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Electronic structure and stability of binary transition metal compounds
Solid State Communications, Vol. 38, pp. 1219-1222. Pergamon Press Ltd. 1981. Printed in Great Britain.
0038-1098/81/241219-04502.00/0
ELECTRONIC STRUCTURE AND STABILITY OF BINARY TRANSITION METAL COMPOUNDS A. Bieber t'2 and F. Gautier l t LMSES, (LA 306), 4 rue B. Pascal, 67000 Strasbourg, France 2 CRM, CNRS, 6 rue Boussingault, 67083 Strasbourg Cedex, France
(Received 27 January 1981 by J. Kanamori) It is shown that the ordered structures occurring in binary substitutional transition metal alloys on simple lattices (f.c.c., b.c.c.) can systematically be obtained from the electronic structure of the disordered alloys. Generalized structural maps are introduced; their physical meaning and their relation to the phenomenological structural maps used previously by other authors are discussed.
1. INTRODUCTION
of the disordered alloys; (ii) to define generalized structural maps, to discuss their physical meaning in relation SEVERAL ATTEMPTS to classify the ordered crystal to the previous ones and to present some preliminary structures of binary alloys AeBl_ e, according to the results for the f.c.c. OS. values of various semi-empirical parameters, have been It is now well established both qualitatively [6] and reported recently [ 1 - 3 ] . Two coordinates at least are quantitatively [7, 8] that most of the cohesive energy necessary to construct such phenomenological structural of TM and their alloys comes from the band structure maps (PSM) since we have to take account of both the energy of the " d " electrons. We will therefore only conaverage crystalline potential and the fluctuations of sider this contribution to the total energy. For simplicity potential from site to site. The simplest parameters are we neglect the "'sd'" hybridization and we assume in this then: (i) the average number of electrons per atom 37 = scheme - in agreement with ASA computations [8, 9] cNA + (1 -- c)N 8 which determines the filling of the that each TM has about 0.8 "s" electron and Ndi " d " bands and (ii) the valence difference £~N = N a - - N a electrons per atom (Ni = Nei + 0.8); the energies El are which characterizes the magnitude of the disorder. We then determined by: (/) the center of gravity ei(~i), the have determined the PSM (37, AN) for transition metal width Wi(~21), and the filling of the " d " bands (i.e. Ndi); (TM) alloys and for particular concentrations (c = 0.25, (ii) the lattice structure. In a similar way, the energy 0.33, 0.5); the separation of the different types of Edis (c) and the energy spectrum of a completely disordered structures (OS) into different regions of the ordered state are determined by (i) the averaged band plane (37, AN) is at least as significant [4, 5] as the separenergy g(~) and the bandwidth W(~2)[g(~) = ation obtained by using other coordinates [ 1 - 3 ] . For "2icie~(~2), W(~2) = Zic i Wi(~Z)], (ii) the diagonal (DD) example the PSM for e = 0.25 (see Fig. 1) shows that: and off-diagonal (ODD) disorder defined by 5a(~) = (1) Cu3Au type OS occur for 7.5 ~ 37< 8.5 when the 2(ea --en)/W and 5nd(~2) = (Wa -- Wn)/ff/respectively; alloys are non magnetic and for 8.5 ---<37~< 10 when the is the average atomic volume of the alloy. These alloys are magnetically ordered; (2) A13Ti occurs only parameters can be determined from band structure calfor37 = 8.75 and AN = -- 5. The coordinates 37, AN are culations: we use here the ASA results [8, 9], ~ being thus relevant but they are not sufficient to determine given either by Vegard's law or by experiment when the most stable OS since the separation between the available. Reliable mixing energies A.Eais(c ) = Eals(c) -various OS is not perfect: for example the (six) possible cEa --(1 - - c ) E n for disordered alloys can then be comsystems corresponding to the same point (37, AN) and puted using a simple non self-consistent CPA [I0, 11]; to TM of different series do not present the same type moreover, the variation of AEdis with 37 and AN can be of OS. qualitatively interpreted in terms of " d " bands broadening and deformation induced by the DD and ODD 2. METHOD respectively [12]. The fluctuations of the energy levels We have undertaken a systematic study of the OS according to their local environment [ 12] and the selfbased on f.c.c, and b.c.c, lattices for TM alloys [5]. Our consistent determination of the charge transfers do not aim in this paper is: (i) to show that the possible OS can change qualitatively the results [ 11, 12] ; we have therebe derived systematically from the electronic structure fore neglected these effects at the present stage. 1219
1220
STABILITY OF BINARY TRANSITION
Fig. 1. Phenomenological structural maps (1, Alv) for transition metal alloys for c = 0.25. We have retained all the unambiguous ordered structures determined from equilibrium phase diagrams [26-291. CUJAU (M) and Cu3Au (NM) corresponds to magnetically ordered and non magnetic ordered Cu3Au compounds respectively. The figures indicate the number of systems in which each ordered structure has been found.
