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A different approach to the generalized coherent-potential aproximation; Multiple scattering formalism
Solid State Communications, Vol. 12, pp. 267—270, 1973. Pergamon Press.
Printed in Great Britain
A DIFFERENT APPROACH TO THE GENERAUZED COHERENT-POTENTIAL APPROXIMATION; MULTIPLE SCATTERING FORMALISM Komajiro Niizeki Department of Physics, Tohoku University, Sendal, Japan (Received 2 November 1972 by Y. Toyozawa)
The single site coherent-potential approximation (SCPA) is generalized in order to treat effects due to clusterization of scatterers, off-diagonal disorder, crystal field and/or short range order. Present formalism is a straight. forward generalization of Soven’s original method i.e. the multiple scattering formalism by which he derived the SCPA. RECENTLY, Nickel and Krumhansl1 have derived, within the conventional simplified model of dis~rdered system, the generalized coherent-potential approximation (CPA(n)) which is a direct generalization of the single-site coherent-potential approximation (SCPA) of Soven2 and other,~7in such a way that contributions from scatterings due to n-site clusters are exactly taken into account. They derived it by means of the self.contained cumulant expansions and the self-consistent diagramniatic-resummation techniques developed by Yonezawa6 and Leath.7 Howewr their method is difficult to generalize so as to treat off-diagonal disorder, crystal field and/or short range order.8 It is shown in the present letter that the SCPA can be easily generalized in such a way that these effects as well as cluster effect can be treated, by means of a straight-forward generalization of Soven’s original method, i.e. the multiple scatteringformalism.
system and possesses the symmetry of the crystal group of the averaged system. The second part ~ to be referred to as the potential depends on configurations of the system. We here assume that t~ can be divided into contnbutions from every possible set of sites of the ‘crystal’, i.e. 2 c where Cstands for a set of sites, i.e. a cluster of sites (or simply a cluster) and the summation is extended over all possible clusters of the infinite crystal. 0 (C) is assumed to depend on atomic configurations of sites in C only, but not always assumed to be localized within sites in C, that is, the potential can be extended beyond the cluster.
c
‘-
~ (C) (ICI ~2) is referred to as a many-site (center or body) potential, where CI stands for the number of sites in C The simplest model Hamiltonian adopted by Nickel and Krumhanslm and by others~7 is a special case, where 0 is a sum of single-site potentials localized within each site. The model Hamiltonian adopted by Foo et aL9 in their investigation of offdiagonal disorder can be incorporated in our scheme if two site potentials due to nearest neighbour pairs are included in addition to single site ones. If a special
We deal with the motion of a quasi-particle such as an electron, a phonon, a Frenkel exciton or a magnon under a random potential of an alloy or a mixed crystal, within the single particle approximation, It is governed by a single particle Hamiltonian which depends on configurations of the system. The Hamiltonian can be divided into two parts, i.e. = H +0 (1) 0
‘
The first part H 0 to be referred to as the free Hamiltonian is independent of configurations of the
case that the hopping integral between heterogeneous pair of atoms is equal to the algebraic mean of both 267
268
THE GENERAUZED COHERENT-POTENTIAL APPROXIMATION
homogeneous ones is realized in that model, the offdiagonal disorder can be confined within extended single site potentials.1°A Hamiltonian in which crystal field plays nonnegligible part can be treated if extended single site potentials and/or many site potentials which are localized or extended are introduced.
where the complex parameter is supressed for convenience. Thus equation (8) can be taken as a defining equation for the effective potential. Iterating equation (9), we have the following perturbation series for the scattering matrix =
The configuration dependent one-particle Green’s function is defined as follows ~ (z)
=
(z
—
JC)~’
(3)
,
where z is a complex parameter. Single particle properties of the system are fully described by the averaged one-particle Green’s function defined as follows G(z) = <~(z)>,
(4)
where the bracket stands for configuration average. It is assumed that there are no long range orders in atomic configuration but not always assumed that there are no short range orders. Hereafter a configuration dependent quantity is represented by a script letter and a configuration independent quantity corresponding to the precedent is represented by the corresponding lightface italic.
Vol. 12, No.4
0,. +0,.G0,.
+0fGZ~fG0f +....
Now the essential step of the present approach is to assume that the effective potential can be also divided into contributions from every possible cluster of the crystal and therefore the potential fluctuation does too, i.e. and
=
Of
(11)
C-
=
c ~ (12) where V(C) which is referred to as Cl-site coherentpotential is independent of atomic configuration and 0~(C)= C(C) V(C). —
Replacing Of in equation (10) in terms of the right hand side of.equation (12), we have ~f(Cl)G~f(C
=
2)G k=I
The effective Hamiltonian and the effective potential (or the self-energy) are defined by the following equations G(z)
and
=
(z H(z))’ 11o + V(z)
(5) (6)
—
11(z) = where 11(z) and V(z) which depend on the parameter are configuration independent and possess the symmetry of the crystal.
