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Gap anisotropy in superconducting alloys with localized states within the energy gap
Solid State Communications, Printed in Great Britain.
Vol. 52, No. 6, pp. 623-626,
GAP ANISOTROPY
00381098/84 $3.00 + .OO Pergamon Press Ltd.
1984.
IN SUPERCONDUCTING ALLOYS WITH LOCALIZED STATES WITHIN THE ENERGY GAP F. Reithofer and E. Schachinger
Institut
fur Theoretische
Physik, Technische
Universitit
Graz, Austria
(Received 26 April 1984 by M.F. Collins) The effect of paramagnetic impurities on the gap function of an anisotropic superconducting alloy is studied within the framework of BCS and Shiba Rusinov theory. It is found that the anisotropy of the gap function increases with increasing impurity concentration but the increase is additionally affected by the parameter e. which describes the relative position of the localized state within the gap.
that Shiba used a contact potential for his t-matrix and thus ended up with only one localized state described by a parameter e. which can easily be identified from Rusinov’s work to be equal to cos (S,’ -SO) where S,’ and S, are the spin up and spin down phase shifts of the zeroth-order partial wave chanel respectively. Shiba’s approach was most recently used by Wanier and Nagi [ 121 as well as Okabe and Nagi [ 131 to investigate the effect of such impurities on the thermodynamics of an anisotropic superconducting alloy. They were able to show that the thermodynamics of the alloy is additionally affected by tne parameter e. and it is therefore of interest to study how the anisotropy of the gap function and as a result how the quasiparticle density of states will be affected by this kind of impurities.. The calculations presented herewith are also restricted to the Shiba approach (i.e. only one localized state in the gap) and the non spin flip part of the impurity potential is treated according to Abrikosov and Gor’kov [ 141. (Nevertheless it was shown by Okabe and Nagi [ 151 that this part of the interaction potential should also be treated within the t-matrix framework. As we need the momentum scattering part only to smear out the anisotropy of the system, it is not essential for the results to be presented which of the two formalisms is used throughout the calculation.) The electron 4 x 4 matrix Green’s function is determined from the Dyson equation
THE ANISOTROPY in the energy gap brings interesting effects into superconductivity, namely if paramagnetic impurities are involved. So Fulde [ 1] studied the effect of such impurities on the system’s critical temperature using BCS formalism and a separable model [2] to describe the anisotropic pairing parameter gk,k’. The electron-impurity mechanism was described by the Abrikosov-Gor’kov (AG) [3] model assuming the electron-impurity interaction to be weak and using a first order Born’s approximation to describe the scattering terms in the Hamiltonian. Their scheme was later on extended by Schachinger and Carbotte [4] and Blezius and Carbotte [5] to investigate how paramagnetic impurities affect the anisotropy of the gap function itself. They showed that the gap anisotropy is an increasing function of the concentration of paramagnetic impurities. A strong coupling calculation performed by Daams et al. [6] confirmed the BCS results qualitatively but reported pronounced quantitative deviations from BCS. One of the most interesting results of these studies was, that close to the critical impurity concentration of the anisotropic alloy superconductivity exists only by virtue of anisotropy. This gave rise to some interesting work about the possibility that anisotropy in a metal may always lead to superconductivity at finite temperatures [7-91. However, it was proved that for superconducting alloys containing dilute concentrations of transition metal paramagnetic impurities, the AG approach does not suffice because it is no longer possible to regard the electron-impurity interaction to be weak. Thus Shiba [lo] and Rusinov [ 111 (SR) independently developed a theory in which the electron-impurity interaction was described by a t-matrix leading to the important effect that localized states exist within the original (pure metal) superconducting gap. The main difference between Shiba’s and Rtisinov’s approach was
d-‘(k,
iq)
= &‘(k,
iq)
- Z(k, ion),
(1)
where tio(k, iwn) is the Green’s function of the unperturbed system, and the self-energy kf,k, ic+) consists of three parts: Qk, ic+)
= CBCS(k, iq)
^ The BCS part of &oa 623
+ XN(k, iwn) + X,f,k, ion). (2)
and the contribution
of normal
GAP ANISOTROPY
624
IN SUPERCONDUCTING
scattering impurities iN are well known for a separable model anisotropy gk,k’ = [ 1 + a(n)] g[ 1 + a(!X)]’ : zBCS
6.