The systematic study of the OS on a given lattice and of their variation with temperature and composition [ 13, 141 is a priori much more difficult. It is always possible - at least in principle - (i) to determine the band structure for each “ reasonable” OS, (ii) to compute their energies E,, (c) = Otis (c) + 3E,i, (c) and (iii) to determine the ground state and its long range order by comparison of Edis and of all the computed Eord. However, this numerical method is heavy and untractable for complicated structures. Moreover, it does not allow to describe the results in terms of simple physical parameters. For this reason, we suggested a systematic method [15-171 which reduces this problem to the determination of the ground state of a generalized 3D Ising model H = En V,p, where p,, is the number of nth neighbouring pairs and V, the corresponding pair interaction defined from the average medium of the completely disordered state as determined by the CPA (KKR [ 18, 191 or TB CPA [20]); for example, V, is given approximately in the tight-binding (TB) CPA and for &nd =0 1151 by:
x L,,
m’
W’m
[z -C(z)]}’
dE.
In this relation, At = f A - t* where t’ is the matrix associated to an atom i in the average medium; G,““’ (z) is the interatomic Green function of the metal A taken between two nth neighbouring sites; m labels the basis orbitals (m= 1, . . . , 5) and C (z) is the CPA self energy. A more complicated but similar expression can be
METAL COMPOUNDS
Vol. 38, Nd. 12
derived when 6 nd is included. The previous Ising model is the first term of the generalized perturbation method which yields the ordering energy AE,, as a cluster expansion [ 151. This expansion is valid for all values of 6 = (6,, And): this is an essential feature since most of the observed OS occur for large F values; direct comparison with band structure computations in some simple cases has shown that this expansion is rapidly convergent [5]. Therefore we consider here only the pair interactions V, and neglect the higher order terms. The properties of the pair interactions are discussed elsewhere [5] ; we just recall here their salient features: (1) they are defined as functions of the distance between neighbours (i.e. n), the disorder 6, the concentration c and the band filling~v,: V, = i-v,, (6,, hnd, c,Nd), the bandwidth p being only a scaling factor; (2) the ZJ, are strongly damped by the disorder (i.e. mean free path effects) so that the range of these interactions becomes smaller when the disorder increases; (3) the vu, can change of sign when 6 increases so that a classical perturbation theory (retaining only terms proportional to S’) cannot be applied for most of the physical systems; (4) the ZI,,must change of sign with the band filling (i.e. with fld), for example, it has been show using the moments of the pair interactions that u, must have at least two zeros, 71~four zeros . . [5] ; (5) the u, can vary strongly with concentration and in view of (4) they must change of sign.for some systems the tilling of which is such that ZJ,,= 0. 3. RESULTS AND DISCUSSION For each concentration c and for each lattice structure the values of ZI, obtained from the previous computations can be used to determine the most stable OS of the above Ising model [2 1,221. However, the pair interactions being concentration dependent, it is easily seen [21] that this homogeneous state can be unstable against a mixing of two (ordered or disordered) phases of different compositions. The most stable homogeneous OS for TM alloys can be obtained as follows: (1) for each concentration we define in the 3D parameter space (dd, S,,,,fl,J the regions of stability for each possible OS, a peculiar system being represented by a point in this space; (2) then the physical systems {ii>, the constituents of which are TM of the ith and jth series (i, j = 1, 2,3), define for this concentration a subset of points in the parameter space. Their positions in the previous regions of stability define the possible homogeneous stable OS. The regions we introduced in the parameter space and the possible OS for all the {ij ) subsets define generalized structural maps (GSM) which are different from the previous PSM (fl, AN): (i) the number of dimensions of the parameter space is larger so
