnm an b0nm am and b0nm is a non-random o-diagonal quantity independent of species located at sites n and m, and common to all components of the alloy. an ; n are diagonal random quantities, depending on the occupation P species on site n : cn i pni cin (with c a or ). jn > is an atomic orbital centered on site n; pni is equal to one if site n is occupied by the atomic species i (having concentration ci ) and to zero otherwise, and in and ain are the on-site energy and a bandwidth parameter, respectively, associated with species i centered on site n and, for simplicity, independent on n in the following. The locator equation of motion for a site-diagonal element of the GF takes the form Gnn gn gnD^n Gnn ;
2
X c~i i
1 1 ~ G z: z ÿ i ÿ a2i D z z ÿ r z ÿ a2 D z 5
This self-consistent equation has to be solved to get the eective interactor, from which it is possible to obtain the physical i-component projected ÿ1 GF Gi z z ÿ i ÿ a2i D z i PA; B and the true alloy average GF, hG zi i ci Gi z. The expression for D associated with the average medium is given by an expression analogous to Eq. (3) but with each gm replaced by z ÿ rÿ1 , and am by a. The coherent potential r z is a complex energy-dependent on-site energy leading to the eective P Hamiltonian ofPthe average medium: Heff z n r zjnihnj n;m6n a2 b0nm jnihmj. Because of their analytical properties D z and r z can be expanded by CF
where gn z ÿ n ÿ1 is the locator associated with site n, and the interactor
r z A0 B21 =z ÿ A1 ÿ B22 =z ÿ A2 ÿ B23 =z ÿ
X X bnm gm bmn bnl gl blm gm bmn D^n
and
m6n
an
m6n l6m
X m6n
D z b21 =z ÿ a1 ÿ b22 =z ÿ a2 ÿ b23 =z ÿ
!
b0nm am gm am b0mn an a2n Dn :
3
By multiplying on right- and left-hand side the equation of motion by an , using the notation an An an A^n A g; G or D and further inversion, one gets ^nn ^ gnÿ1 ÿ Dn ÿ1 G
z ÿ n ÿ Dn a2n
6
ÿ1 :
4
The idea of the CPA is to build an eective medium which is determined self-consistently in such a way that the averaged occupancy of an impurity on the substituted site, embedded in the eective medium, reproduces the eective medium itself. Coming back to the original quantities and in agreement with this single-site assumption, the average medium surrounding site n 0 has to be build according to the following equation (using P c~i ci a2i =a2 with a2 i ci a2i )
7
Thus, the basic principle of the method is to determine the CF coecients fAp ; Bp g for r and faq ; bq g for D, rather than calculating the value of these functions at each complex energy z. The CF expansion of r z (or D z) is equivalent to the schematic representation of the associated TB Hamiltonian by a semi-linear chain whose onsite energies are the Aq (or ap ) and the nearestneighbor hopping integrals are the Bq (or bp ). This equivalence immediately allows us to replace the eective Hamiltonian Heff z for the disordered alloy by a Hamiltonian H~ represented by a semilinear chain Aq ; Bq attached to each orbital of the lattice on which the alloy is based. These chains exactly represent the eect of the self-energy r z as presented in Fig. 1 in the case of an alloy based on an in®nite linear chain lattice. The advantage of this formulation resides in the fact that methods developed for solving TB Hamiltonians, such as the recursion technique [9], can now be directly used with this energy-independent eective Hamiltonian H~ . If one calculates D z by the recursion
J.P. Julien et al. / Computational Materials Science 17 (2000) 217±223
219
~ Fig. 2. Schematic representation of Gi z and G z associated ~ respectively. with Hi and H,
Fig. 1. Equivalent representations of the eective Hamiltonian describing chemical disorder, within the CPA, here for an alloy based on an in®nite linear chain (thick solid lines).