=
(3)
(1 +@))60~0*uO
r
= - fIv(O)g
[ 1 + ~(a’)]
1 2
dw tanh
-co x Im
(4)
r,=P.
l
6-‘(k,
z) = [zZ(& z)TO*oO -fkTO’o,-J -Z(~,z)A(~,z)~,~u,]-‘,
(6)
with the so far unknown renormalization functions Z(a, z) and A(a, z). The self-energy contribution of the paramagnetic impurities is described by the Shiba f-matrix element [lo] which can easily be shown to be identical to Rusinov’s [ 111 results if one restricts the calculation to s-wave scattering only Q$,z)
(10)
Equation (9) together with the self consistency equation (4) determines completely the physical behaviour of the system. Close to the critical temperature Tz A(!d, z) becomes infinitesimally small and only terms linear in A(a, z) are kept in the equations (9) and (4). In their linear form these equations imply the following ansatz for the gap function
z>= A,(z) + 4WLtz),
(11)
where the isotropic [A,(z)] and anisotropic [A,(z)] parts of the anisotropy function are now independent of the angle a. Impurity scattering affects the degree of anisotropy through A but not the anisotropy function a(n) itself. The combination of equation (11) with the linearized equations (9) and (4) result in two equations which determine A, and A, : A,(z)
= So
1 -isig:Imzrl(l
A,(z)
= a0
1-
i
1 (12a
+eo)
sign Im z
(W
(r, + r2)
Z
I
Schachinger and Carbotte [4] introduced the ratio Re [A,(z)]/Re [A,(z)] as a measure of gap anisotropy and the equation (12) give for the SR-treatment the interesting result: lim r2=o;r1-t-
= M(z)
Vol. 52, No. 6
r2 = n7rN(0)V2.
1 +a2
A(& Here Ti Oj denotes the direct product of two Pauli spin matrices, where the different symbols 7 and u were chosen to make the notation clearer, V is the momentum scattering potential which is assumed to be a contact potential for matters of simplicity, and N(0) is the quasiparticle density of states at the Fermi energy. The following standard ansatz was used for the Green’s function:
ALLOYS
z
(13)
= (1 -eo)2. 0
=--n----
da ZT~.U~ + eoA(Q z)~~*u~ P 1 +cw2 I 47T e;A2(a2, z) -z2
x dA2(Q,z)-z2
,
(7)
with p = N(O)n
f’;
0
(Y = N(O)+;
1 --(IL2 Eg = 1+oJ2’
(8)
J denotes the exchange integral, S is the classical spin of the impurity atom and n is the impurity concentration. Solving equation (1) finally gives one equation which determines the so far arbitrary function A(Sl, z) which obviously plays the role of the gap function in the superconducting alloy : A(a, Z) = [ 1 + ~(a)] 60 + r2 j g
- r2 j
2
The reported AG-limit of Re A,(z)/Re A,(z) = 4 is found for e. = 1 which corresponds to the AG-limit of the SR-theory. Thus we find as a first result that the anisotropy of the gap function is less affected by impurities of the SR kind. Using the ansatz (11) in the nonlinear equations (9) and (4) leads to: A(u) = A, +aA, @qq=-s Ao
=
60 -
rl(l
+
EO)&
j
j
daaWz)
EoA2ca>_z2
E2A2ca)_z2 0
A’~$~,vz~f;f’ ,
[W-22, z) -co A@, z>l
daP(a)
dE@=s -rlEoA,
A1
=
4-j
&o---IA,
(9) with
(14a)
--F2Al
j
j
d@(a)
d-gp-+ 3
e2A2~u~_z2 0
(14c)
GAP ANISOTROPY
Vol. 52, No. 6
IN SUPERCONDUCTING
ALLOYS
625
1
1 -
Re6o.t) --
-1miiiJ
1
F&. 1. Isotropic (Ae = Ae /6 e) and anisotropic part (A, = A,/&,) of the gap function for different values of yp and for yN = 0 as a function of the reduced frequencies G = w/6. The anisotropy parameter (a’) = 0.04 and the localized state is described by e. = 0.7. where we introduced the anisotropy distribution P(u) following Clem [ 161. P(u) is defined as the fraction of the Fermi surface at which a
= 1;
j- d&‘(a)
= (a?).