Vo[. 38, No. 12
STABILITY OFBINARY TRANSITION METAL COMPOUNDS
that the systems corresponding to different {i/} subsets and the same values of (-~, AN) can present different OS; (ii) the set of the experimentally observed OS reported in the PSM are necessarily a subset of the predicted OS for several reasons: (1) the observed OS are, in principle, equilibrium states which are stable against phase separation; (2) some equilibrium states cannot easily experimentally be attained, the ordering energies and order-disorder critical temperatures being too small. Let us now discuss the results we have obtained for f.c.c, lattices when the ODD is neglected. The v~ being strongly damped and very small for n > 4 [5] we have used the results obtained by Kanamori and Kakehashi [22] for the Ising model when v, = 0 for n > 4: we believe that the neglected pair (or cluster) interactions do not modify significantly our results except in the regions of the parameter space for which several OS are nearly degenerate. In general, v~ is much larger than v~ (n > 1); therefore, the regions in which OS can be stable are roughly determined by the sign of v t whereas the relative stability of the OS is governed by the signs and the relative orders of magnitude of v2, v3, v4. The regions of stability of the OS are roughly limited by the lines N~ (Ga) for which v~ vanishes; segregation occurs for nearly filled or empty bands whereas order occurs for half filled bands. This result is well known in the field of itinerant magnetism: magnetic pair interacton are antiferromagnetic for half-filled bands and ferromagnetic for nearly empty (or filled) bands [23]. As previously noted the v, (n > 1) determine the relative stability of the OS; for example, the lines N~ (Sa) for which the antiphase boundary energy ~"of CuAul (~" = v2 --4v3 + 4v4) [24] vanishes, determine domains in the regions of stability [defined by the lines iV~ (Sa)] in which either OS of the CuaAu (~"< 0) or OS of the AI3Ti (~"> 0) families [22] are stable (see Fig. 2). We have studied the existence of the OS for vt > 0 and for several concentrations (namely c = I/5, 1/4, 2/7, 1/3, 1/2, 4/7, 3/4 and 4/5) corresponding to the occurence of stoichiometric ordered structures on the Cc.c. lattice. Let us briefly summarize the results we obtained for some typical concentrations: (1) For c = 0.25 the Cu3Au OS is stable essentially in a broad region: 4.5 ~
c= 0.5
e.~a
1221 c:0.33
9 ~d
c=0.25
I-
6
d
~
~ IIII
\\\\ "~q~b ,, ,, ,',.',.',
,
.~-
,?.uAu [
;,0 ~,u3 Au
AL3Ti
\\\\\\\\
-- I~d=Nl(Gdl(Vl=0} --- I~d=N~(6d} ( ~ =0l .....t~d=~t(Gd)(Vt.--0) .I
0
Fig. 2. Regions of existence of ordered ground states for binary AeB~_ c transition metal alloys. N a = CNaA + (1 -- c)Naa in the average number of " d " electrons per atom and Ga is the diagonal disorder (the off-diagonal disorder ~5na = 0). (4.5 < Na < 6.5) but the structure 40 introduced by Kanamori and Kakehashi [22] is the most stable for lower values of Na (i.e. 3.5 < ~7a < 4.6). (3) For c = 0.33 the situation is more complicated. The tworegions for which the Pt2Mo is the most stable (3.2 ~
1222
STABILITY OF BINARY TRANSITION METAL COMPOUNDS
Fig. 2 corresponds to an average number of minority spins 2Nd~ = 7.2 and to a positive value of~a~ (54~ 0.5). This implies that the Cu3Au OS is the ground state for this system. This work is the first attempt to determine the possible OS on a given lattice for TM alloys. The good agreement with experiment is surprising but, in view of the approximation we have made, it must be considered only as a first step towards the prediction of the OS occuring in the phase diagrams. We believe that the qualitative features we obtained will remain valid despite these approximations. However (i) if the ODD does not change the sequence of OS for small AN [5], the competition between ODD and relaxation effects may affect the results and must be studied carefully; (ii) the order of magnitude and the variations of the ordering energies with the disorder is not qualitatively affected by the self-consistency (charge transfer) [ 13, 16]. However the effect of self-consistency on the relative stability of the various OS has not been considered up to now. The study of these effects is now under progress.