C1,C2
...
1=
G(z) +G(z)~T(z)G(z).
(7)
~ ~T~c’~
(14)
C
and ~ji
c Ck(C1UC2U...
C f(C1 ) GC,.(C’2 ) G
—
=
G(~(C~).
C1 U C2 U U Ck only. Therefore the summation in steps equation (13) can be divided into the following two
1.c2
~(z)
...
Ck
(13) Each term of the right hand side of equation (13) depends on atomic configurations of a cluster C~
5’(C)=
Let the potential fluctuation in a given atomic configuration be defined byO f(z) = 0 V(z). Then the scattering matrix of the quasi-particle, due to the potential fluctuation, is defined by the following equation
(10)
...
UCk.~C)
GCf(C’k).
(15)
Next, physical meaning of f1(C) is investigated. Let ~J(C)
~
fJ(C’)
(16)
C,cc
Because of equation (4), the following self.consistency condition must be satisfied, i.e., (CS(z)) = 0. (8) While the scattering matrix satisfies the following integral equation (9)
and define 0(C), V(C), and 0~(C~ by similar equations in terms of C(C’)’s, P~(C’)’s,and 0,.(C’)’s, respectively. Then we have
Vol. 12, No.4
THE GENERALIZED COHERENT-POTENTIAL APPROXIMATION I
h.~1 C~.C
2
0,(C1 )GC,.(C2 )G..GCf(Ck)
CkCC
=0,(C)+0r(C)GOf(C)+Of(C)GOf(C)G0f(C)+... 0,.(C) (1 GO,.(C))’. (17) —
Accordingly, ~1(C)is nothing but the scattering matrix of the quasi-particle when it passes the potential fluctuation due to the cluster C which is embedded in the averaged medium characterized by the effective potential V. In fact,0 ,.(C) is given by 0(C) v(C), where 0(C) is the total contributions of random potentials due to sites in C only and V(C) is the counterpart of the coherent-potential. —
Equation (14) and the self-consistency condition: equation (8) allow us to determine coherent-potentials which are unspecified as yet, through the following infinite set of partial self-consistency conditions (~(C))= 0 for all C.
(18)
This set of equations is obviously equivalent with the following infinite set of equations, owing to equation (16), i.e.
=
0 for all C,
(19)
where ~T(C)’sare evaluated by means of equation(17). That is, a scattering matrix due to a potential fluctuation of any cluster must vanish on the average. The set of equations (19) obviously imposes a necessary arid sufficient condition for unique division of the effective potential in such a way as shown in equation (11). Thus the assumption is justified. So far we have not resorted to any approximation. Accordingly solutions of the set of equations (19) would provide the exact averaged Green’s function. In practice we can not solve such an infinite set of equations. So, some approximations are needed. One reasonable approximation is to restrict the number of sites in allowed clusters which contribute to the effective potential. That is, let V
V~
I
V(C)
(1 ib)
C(lCj~n)
and require that =
0forallC(lCi~n),
(l9b)
where ~T(C)’smust be evaluated in ~crmsof the approximate effective potential. Then a self-consistent solution of the set of equations (Ilb) and (19b) will provide us with an approximate averaged Green’s function. This approximation which is a quite natural generalization
269 2 is nothing but the
of the SCPA proposed by Sown CPA(n). In fact, if it isadopted applied by for Nickel the simplest model of disordered system, and Krumhansi,’ it is easy to show that their result is exactly reproduced, since V(C) reduces to a CI X Cl-matrix in the site representation of the Hilbert space, in that model. However the CPA(n ) is not the unique possibility of approximation nor the most convenient one, because it retains in many (infinite) clusters which have so large a diameter that they contribute negligibly to the effective potential. Various different approximations in which various sets of clusters are retained are also possible. They are referred to as generalized coherent-potential approximation (GCPA). The simplest nontrivial generalization over the SCPA is the nearest neighbour pair coherent-potential approximation (NNPCPA) in which all nearest neighbour pairs of sites are retained, in addition to single site ‘clusters’. If the NNPCPA is applied for the model of Foo et aL, the off-diagonal disorder and/or shortrange order, if it ever exists, can be treated. Then it is easy to show that the problem is reduced to a trancendental simultaneous equation with three unknowns, one of which refers to a single site coherentpotential and the other two refer to a nearest neighbour pair coherent-potential. If the special case mentioned previously is realized in that model, the off-diagonal disorder can be treated within the SCPA with extended single site potentials. We note here that the TCPA of Foo et al. is different from our NNPCPA. The latter saves the failure in reproducing the center of the energy band in the former treatment. Formulation of two-particle Green’s function is also possible in a similar way, if the irreducible kernel is assumed to be divided into contributions from every possible cluster. Detailed considerations and numerical investigations on the present approach will be presented in a forthcoming paper. Acknowledgements The author thanks Professor M. Watabe for critical reading of this manuscript. He also thanks Professors A. Morita, S. Ogawa and A. Yanase for encouragement. —
270
THE GENERALIZED COHERENT-POTENTIAL APPROXIMATION
Vol. 12, No.4
REFERENCES 1. 2.