= 0; (15)
It is also obvious from the structure of the equations (14) that they can be renormalized by dividing by ho. It is rather easy to derive an approximate solution of the equations (14) assuming the anisotropy of the system to be small. In this case it is possible to develop the square root in a power series in a keeping only terms of maximum order (u2). The resulting system of equations can be solved numerically using either a Newton algorithm or an algorithm of successive approximation. Figure 1 presents the results for a system with (0’) = 0.04 and e. = 0.7 for various values of the parameter yp = F2/60 and yN = F1/ho set equal to zero. Thus only paramagnetic scattering is regarded in this figure. The result differs distinctively from the one found for the AG-theory [4]. Around o/Se = ee the
Fig, 2. Same as Fig, 1 but for an almost isotropic system with YN = 10.
imaginary part of both A, and A, becomes finite and this region corresponds to the localized state within the energy gap and a comparison with Fig. 2 which shows the results for the identical but isotropic system (&,, L 10) reveals immediately the influence of anisotropy. The localized state is distinctively broader in the anisotropic case and there is an additional structure found for both the A, and the A, curve. This structure is a result of our separable model and should not be observable by experiment. These results are found to be even more pronounced in the quasiparticle density of states which is determined from the gap function by N(w) N(0)
=
da 5 z
= I
0 Re d/w’ - A2(s2, w)
d&‘(u) Re
w dw”
- A2(Q, o) ’
In Figs. 3 and 4 the tunneling characteristic according to equation (16) is presented for the anisotropic system (YN = 0) and the isotropic one (UN = 10) respectively. A comparison of the two figures depicts the main influence of anisotropy on the quasiparticle density of states: (i) The localized state within the energy gap is
(16)
626
GAP ANISOTROPY
IN SUPERCONDUCTING
ALLOYS
Vol. 52, No. 6
n
Fig. 3. Quasiparticle density of states for the system with ee = 0.7 and (a’> 0 0.04 with no normal scattering impurities (yN = 0.0) as a function of the reduced frequencies z = o/S. significantly broader in the anisotropic system. Nevertheless this broadening becomes less and less pronounced when the parameter EOgoes to zero. (ii) The maximum in the quasiparticle density of states in the region of the localized state is larger in the anisotropic system. (iii) The onset of the gapless regime is observed for smaller impurity concentrations in an anisotropic system. Acknowledgements - This research was supported by Fonds zur Forderung der wlssenschaftlichen Forschung, contract Nr. 4440. All numerical calculations were performed on the UNIVAC 1 lOO/Sl computer of EDVZentrum der Technischen Universitft Graz, Austria. REFERENCES 1. 2. 3.
P. Fulde,Phys. Rev. 139, A726 (1965). D. Markowitz & L.P. Kadanoff, Phys. Rev. 131, 563 (1963). A.A. Abrikosov & L.P. Gor’kov, Sov. Phys. -JETP 9,220 (1959).
Fig. 4. Same as Fig. 3 but for the almost isotropic system (ye = 10).
15.
E. Schachinger & J.P. Carbotte, J. Low Temp. Phys. 42,81 (1981). J.W. Belzius & J.P. Carbotte, Can. J. Phys. 62, 158 (1984). J.M. Daams, E. Schachinger & J.P. Carbotte, J. Low Temp. Phys. 42,69 (1981). M.D. Whitmore & J.P. Carbotte, Phys. Rev. B23, 5782 (1981). C.R. Leavens, A.H. Macdonald & D.J.W. Geldar, Phys. Rev. B26,3960 (1982). M.D. Whitmore, J.P. Carbotte & E. Schachinger, Phys. Rev. B29,2510 (1984). H. Shiba, Prop. Theor. Phys. 40,435 (1968). A.I. Rusinov, Zh. Eksp. Teor. Fiz. 56,2047 (1969); [Sov. Phys.-JETP 29,llOl (1969)]. K.B. Wanier & A.D.S. Nag&J. Low Temp. Phys. 4597 (1981). Y. Okabe & A.D.S. Nagi, Phys. Rev. B28, 1323 (1983). A.A. Abrikosov, L.P. Gor’kov & I.E. Dzyaloskinskii,Methods of Quantum Field Theory in Statistical Physics, Dover Publications, New York (1975). Y. Okabe & A.D.S. Nagi, - Phys. - Rev. B28,1320
16.
f!?%m,
4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
Phys. Rev. 148,392
(1966).
Vol. 52, No. 6, pp. 623-626,
GAP ANISOTROPY
00381098/84 $3.00 + .OO Pergamon Press Ltd.