Acknowledgements - It is a pleasure to thank Drs Treglia and Ducastelle for stimulating discussions and to Drs M. and H. Henion for communicating their results on ferromagnetic nickel alloys. The computations have been performed on the Univac 1110 computer of the Computer Centre of Strasbourg Cronenbourg; the authors are deeply indebted to Mr M. Gendner and Mr G Weill for their invaluable technical assistance. REFERENCES O. Kubachewski, Phase Stability in Metals and Alloys (Edited by P.S. Rudman et al.). McGrawHill, New York (1967). 2. J. St. John & A.N. Bloch, Phys. Rev. Lett. 33, 1095 (1974), G. Simons & A.N. Bloch, Phya Rev. BT, 2754 (1973); E.S. Machlin, T.P. Chow & J.C. Phillips, Phys. Rev. Lett. 38, 1292 (1977); A. Zunger, Phys.Rev. Lett. 44, 582 (1980); A. Zunger, Phys. Rev. B12, 5839 (1980). 3. R.E. Watson & L.H. Bennet, Phys. Rev. BI8, 6439 (1978);J. Phys. Chem. Solids 39, 1235 (1978). 4. These plots have been suggested independently by E.S. Machlin & B. Loh,Phys. Rev. Lett. 45, 1642 (1980). 5. A. Bieber, F. Gautier, G. Treglia & F. Ducastelle lnt. Conj'. TransitionMetals. Leeds (August 1980) (and to be published). 6. J. Friedel, Phys&s of Metals (Edited by J.M. Ziman), Vol. 1. Cambridge University Press, London (1978). 7. J.F. Janak, V.L. Moruzzi & A.R. Williams, Phya Rev. B12, 1257 (1975); V.L. Moruzzi, J.F. Janak & A.R. Williams, Calculated Electronic Properties 1.
Vof. 38, No7 12
of Metal~ Pergamon, New York (1978); C.D. Gelatt Jr., H. Ehfenreich & R.E. Watson, Phyg Reu. B15, 1613 (1977). 8. D.G. Pettifor, J. Phys. F7,613 (1977). 9. O.K. Andersen, Itinerant Electron Magnetism, (Edited by R.D. Lowde and E.P. Wolfarth), p. 317. North Holland, Amsterdam (1977). 10. F. Gautier, J. Van Der Rest & F. Brouers, J. Phys. FS, 1884 (1975). I 1. J. Van Der Rest, F. Gautier & F. Brouers, J. Phy~ FS, 2283 (1975); M. Cyrot & F. Cyrot-Lackmann, J. Phys. F6, 2257 (1976). 12. D.G. Pettifor, Phya Rev. Lett. 42,846 (1979); SolidState Commun. 28,621 (1978); C.M. Varma, SolidState Commun. 31,295 (1979); A.R. Williams, C.D. Gelatt Jr. & V.L. Moruzzi, Phy~ Rev. Lett. 44,429 (1980). 13. J. Giner, F. Brouers, F. Gautier & J. Van Der Rest, J. Phy£ F6, 1281 (1976). 14. R.C. Kittler & L.M. Falicov, Phys~ Rev. B18, 2506 (1978). 15. F. Gautier, F. Ducastelle & J. Giner, PhiZ Mag. 31, 1373 (1975); F. DucasteUe & F. Gautier, J. Phys. F6, 2039 (1976). 16. G. Treglia, F. Ducastelle & F. Gautier, J. Phys. F8, 1437 (1978). 17. G. Treglia & F. Ducastelle,J. Phys. FIO, 2137 (1980). 18. B.L. Gyorffy & G.M. Stocks, J. Phys. Paris 35, C4-75 (1974). 19. W.M. Temmerman, B.L. Gyorffy & G.M. Stocks, J. Phys. F8, 2461 (1978). 20. B. Vehcky, S. Kirkpatrick & H. Ehrenreich, Phys. Rev. 175,747 (1968). 2 I. D. de Fontaine, Solid State Phys. (Edited by H. Ehrenreich, F. Seitz and D. Turnbull), Vol. 34, p. 73. Academic Press, New York (1979) and references therein (see especially Section 6); M. Sanchez & D. de Fontaine,Phya Rev. B21,216 (1980). 22. J. Kanamori & Y. Kakehashi, J. Phys. (Paris) 38, C7-274 (I 977). 23. T. Moriya, Int. School of Physics Enrico Fermi Course 37, p. 206. Academic Press, New York (1967). 24. J. Kanamori, Modulated Structures (Proc. of AIP Conf.) (Edited by J.M. Cowley, J.B. Salamon and B.J. Wuensch). AlP (1979). 25. R.V. Kolubaev, N.S. Golosov & V.E. Panin, Sov. Phys. SolidState 19, 1863 (1978). 26. M. Hansen, K. Anderko, Constitution o f Binary Alloys. McGraw-Hill, New York (1958). 27. R.P. Elliott, Constitution of Binary Alloys, First Supplement. McGraw-Hill, New York (1965). 28. F.A. Shunk, Constitution of Binary Alloys, Second Supplement. McGraw-Hill, New York (1969). 29. Metals Handbook, Metallography, Structures and Phase Diagrams, ASM, Vol. 8, p. 232 (1973).