NICKEL B.G. and K.RUMHANSL J.A., Phys. Rev. B4, 4354 (1971). SOVEN P.,Phys. Rev. 156, 809(1967).
3.
TAYLOR D.W., Phys. Rev. 156, 1017 (1967).
4.
ONODERA Y. and TOYOZAWA Y.,J. Phys. Soc. Japan 24, 1341 (1968).
5.
VEUCKY B., KIRKPATRICK S. and EHRENREICH H.,Phys. Rev. 175, 747 (1968).
6.
YONEZAWA F.,J~gr.theor. Phys. (Kyoto)40, 734(1968).
7.
LEATH P.L.,Phys. Rei’. 171, 725 (1968).
8. 9.
The crystal field makes atomic level of an atom occupying a site depend on configurations of sites surrounding FOO E-N1., AMAR Hand AUSLOOS M., Phys. Rev. 84, 3350 (1971).
10.
It is this advantage that the weak-coupling theory of off-diagonal disorder due to BERK [Phj’s.Rev. Bl, 1336 (1970)] is simplified in the special case. The present theory to be developed need not assume a weak-coupling Hamiltonian.
Die single site coherent-potential approximation (SCPA) wird verailgemeinert, urn den Haufen des Sreungzentrum, die nichtdiagonale Unordnung, das Kristalfeld, und/oder die Wirkung der Nahordnung zu behandein. Die vorgelegte Methode ist eine d.irekte Verailgemeinerung der originalen Methode den mehrfachen Streuung von Soven fur SCPA.
it.
Printed in Great Britain
A DIFFERENT APPROACH TO THE GENERAUZED COHERENT-POTENTIAL APPROXIMATION; MULTIPLE SCATTERING FORMALISM Komajiro Niizeki Department of Physics, Tohoku University, Sendal, Japan (Received 2 November 1972 by Y. Toyozawa)
The single site coherent-potential approximation (SCPA) is generalized in order to treat effects due to clusterization of scatterers, off-diagonal disorder, crystal field and/or short range order. Present formalism is a straight. forward generalization of Soven’s original method i.e. the multiple scattering formalism by which he derived the SCPA. RECENTLY, Nickel and Krumhansl1 have derived, within the conventional simplified model of dis~rdered system, the generalized coherent-potential approximation (CPA(n)) which is a direct generalization of the single-site coherent-potential approximation (SCPA) of Soven2 and other,~7in such a way that contributions from scatterings due to n-site clusters are exactly taken into account. They derived it by means of the self.contained cumulant expansions and the self-consistent diagramniatic-resummation techniques developed by Yonezawa6 and Leath.7 Howewr their method is difficult to generalize so as to treat off-diagonal disorder, crystal field and/or short range order.8 It is shown in the present letter that the SCPA can be easily generalized in such a way that these effects as well as cluster effect can be treated, by means of a straight-forward generalization of Soven’s original method, i.e. the multiple scatteringformalism.
system and possesses the symmetry of the crystal group of the averaged system. The second part ~ to be referred to as the potential depends on configurations of the system. We here assume that t~ can be divided into contnbutions from every possible set of sites of the ‘crystal’, i.e. 2 c where Cstands for a set of sites, i.e. a cluster of sites (or simply a cluster) and the summation is extended over all possible clusters of the infinite crystal. 0 (C) is assumed to depend on atomic configurations of sites in C only, but not always assumed to be localized within sites in C, that is, the potential can be extended beyond the cluster.