method with an initial vector j0g which is taken to be the atomic orbital j0i centered on site 0, then it is well known that the successive vectors jpg of the recursion extend away from site 0. In particular, jpg will have non-zero components on sites q of the chain attached to the sites of the real crystal up to a ®nite value of q. Thus, in order to calculate jpg and also ap ; bp , we do not need to know all the Aq ; Bq . By a precise analysis of the recusion scheme, one can obtain an expression for the ap and bp in terms of the Aq ; Bq of the following type: ap ap Apÿ1 ; Bpÿ1 ; . . . ; B1 ; A0 ; bp bp Bpÿ1 ; Apÿ2 ; . . . ; B1 ; A0 :
Fig. 3. Upper panel: Schematic representation of the Hamiltonian H~ expressed in the new basis fjpgs; jpgAS g (b), and compared with its representation in the original basis fjpgA ; jpgB g (a) for a binary alloy. Lower panel: Equivalent representations of the self-energy r z.
8
In practice, the Aq and Bq are obtained from the ap ; bp once again by recursion in the eective medium. Using the representation of r and D in terms of semi-in®nite linear chains, we are interested in the evaluation of the diagonal elements of the Green ~ associated operators si f0jGi j0gsi and f0jG zj0g with Hi and H~ , as in Fig. 2. The Hamiltonian H~ is given by the sum of the two Hamiltonians HA and HB which are totally uncoupled. Using the properties of the sum-space [12], H~ can be conveniently expressed in a new basis, dierent from the original basis fjpgSA ; jpgSB ; p 0; 1; 2; . . .g. For each p, we de®ne two orthonormal states
q c~A d0p cA 1 ÿ d0p jpgSA q c~B d0p cB 1 ÿ d0p jpgSB ; q c~B d0p cB 1 ÿ d0p jpgSA q ÿ c~A d0p cA 1 ÿ d0p jpgSB ;
jpgS
jpgAS
9
which generates the sum-space S associated with H~ . After analytically computing the matrix elements of H~ in the new basis and by simple identi®cation of the two representations (in the old and new basis) of the Hamiltonian H~ given in Fig. 3 (upper picture), one sees that r can be represented
220
J.P. Julien et al. / Computational Materials Science 17 (2000) 217±223
by the object on the lower part of Fig. 3 (righthand side representation). An additional recursion on that object, starting from the state j0gs, will provide the CF coecients of r leading to its semiin®nite chain (left-hand side representation). Using a similar procedure, the Aq and Bq can be easily obtained from ap ; bp for higher order multicomponent alloys and/or multi-orbital and multisite cases [8]. 3. Embedded cluster method and eective interaction When dealing with the atomistic description of chemical order in alloys, one has to combine electronic structure calculations with a statistical treatment, both requiring a high degree of accuracy. One way to meet this challenge is to cast the quantum mechanical description of the energetics of an alloy in the form of an Ising model which is most appropriate for a subsequent statistical mechanics treatment of order-disorder phenomena in alloys as functions of temperature and concentration [4]. The mapping of the energetics resulting from the solution of the Hamiltonian (1) (in its multi-orbital generalization), onto an Ising form has been originally achieved within the so-called generalized perturbation method (GPM) [5], which is a perturbation treatment applied to the CPA reference medium, and was generalized in the ECM [6] to account for the correlations inside ®nite clusters embedded in the CPA medium. In the ECM, the con®gurational (or ordering) energy for a binary alloy is expressed by an expansion in terms of eective pair and multi-site (concentration-dependent) interactions X X 2 Vn 1 dcn Vnm dcn dcm ; DEord fpn g n
n;m6n
10 where dcn refers to the ¯uctuation of concentration on site n; dcn pn ÿ c, where c is the concentration in B-species, and pn is an occupation number associated with site n, equal to 1 or 0 depending on whether or not site n is occupied by a B-species. The eective single interaction Vn 1 , which plays
only a role if there exist inequivalent sites, associated with the interchange of a B with an A species at site n is given by Vn 1
Z
EF
ÿ1
ÿ EF nBn ÿ nA n d;
11
where EF is the Fermi energy of the CPA medium, and, with i referring to the species and ÿ1 The nin ÿI=p limg!0 z ÿ in ÿ ai2 n Dn z . 2 eective pair interaction (EPI), Vnm , given by the dierence in single site interaction at site n when site m 6 n is occupied by an A or a B species, can be computed with standard recursions within the so-called orbital peeling method (OPM) [7] according to i; j A; B pÿ1 p n0 X X X 2 X 1ÿd 2 ÿ I ÿ1 ij zk;ij ÿ pak;ij Vnm a p ij a1 a1 k1 !