0038-1098/81/241219-04502.00/0
ELECTRONIC STRUCTURE AND STABILITY OF BINARY TRANSITION METAL COMPOUNDS A. Bieber t'2 and F. Gautier l t LMSES, (LA 306), 4 rue B. Pascal, 67000 Strasbourg, France 2 CRM, CNRS, 6 rue Boussingault, 67083 Strasbourg Cedex, France
(Received 27 January 1981 by J. Kanamori) It is shown that the ordered structures occurring in binary substitutional transition metal alloys on simple lattices (f.c.c., b.c.c.) can systematically be obtained from the electronic structure of the disordered alloys. Generalized structural maps are introduced; their physical meaning and their relation to the phenomenological structural maps used previously by other authors are discussed.
1. INTRODUCTION
of the disordered alloys; (ii) to define generalized structural maps, to discuss their physical meaning in relation SEVERAL ATTEMPTS to classify the ordered crystal to the previous ones and to present some preliminary structures of binary alloys AeBl_ e, according to the results for the f.c.c. OS. values of various semi-empirical parameters, have been It is now well established both qualitatively [6] and reported recently [ 1 - 3 ] . Two coordinates at least are quantitatively [7, 8] that most of the cohesive energy necessary to construct such phenomenological structural of TM and their alloys comes from the band structure maps (PSM) since we have to take account of both the energy of the " d " electrons. We will therefore only conaverage crystalline potential and the fluctuations of sider this contribution to the total energy. For simplicity potential from site to site. The simplest parameters are we neglect the "'sd'" hybridization and we assume in this then: (i) the average number of electrons per atom 37 = scheme - in agreement with ASA computations [8, 9] cNA + (1 -- c)N 8 which determines the filling of the that each TM has about 0.8 "s" electron and Ndi " d " bands and (ii) the valence difference £~N = N a - - N a electrons per atom (Ni = Nei + 0.8); the energies El are which characterizes the magnitude of the disorder. We then determined by: (/) the center of gravity ei(~i), the have determined the PSM (37, AN) for transition metal width Wi(~21), and the filling of the " d " bands (i.e. Ndi); (TM) alloys and for particular concentrations (c = 0.25, (ii) the lattice structure. In a similar way, the energy 0.33, 0.5); the separation of the different types of Edis (c) and the energy spectrum of a completely disordered structures (OS) into different regions of the ordered state are determined by (i) the averaged band plane (37, AN) is at least as significant [4, 5] as the separenergy g(~) and the bandwidth W(~2)[g(~) = ation obtained by using other coordinates [ 1 - 3 ] . For "2icie~(~2), W(~2) = Zic i Wi(~Z)], (ii) the diagonal (DD) example the