c
‘-
~ (C) (ICI ~2) is referred to as a many-site (center or body) potential, where CI stands for the number of sites in C The simplest model Hamiltonian adopted by Nickel and Krumhanslm and by others~7 is a special case, where 0 is a sum of single-site potentials localized within each site. The model Hamiltonian adopted by Foo et aL9 in their investigation of offdiagonal disorder can be incorporated in our scheme if two site potentials due to nearest neighbour pairs are included in addition to single site ones. If a special
We deal with the motion of a quasi-particle such as an electron, a phonon, a Frenkel exciton or a magnon under a random potential of an alloy or a mixed crystal, within the single particle approximation, It is governed by a single particle Hamiltonian which depends on configurations of the system. The Hamiltonian can be divided into two parts, i.e. = H +0 (1) 0
‘
The first part H 0 to be referred to as the free Hamiltonian is independent of configurations of the
case that the hopping integral between heterogeneous pair of atoms is equal to the algebraic mean of both 267
268
THE GENERAUZED COHERENT-POTENTIAL APPROXIMATION
homogeneous ones is realized in that model, the offdiagonal disorder can be confined within extended single site potentials.1°A Hamiltonian in which crystal field plays nonnegligible part can be treated if extended single site potentials and/or many site potentials which are localized or extended are introduced.
where the complex parameter is supressed for convenience. Thus equation (8) can be taken as a defining equation for the effective potential. Iterating equation (9), we have the following perturbation series for the scattering matrix =
The configuration dependent one-particle Green’s function is defined as follows ~ (z)
=
(z
—
JC)~’
(3)
,
where z is a complex parameter. Single particle properties of the system are fully described by the averaged one-particle Green’s function defined as follows G(z) = <~(z)>,
(4)
where the bracket stands for configuration average. It is assumed that there are no long range orders in atomic configuration but not always assumed that there are no short range orders. Hereafter a configuration dependent quantity is represented by a script letter and a configuration independent quantity corresponding to the precedent is represented by the corresponding lightface italic.
Vol. 12, No.4
0,. +0,.G0,.
+0fGZ~fG0f +....
Now the essential step of the present approach is to assume that the effective potential can be also divided into contributions from every possible cluster of the crystal and therefore the potential fluctuation does too, i.e. and
=
Of
(11)
C-
=
c ~ (12) where V(C) which is referred to as Cl-site coherentpotential is independent of atomic configuration and 0~(C)= C(C) V(C). —
Replacing Of in equation (10) in terms of the right hand side of.equation (12), we have ~f(Cl)G~f(C
=
2)G k=I
The effective Hamiltonian and the effective potential (or the self-energy) are defined by the following equations G(z)
and
=
(z H(z))’ 11o + V(z)
(5) (6)
—
11(z) = where 11(z) and V(z) which depend on the parameter are configuration independent and possess the symmetry of the crystal.
C1,C2
...
1=
G(z) +G(z)~T(z)G(z).
(7)
~ ~T~c’~
(14)
C
and ~ji
c Ck(C1UC2U...
C f(C1 ) GC,.(C’2 ) G
—
=
G(~(C~).
C1 U C2 U U Ck only. Therefore the summation in steps equation (13) can be divided into the following two
1.c2
~(z)
...
Ck
(13) Each term of the right hand side of equation (13) depends on atomic configurations of a cluster C~
5’(C)=
Let the potential fluctuation in a given atomic configuration be defined byO f(z) = 0 V(z). Then the scattering matrix of the quasi-particle, due to the potential fluctuation, is defined by the following equation
(10)
...
UCk.~C)
GCf(C’k).
(15)
Next, physical meaning of f1(C) is investigated. Let ~J(C)
~
fJ(C’)
(16)
C,cc
Because of equation (4), the following self.consistency condition must be satisfied, i.e., (CS(z)) = 0. (8) While the scattering matrix satisfies the following integral equation (9)
and define 0(C), V(C), and 0~(C~ by similar equations in terms of C(C’)’s, P~(C’)’s,and 0,.(C’)’s, respectively. Then we have
Vol. 12, No.4
THE GENERALIZED COHERENT-POTENTIAL APPROXIMATION I
h.~1 C~.C
2
0,(C1 )GC,.(C2 )G..GCf(Ck)
CkCC
=0,(C)+0r(C)GOf(C)+Of(C)GOf(C)G0f(C)+... 0,.(C) (1 GO,.(C))’. (17) —
Accordingly, ~1(C)is nothing but the scattering matrix of the quasi-particle when it passes the potential fluctuation due to the cluster C which is embedded in the averaged medium characterized by the effective potential V. In fact,0 ,.(C) is given by 0(C) v(C), where 0(C) is the total contributions of random potentials due to sites in C only and V(C) is the counterpart of the coherent-potential. —
Equation (14) and the self-consistency condition: equation (8) allow us to determine coherent-potentials which are unspecified as yet, through the following infinite set of partial self-consistency conditions (~(C))= 0 for all C.
