- Email: [email protected]
A general approach to the stochastic impurity problem in crystals
Volume 39A, number 1
A GENERAL
PHYSICS LETTERS
10 April 1972
APPROACH TO THE STOCHASTIC IMPURITY PROBLEM IN CRYSTALS W. HEHL lnstitut fur Theoretische und Angewandte Physik. Universitat Stuttgart, and MPI fur Metallforschung, Institut fur Physik, Stuttgart, Germany Received 21 February 1972
Using a recursive method, a procedure is developed for calculating the effect of multiple-scattering in stochastic and random impurity problems in terms of an exact expansion in powers of the concentration In order to calculate the physical properties of crystals with impurities the Green function has to be configuration-averaged. We describe a systematic scheme capable of including all cluster effects for stochastically distributed impurities with arbitrary potentials. Regarding only pairs of defects characterized by 6-function potentials without any statistical correlation of the sites our result agrees with a formula given by Aiyer and co-workers [ 1]. We start with the averaged scattering matrix (T) in the one-particle case, defined by the standard series [2]
= ~
m,n
n~:m
+
~J
(I)
and collect all contributions of individual scatterers in (T)(1), of pairs in (T) (2) or of a cluster of n different scatterers in (T) (n),
(2)
Of course (T) (1) is simply C ~ R t t ( R 1 ) -- c ~ R ~ t 1, but note that all terms in eq. (1) from the second term onwards contain contributions of pairs of scatterers. The quantity (T) (n) includes thus all terms which obtain the same correlation function P(n)(R 1 ..... Rn) =~ cno(R1 ..... Rn)* as a result of the averaging process. We may write (T)(n)=c n
~' g(R 1 ..... R n ) ' S ( R 1 ..... R n ) , RI...Rn
(3)
where S(R 1 ..... R n ) =- S(n) is an operator summing all * In the random case g = 1.
graphs [3, 4] with n different scatterers and the first scatterer at R1, the second at R2, the third scatterer at R3, and so on. The prime means exclusion of all cases where at least two lattice sites R i are identical ("exclusion effect" [5]). We introduce a further operator n
u(R t ..... R . ) - u ( n ) = ~
~
m = 1 P(n,m)
s(R; 1..... R~,.),
where the inner sum is carried out over all m-permutations of the n distinct sites R i [6]. Then this operator contains all possible graphs with no more than n different impurities. The operator S(n) will be generated by replacing all internal lines in the skeleton diagrams of fig. la by the operator G o O ( n - 1)Go, and by multiplying with S(n - 1)G o and (1 + GoU(n - 1)) on the left and right hand sides of each graph, respectively. Summing up gives us
1 +G OU ( n - 1) S(n) = S ( n - l)Got n 1 - G o U ( n - 1)Got n
(4)
All operators S(n) are generalizations of the t-matrices for n different scatterers. With the trivial operators $(1) -'- U(1 ) = tl, we find, e.g. S(R 1,R2) = t 1Got2(1 - Got 1G o t 2 ) - 1(1 +Got 1) U(R 1,R2) = t 1 + S(R 1,R2) + t 2 + S(R 2,R 1) . In the case of strongly localized defects (S(RI, R2) + S(R2,RI) ) yields in Wannier representation the matrix t 2 defined in ref. [ 1]. In Bloch representation the contributions of reducible graphs lead to divergencies in the T matrix
(5)
Volume 39A, number 1
PHYSICS LETTERS
10 April 1972
approximation including the effects of the stochastic distribution of impurities with arbitrary potentials.
b) >'~-''--
/"
./
_
f
Fig. 1. Graphical representation of eq. (4). Tile vertices denote scattering at lattice site (i.e., an operator tit) . expansion while the corresponding self-energy expansion remains well defined. This should provide us with a direct access to the fine structure in the density of states [71 of a three-dimensional stochastic alloy. We have extended this procedure to products of Green operators which are important for calculating the electrical resistivity. We obtained the irreducible scattering vertex 114] which is exact up to the second power of the defect concentration, The present method may serve not only for calculating properties of moderately dilute alloys, but, following the work of Nickel and Krumhansl [8], also for an extension of the usual CPA method [2] to a CP n-site
The author wishes to thank Professor Dr. A. Seeger for his stimulating interest in this work. tie also acknowledges discussions with Dr. E. Mann, Dr. H. Teichler and Dipl. Phys. R. Kloss.
R CJe£~FelICCS [11 R.N. Aiyer, R.J. Elliott, J.A. Krumhansl and I'.k. Leath, Phys. Rev. 181 (1969) 1006. [2] B. Velick¢,,S. Kirkpatrick and H. Ehrenreich, Phys. Rev. 175 (1968) 747. [3] J.S. Langer, J. Math. Phys. 2 (1961) 584. [4] J.M. Luttinger, in: Mathematical methods in solid state and superfluid theory (Oliver and Boyd, Edinburgh, 1969) p, 185. [5] F. Yonezawa, Prog. Theor. Phys. 40 (1968) 734. [6] J. Riordan, An introduction to combinatorial analysis (Wiley, New York, 1958) p. 2. [7] R.L. Agacy and R.E. Borland, Proc. Phys. Soc. 84 (1964) 320. [8] B.G. Nickel and J.A. Krumhansl, Phys. Rev. B4 (1971) 4354.