PSM for e = 0.25 (see Fig. 1) shows that: and off-diagonal (ODD) disorder defined by 5a(~) = (1) Cu3Au type OS occur for 7.5 ~ 37< 8.5 when the 2(ea --en)/W and 5nd(~2) = (Wa -- Wn)/ff/respectively; alloys are non magnetic and for 8.5 ---<37~< 10 when the is the average atomic volume of the alloy. These alloys are magnetically ordered; (2) A13Ti occurs only parameters can be determined from band structure calfor37 = 8.75 and AN = -- 5. The coordinates 37, AN are culations: we use here the ASA results [8, 9], ~ being thus relevant but they are not sufficient to determine given either by Vegard's law or by experiment when the most stable OS since the separation between the available. Reliable mixing energies A.Eais(c ) = Eals(c) -various OS is not perfect: for example the (six) possible cEa --(1 - - c ) E n for disordered alloys can then be comsystems corresponding to the same point (37, AN) and puted using a simple non self-consistent CPA [I0, 11]; to TM of different series do not present the same type moreover, the variation of AEdis with 37 and AN can be of OS. qualitatively interpreted in terms of " d " bands broadening and deformation induced by the DD and ODD 2. METHOD respectively [12]. The fluctuations of the energy levels We have undertaken a systematic study of the OS according to their local environment [ 12] and the selfbased on f.c.c, and b.c.c, lattices for TM alloys [5]. Our consistent determination of the charge transfers do not aim in this paper is: (i) to show that the possible OS can change qualitatively the results [ 11, 12] ; we have therebe derived systematically from the electronic structure fore neglected these effects at the present stage. 1219
1220
STABILITY OF BINARY TRANSITION
Fig. 1. Phenomenological structural maps (1, Alv) for transition metal alloys for c = 0.25. We have retained all the unambiguous ordered structures determined from equilibrium phase diagrams [26-291. CUJAU (M) and Cu3Au (NM) corresponds to magnetically ordered and non magnetic ordered Cu3Au compounds respectively. The figures indicate the number of systems in which each ordered structure has been found.
The systematic study of the OS on a given lattice and of their variation with temperature and composition [ 13, 141 is a priori much more difficult. It is always possible - at least in principle - (i) to determine the band structure for each “ reasonable” OS, (ii) to compute their energies E,, (c) = Otis (c) + 3E,i, (c) and (iii) to determine the ground state and its long range order by comparison of Edis and of all the computed Eord. However, this numerical method is heavy and untractable for complicated structures. Moreover, it does not allow to describe the results in terms of simple physical parameters. For this reason, we suggested a systematic method [15-171 which reduces this problem to the determination of the ground state of a generalized 3D Ising model H = En V,p, where p,, is the number of nth neighbouring pairs and V, the corresponding pair interaction defined from the average medium of the completely disordered state as determined by the CPA (KKR [ 18, 191 or TB CPA [20]); for example, V, is given approximately in the tight-binding (TB) CPA and for &nd =0 1151 by:
x L,,
m’
W’m
[z -C(z)]}’
dE.