nPk;ij ÿ nZk;ij EF : 12 and pak;ij (with their respective numbers, nk;ij zk;ij Z a and nk;ij P ) are the zeros and the poses, up to EF , of k;ij Gk;ij nm , the resolvent of a Hamiltonian Hnm which de®nes a system where the two species i and j located at sites n and m, respectively, are embedded in the CPA medium, and for which the orbitals from 1 to k ÿ 1 are omitted at site n. In practice, since the self-energy rn z is known by its continued fraction expansion, it is a simple matter to determine the continued fraction expansion of Gk;ij nm with a recursion scheme by (i) branching on each site of the real lattice, except at sites n and m, a semi-linear chain representing rn z or n0 chains representing rnl z, in the multiorbital case; (ii) locating the species i and j at sites n and m, respectively; and (iii) setting the projection of the recursion vector on site n, at each step of the recursion, to zero for the orbitals 1 to k ÿ 1, in accordance with the OPM. Once these interactions are known, the ground-state properties of the alloy at zero-temperature, i.e., the possible ordered state which is stable at each concentration, can be predicted. Finally, combined with a
J.P. Julien et al. / Computational Materials Science 17 (2000) 217±223
statistical model such as the cluster variation method or with Monte Carlo (MC) simulations, the con®gurational part of the free energy can be computed, and hence, the phase diagram of an alloy which summarizes the phase stability properties as functions of temperature and concentration [4].
4. Applications To compare and show the in¯uence of ODD in a simple case, we have calculated the DOS for the d-band of CuPd equi-concentration alloy based on a fcc lattice, using the Slater±Koster parametrization proposed by Harrison [10] and used in Ref. [13] for comparison: ®rst within the diagonal CPA neglecting the ODD, by simply averaging hopping integrals according to the so-called virtual crystal bandwidth approximation and applying our method by setting equal a's and second, within the full approach to Shiba's ODD (see DOS on Fig. 4). The DOS calculated with ODD has a slightly larger bandwidth and the high sharp peak, close to the top of the band in the diagonal CPA calculation has disappeared in this more elaborated computation, showing the in¯uence of the ODD. Our real space results agrees very well with the results of [13], obtained by a k-space solution for each energy. We now present some results for amorphous Zr±Ni alloys, for which, after determining the
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CPA medium, the eective interactions have been calculated with the method described above. A full spd multi-orbital basis was used and the TB parameters were determined within the TB±LMTO approach [11] for the alloy composition Zr2 Ni. For a reliable amorphous structure [14] of 96 atoms, we determined the CPA±DOS of the chemically random alloy and compared it with the same composition chemically random alloy based on a bcc lattice (Fig. 5). As expected, the DOS of the amorphous phase is less structured than its bcc lattice counterpart, but in both cases, EF is located close to minimum of DOS where partial Ni-3d and Zr-4d bands cross. On these CPA results, the effective interactions were determined within the
Fig. 4. Total average DOS of the Cu50 Pd50 alloy calculated in the CPA recursion approach with no ODD (dashed line) and with Shiba's ODD (solid line).
Fig. 5. DOSs of bcc-based (a) and amorphous (b) Zr2 Ni (solid line: total; dashed line: Ni; dotted line: Zr).