A GENERAL
PHYSICS LETTERS
10 April 1972
APPROACH TO THE STOCHASTIC IMPURITY PROBLEM IN CRYSTALS W. HEHL lnstitut fur Theoretische und Angewandte Physik. Universitat Stuttgart, and MPI fur Metallforschung, Institut fur Physik, Stuttgart, Germany Received 21 February 1972
Using a recursive method, a procedure is developed for calculating the effect of multiple-scattering in stochastic and random impurity problems in terms of an exact expansion in powers of the concentration In order to calculate the physical properties of crystals with impurities the Green function has to be configuration-averaged. We describe a systematic scheme capable of including all cluster effects for stochastically distributed impurities with arbitrary potentials. Regarding only pairs of defects characterized by 6-function potentials without any statistical correlation of the sites our result agrees with a formula given by Aiyer and co-workers [ 1]. We start with the averaged scattering matrix (T) in the one-particle case, defined by the standard series [2]
m,n
n~:m
+
~J
(I)
and collect all contributions of individual scatterers in (T)(1), of pairs in (T) (2) or of a cluster of n different scatterers in (T) (n),
(2)
Of course (T) (1) is simply C ~ R t t ( R 1 ) -- c ~ R ~ t 1, but note that all terms in eq. (1) from the second term onwards contain contributions of pairs of scatterers. The quantity (T) (n) includes thus all terms which obtain the same correlation function P(n)(R 1 ..... Rn) =~ cno(R1 ..... Rn)* as a result of the averaging process. We may write (T)(n)=c n
~' g(R 1 ..... R n ) ' S ( R 1 ..... R n ) , RI...Rn
(3)
where S(R 1 ..... R n ) =- S(n) is an operator summing all * In the random case g = 1.
graphs [3, 4] with n different scatterers and the first scatterer at R1, the second at R2, the third scatterer at R3, and so on. The prime means exclusion of all cases where at least two lattice sites R i are identical ("exclusion effect" [5]). We introduce a further operator n
u(R t ..... R . ) - u ( n ) = ~
~
m = 1 P(n,m)
s(R; 1..... R~,.),
where the inner sum is carried out over all m-permutations of the n distinct sites R i [6]. Then this operator contains all possible graphs with no more than n different impurities. The operator S(n) will be generated by replacing all internal lines in the skeleton diagrams of fig. la by the operator G o O ( n - 1)Go, and by multiplying with S(n - 1)G o and (1 + GoU(n - 1)) on the left and right hand sides of each graph, respectively. Summing up gives us
1 +G OU ( n - 1) S(n) = S ( n - l)Got n 1 - G o U ( n - 1)Got n
(4)
All operators S(n) are generalizations of the t-matrices for n different scatterers. With the trivial operators $(1) -'- U(1 ) = tl, we find, e.g. S(R 1,R2) = t 1Got2(1 - Got 1G o t 2 ) - 1(1 +Got 1) U(R 1,R2) = t 1 + S(R 1,R2) + t 2 + S(R 2,R 1) . In the case of strongly localized defects (S(RI, R2) + S(R2,RI) ) yields in Wannier representation the matrix t 2 defined in ref. [ 1]. In Bloch representation the contributions of reducible graphs lead to divergencies in the T matrix
(5)
Volume 39A, number 1
PHYSICS LETTERS
10 April 1972
approximation including the effects of the stochastic distribution of impurities with arbitrary potentials.
b) >'~-''--
/"
./
_
f
Fig. 1. Graphical representation of eq. (4). Tile vertices denote scattering at lattice site (i.e., an operator tit) . expansion while the corresponding self-energy expansion remains well defined. This should provide us with a direct access to the fine structure in the density of states [71 of a three-dimensional stochastic alloy. We have extended this procedure to products of Green operators which are important for calculating the electrical resistivity. We obtained the irreducible scattering vertex 114] which is exact up to the second power of the defect concentration, The present method may serve not only for calculating properties of moderately dilute alloys, but, following the work of Nickel and Krumhansl [8], also for an extension of the usual CPA method [2] to a CP n-site
The author wishes to thank Professor Dr. A. Seeger for his stimulating interest in this work. tie also acknowledges discussions with Dr. E. Mann, Dr. H. Teichler and Dipl. Phys. R. Kloss.
R CJe£~FelICCS [11 R.N. Aiyer, R.J. Elliott, J.A. Krumhansl and I'.k. Leath, Phys. Rev. 181 (1969) 1006. [2] B. Velick¢,,S. Kirkpatrick and H. Ehrenreich, Phys. Rev. 175 (1968) 747. [3] J.S. Langer, J. Math. Phys. 2 (1961) 584. [4] J.M. Luttinger, in: Mathematical methods in solid state and superfluid theory (Oliver and Boyd, Edinburgh, 1969) p, 185. [5] F. Yonezawa, Prog. Theor. Phys. 40 (1968) 734. [6] J. Riordan, An introduction to combinatorial analysis (Wiley, New York, 1958) p. 2. [7] R.L. Agacy and R.E. Borland, Proc. Phys. Soc. 84 (1964) 320. [8] B.G. Nickel and J.A. Krumhansl, Phys. Rev. B4 (1971) 4354.