In this relation, At = f A - t* where t’ is the matrix associated to an atom i in the average medium; G,““’ (z) is the interatomic Green function of the metal A taken between two nth neighbouring sites; m labels the basis orbitals (m= 1, . . . , 5) and C (z) is the CPA self energy. A more complicated but similar expression can be
METAL COMPOUNDS
Vol. 38, Nd. 12
derived when 6 nd is included. The previous Ising model is the first term of the generalized perturbation method which yields the ordering energy AE,, as a cluster expansion [ 151. This expansion is valid for all values of 6 = (6,, And): this is an essential feature since most of the observed OS occur for large F values; direct comparison with band structure computations in some simple cases has shown that this expansion is rapidly convergent [5]. Therefore we consider here only the pair interactions V, and neglect the higher order terms. The properties of the pair interactions are discussed elsewhere [5] ; we just recall here their salient features: (1) they are defined as functions of the distance between neighbours (i.e. n), the disorder 6, the concentration c and the band filling~v,: V, = i-v,, (6,, hnd, c,Nd), the bandwidth p being only a scaling factor; (2) the ZJ, are strongly damped by the disorder (i.e. mean free path effects) so that the range of these interactions becomes smaller when the disorder increases; (3) the vu, can change of sign when 6 increases so that a classical perturbation theory (retaining only terms proportional to S’) cannot be applied for most of the physical systems; (4) the ZI,,must change of sign with the band filling (i.e. with fld), for example, it has been show using the moments of the pair interactions that u, must have at least two zeros, 71~four zeros . . [5] ; (5) the u, can vary strongly with concentration and in view of (4) they must change of sign.for some systems the tilling of which is such that ZJ,,= 0. 3. RESULTS AND DISCUSSION For each concentration c and for each lattice structure the values of ZI, obtained from the previous computations can be used to determine the most stable OS of the above Ising model [2 1,221. However, the pair interactions being concentration dependent, it is easily seen [21] that this homogeneous state can be unstable against a mixing of two (ordered or disordered) phases of different compositions. The most stable homogeneous OS for TM alloys can be obtained as follows: (1) for each concentration we define in the 3D parameter space (dd, S,,,,fl,J the regions of stability for each possible OS, a peculiar system being represented by a point in this space; (2) then the physical systems {ii>, the constituents of which are TM of the ith and jth series (i, j = 1, 2,3), define for this concentration a subset of points in the parameter space. Their positions in the previous regions of stability define the possible homogeneous stable OS. The regions we introduced in the parameter space and the possible OS for all the {ij ) subsets define generalized structural maps (GSM) which are different from the previous PSM (fl, AN): (i) the number of dimensions of the parameter space is larger so
Vo[. 38, No. 12
STABILITY OFBINARY TRANSITION METAL COMPOUNDS
that the systems corresponding to different {i/} subsets and the same values of (-~, AN) can present different OS; (ii) the set of the experimentally observed OS reported in the PSM are necessarily a subset of the predicted OS for several reasons: (1) the observed OS are, in principle, equilibrium states which are stable against phase separation; (2) some equilibrium states cannot easily experimentally be attained, the ordering energies and order-disorder critical temperatures being too small. Let us now discuss the results we have obtained for f.c.c, lattices when the ODD is neglected. The v~ being strongly damped and very small for n > 4 [5] we have used the results obtained by Kanamori and Kakehashi [22] for the Ising model when v, = 0 for n > 4: we believe that the neglected pair (or cluster) interactions do not modify significantly our results except in the regions of the parameter space for which several OS are nearly degenerate. In general, v~ is much larger than v~ (n > 1); therefore, the regions in which OS can be stable are roughly determined by the sign of v t whereas the relative stability of the OS is governed by the signs and the relative orders of magnitude of v2, v3, v4. The regions