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J.P. Julien et al. / Computational Materials Science 17 (2000) 217±223
Fig. 6. On-site interactions (a) and pair interactions (b) for a 96 atom-cell of amorphous Zr2 Ni as function of site number n and distance, respectively.
ECM with the OPM technique as described above. Fig. 6 shows results for the species-decomposed on-site terms Vn 1;Zr and Vn 1;Ni as a function of site number n and the EPIs which, here, are determined for all neighbors of each of the 96 sites up to a cut o distance of 4 A. Note that the on-site terms show a signi®cant drop beyond n 62, which is indicative of a dierent environment for Zr and Ni, since originally, when the structural model was constructed, the sites were sorted out, with the ®rst 64 being occupied by Zr and the last 22 by Ni. These terms will play a major role in MC simulations. Although the EPIs overall decrease with distance, large scatter in magnitude should be noticed, at a particular distance, their sign can even change. This must be attributed to the renormalized character of the interactions within the ECM, and the nature of the environment (topology) in which a given pair of atoms is embedded. However, as in the crystalline case for which the ®rst and second EPIs are equal to 14:1 and ÿ6:3 mRy/atom at distances of 2.88 and 3.33 A, respectively, the EPIs are indicative of a clear tendency towards order, that is, for this particular model of amorphous alloy, there is a tendency to have Ni±Zr pairs favoured over Ni±Ni and Zr± Zr pairs. With this energetics, MC simulations can be performed so that the atoms are positioned in agreement with the con®gurational part of the total energy . With this new atomic con®guration, the amorphous structure can be relaxed with TB molecular dynamics before a new cycle of
calculations, and so on until self-consistency is achieved.
5. Conclusion We have shown that the solution of CPA equations including Shiba's ODD, and calculation of the properties and energetics of alloys can be eciently performed in real-space using the recursion technique. The associated CF coecients are valid for the whole energy range, and avoid the search of solution for each energy and the CF expansion is truncated (using standard terminator techniques) at a desired accuracy, at a usually small number of coecients. This new method allows to study the stability and chemical order of alloys based on periodic or amorphous lattices, and opens applications to a wide class of systems with inequivalent sites, where traditional k-space CPA solutions are at least inecient because of numerical instabilities and computing time or even impossible if there are no symmetries on the underlying lattice.
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J.P. Julien et al. / Computational Materials Science 17 (2000) 217±223 [4] P.E.A. Turchi, in: J.H. Westbrook, R.L. Fleischer (Eds.), Intermetallic Compounds: Principles and Practice, vol. 1, Wiley, New York, 1995, pp. 21±54 (Chapter 2). [5] F. Ducastelle, F. Gautier, J. Phys. F: Metals Phys. 6 (1976) 2039. [6] A. Gonis, X.-G. Zhang, A.J. Freeman, P.E.A. Turchi, G.M. Stocks, D.M. Nicholson, Phys. Rev. B 36 (1987) 4630. [7] N.R. Burke, Surf. Sci. 58 (1976) 349. [8] P.E.A. Turchi, D. Mayou, J.P. Julien, Phys. Rev. B 56 (1997) 1726.
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[9] R. Haydock, in: H. Ehrenreich, F. Seitz, D. Turnbull (Eds.), Solid State Physics, vol. 35, Academic Press, New York, 1980, p. 215. [10] W.A. Harrison, Electronic Structure and the Properties of Solids, Freeman, San Francisco, 1980. [11] O.K. Andersen, O. Jepsen, Phys. Rev. Lett. 53 (1984) 2571. [12] D. Mayou, A. Pasturel, D. Nguyen Manh, J. Phys. C 19 (1986) 719. [13] J. Kudrnovsky, J. Masek, Phys. Rev. B 31 (1985) 6424. [14] W.Y. Ching, L.W. Song, S.S. Jaswal, Phys. Rev. B 30 (1984) 544.