(18)
This set of equations is obviously equivalent with the following infinite set of equations, owing to equation (16), i.e.
=
0 for all C,
(19)
where ~T(C)’sare evaluated by means of equation(17). That is, a scattering matrix due to a potential fluctuation of any cluster must vanish on the average. The set of equations (19) obviously imposes a necessary arid sufficient condition for unique division of the effective potential in such a way as shown in equation (11). Thus the assumption is justified. So far we have not resorted to any approximation. Accordingly solutions of the set of equations (19) would provide the exact averaged Green’s function. In practice we can not solve such an infinite set of equations. So, some approximations are needed. One reasonable approximation is to restrict the number of sites in allowed clusters which contribute to the effective potential. That is, let V
V~
I
V(C)
(1 ib)
C(lCj~n)
and require that =
0forallC(lCi~n),
(l9b)
where ~T(C)’smust be evaluated in ~crmsof the approximate effective potential. Then a self-consistent solution of the set of equations (Ilb) and (19b) will provide us with an approximate averaged Green’s function. This approximation which is a quite natural generalization
269 2 is nothing but the
of the SCPA proposed by Sown CPA(n). In fact, if it isadopted applied by for Nickel the simplest model of disordered system, and Krumhansi,’ it is easy to show that their result is exactly reproduced, since V(C) reduces to a CI X Cl-matrix in the site representation of the Hilbert space, in that model. However the CPA(n ) is not the unique possibility of approximation nor the most convenient one, because it retains in many (infinite) clusters which have so large a diameter that they contribute negligibly to the effective potential. Various different approximations in which various sets of clusters are retained are also possible. They are referred to as generalized coherent-potential approximation (GCPA). The simplest nontrivial generalization over the SCPA is the nearest neighbour pair coherent-potential approximation (NNPCPA) in which all nearest neighbour pairs of sites are retained, in addition to single site ‘clusters’. If the NNPCPA is applied for the model of Foo et aL, the off-diagonal disorder and/or shortrange order, if it ever exists, can be treated. Then it is easy to show that the problem is reduced to a trancendental simultaneous equation with three unknowns, one of which refers to a single site coherentpotential and the other two refer to a nearest neighbour pair coherent-potential. If the special case mentioned previously is realized in that model, the off-diagonal disorder can be treated within the SCPA with extended single site potentials. We note here that the TCPA of Foo et al. is different from our NNPCPA. The latter saves the failure in reproducing the center of the energy band in the former treatment. Formulation of two-particle Green’s function is also possible in a similar way, if the irreducible kernel is assumed to be divided into contributions from every possible cluster. Detailed considerations and numerical investigations on the present approach will be presented in a forthcoming paper. Acknowledgements The author thanks Professor M. Watabe for critical reading of this manuscript. He also thanks Professors A. Morita, S. Ogawa and A. Yanase for encouragement. —
270
THE GENERALIZED COHERENT-POTENTIAL APPROXIMATION
Vol. 12, No.4
REFERENCES 1. 2.
NICKEL B.G. and K.RUMHANSL J.A., Phys. Rev. B4, 4354 (1971). SOVEN P.,Phys. Rev. 156, 809(1967).
3.
TAYLOR D.W., Phys. Rev. 156, 1017 (1967).
4.
ONODERA Y. and TOYOZAWA Y.,J. Phys. Soc. Japan 24, 1341 (1968).
5.
VEUCKY B., KIRKPATRICK S. and EHRENREICH H.,Phys. Rev. 175, 747 (1968).
6.
YONEZAWA F.,J~gr.theor. Phys. (Kyoto)40, 734(1968).
7.
LEATH P.L.,Phys. Rei’. 171, 725 (1968).
8. 9.
The crystal field makes atomic level of an atom occupying a site depend on configurations of sites surrounding FOO E-N1., AMAR Hand AUSLOOS M., Phys. Rev. 84, 3350 (1971).
10.
It is this advantage that the weak-coupling theory of off-diagonal disorder due to BERK [Phj’s.Rev. Bl, 1336 (1970)] is simplified in the special case. The present theory to be developed need not assume a weak-coupling Hamiltonian.
Die single site coherent-potential approximation (SCPA) wird verailgemeinert, urn den Haufen des Sreungzentrum, die nichtdiagonale Unordnung, das Kristalfeld, und/oder die Wirkung der Nahordnung zu behandein. Die vorgelegte Methode ist eine d.irekte Verailgemeinerung der originalen Methode den mehrfachen Streuung von Soven fur SCPA.
it.