of stability of the OS are roughly limited by the lines N~ (Ga) for which v~ vanishes; segregation occurs for nearly filled or empty bands whereas order occurs for half filled bands. This result is well known in the field of itinerant magnetism: magnetic pair interacton are antiferromagnetic for half-filled bands and ferromagnetic for nearly empty (or filled) bands [23]. As previously noted the v, (n > 1) determine the relative stability of the OS; for example, the lines N~ (Sa) for which the antiphase boundary energy ~"of CuAul (~" = v2 --4v3 + 4v4) [24] vanishes, determine domains in the regions of stability [defined by the lines iV~ (Sa)] in which either OS of the CuaAu (~"< 0) or OS of the AI3Ti (~"> 0) families [22] are stable (see Fig. 2). We have studied the existence of the OS for vt > 0 and for several concentrations (namely c = I/5, 1/4, 2/7, 1/3, 1/2, 4/7, 3/4 and 4/5) corresponding to the occurence of stoichiometric ordered structures on the Cc.c. lattice. Let us briefly summarize the results we obtained for some typical concentrations: (1) For c = 0.25 the Cu3Au OS is stable essentially in a broad region: 4.5 ~
c= 0.5
e.~a
1221 c:0.33
9 ~d
c=0.25
I-
6
d
~
~ IIII
\\\\ "~q~b ,, ,, ,',.',.',
,
.~-
,?.uAu [
;,0 ~,u3 Au
AL3Ti
\\\\\\\\
-- I~d=Nl(Gdl(Vl=0} --- I~d=N~(6d} ( ~ =0l .....t~d=~t(Gd)(Vt.--0) .I
0
Fig. 2. Regions of existence of ordered ground states for binary AeB~_ c transition metal alloys. N a = CNaA + (1 -- c)Naa in the average number of " d " electrons per atom and Ga is the diagonal disorder (the off-diagonal disorder ~5na = 0). (4.5 < Na < 6.5) but the structure 40 introduced by Kanamori and Kakehashi [22] is the most stable for lower values of Na (i.e. 3.5 < ~7a < 4.6). (3) For c = 0.33 the situation is more complicated. The tworegions for which the Pt2Mo is the most stable (3.2 ~
1222
STABILITY OF BINARY TRANSITION METAL COMPOUNDS
Fig. 2 corresponds to an average number of minority spins 2Nd~ = 7.2 and to a positive value of~a~ (54~ 0.5). This implies that the Cu3Au OS is the ground state for this system. This work is the first attempt to determine the possible OS on a given lattice for TM alloys. The good agreement with experiment is surprising but, in view of the approximation we have made, it must be considered only as a first step towards the prediction of the OS occuring in the phase diagrams. We believe that the qualitative features we obtained will remain valid despite these approximations. However (i) if the ODD does not change the sequence of OS for small AN [5], the competition between ODD and relaxation effects may affect the results and must be studied carefully; (ii) the order of magnitude and the variations of the ordering energies with the disorder is not qualitatively affected by the self-consistency (charge transfer) [ 13, 16]. However the effect of self-consistency on the relative stability of the various OS has not been considered up to now. The study of these effects is now under progress.
Acknowledgements - It is a pleasure to thank Drs Treglia and Ducastelle for stimulating discussions and to Drs M. and H. Henion for communicating their results on ferromagnetic nickel alloys. The computations have been performed on the Univac 1110 computer of the Computer Centre of Strasbourg Cronenbourg; the authors are deeply indebted to Mr M. Gendner and Mr G Weill for their invaluable technical assistance. REFERENCES O. Kubachewski, Phase Stability in Metals and Alloys (Edited by P.S. Rudman et al.). McGrawHill, New York (1967). 2. J. St. John & A.N. Bloch, Phys. Rev. Lett. 33, 1095 (1974), G. Simons & A.N. Bloch, Phya Rev. BT, 2754 (1973); E.S. Machlin, T.P. Chow & J.C. Phillips, Phys. Rev. Lett. 38, 1292 (1977); A. Zunger, Phys.Rev. Lett. 44, 582 (1980); A. Zunger, Phys. Rev. B12, 5839 (1980). 3. R.E. Watson & L.H. Bennet, Phys. Rev. BI8, 6439 (1978);J. Phys. Chem. Solids 39, 1235 (1978). 4. These plots have been suggested independently by E.S. Machlin & B. Loh,Phys. Rev. Lett. 45, 1642 (1980). 5. A. Bieber, F. Gautier, G. Treglia & F. Ducastelle lnt. Conj'. TransitionMetals. Leeds (August 1980) (and to be published). 6. J. Friedel, Phys&s of Metals (Edited by J.M. Ziman), Vol. 1. Cambridge University Press, London (1978). 7. J.F. Janak, V.L. Moruzzi & A.R. Williams, Phya Rev. B12, 1257 (1975); V.L. Moruzzi, J.F. Janak & A.R. Williams, Calculated Electronic Properties 1.
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