1984.
IN SUPERCONDUCTING ALLOYS WITH LOCALIZED STATES WITHIN THE ENERGY GAP F. Reithofer and E. Schachinger
Institut
fur Theoretische
Physik, Technische
Universitit
Graz, Austria
(Received 26 April 1984 by M.F. Collins) The effect of paramagnetic impurities on the gap function of an anisotropic superconducting alloy is studied within the framework of BCS and Shiba Rusinov theory. It is found that the anisotropy of the gap function increases with increasing impurity concentration but the increase is additionally affected by the parameter e. which describes the relative position of the localized state within the gap.
that Shiba used a contact potential for his t-matrix and thus ended up with only one localized state described by a parameter e. which can easily be identified from Rusinov’s work to be equal to cos (S,’ -SO) where S,’ and S, are the spin up and spin down phase shifts of the zeroth-order partial wave chanel respectively. Shiba’s approach was most recently used by Wanier and Nagi [ 121 as well as Okabe and Nagi [ 131 to investigate the effect of such impurities on the thermodynamics of an anisotropic superconducting alloy. They were able to show that the thermodynamics of the alloy is additionally affected by tne parameter e. and it is therefore of interest to study how the anisotropy of the gap function and as a result how the quasiparticle density of states will be affected by this kind of impurities.. The calculations presented herewith are also restricted to the Shiba approach (i.e. only one localized state in the gap) and the non spin flip part of the impurity potential is treated according to Abrikosov and Gor’kov [ 141. (Nevertheless it was shown by Okabe and Nagi [ 151 that this part of the interaction potential should also be treated within the t-matrix framework. As we need the momentum scattering part only to smear out the anisotropy of the system, it is not essential for the results to be presented which of the two formalisms is used throughout the calculation.) The electron 4 x 4 matrix Green’s function is determined from the Dyson equation
THE ANISOTROPY in the energy gap brings interesting effects into superconductivity, namely if paramagnetic impurities are involved. So Fulde [ 1] studied the effect of such impurities on the system’s critical temperature using BCS formalism and a separable model [2] to describe the anisotropic pairing parameter gk,k’. The electron-impurity mechanism was described by the Abrikosov-Gor’kov (AG) [3] model assuming the electron-impurity interaction to be weak and using a first order Born’s approximation to describe the scattering terms in the Hamiltonian. Their scheme was later on extended by Schachinger and Carbotte [4] and Blezius and Carbotte [5] to investigate how paramagnetic impurities affect the anisotropy of the gap function itself. They showed that the gap anisotropy is an increasing function of the concentration of paramagnetic impurities. A strong coupling calculation performed by Daams et al. [6] confirmed the BCS results qualitatively but reported pronounced quantitative deviations from BCS. One of the most interesting results of these studies was, that close to the critical impurity concentration of the anisotropic alloy superconductivity exists only by virtue of anisotropy. This gave rise to some interesting work about the possibility that anisotropy in a metal may always lead to superconductivity at finite temperatures [7-91. However, it was proved that for superconducting alloys containing dilute concentrations of transition metal paramagnetic impurities, the AG approach does not suffice because it is no longer possible to regard the electron-impurity interaction to be weak. Thus Shiba [lo] and Rusinov [ 111 (SR) independently developed a theory in which the electron-impurity interaction was described by a t-matrix leading to the important effect that localized states exist within the original (pure metal) superconducting gap. The main difference between Shiba’s and Rtisinov’s approach was
d-‘(k,
iq)
= &‘(k,
iq)
- Z(k, ion),
(1)
where tio(k, iwn) is the Green’s function of the unperturbed system, and the self-energy kf,k, ic+) consists of three parts: Qk, ic+)
= CBCS(k, iq)
^ The BCS part of &oa 623
+ XN(k, iwn) + X,f,k, ion). (2)
and the contribution
of normal
GAP ANISOTROPY
624
IN SUPERCONDUCTING
scattering impurities iN are well known for a separable model anisotropy gk,k’ = [ 1 + a(n)] g[ 1 + a(!X)]’ : zBCS
6.
=
(3)
(1 +@))60~0*uO
r
= - fIv(O)g
[ 1 + ~(a’)]
1 2
dw tanh
-co x Im
(4)
r,=P.
l
6-‘(k,
z) = [zZ(& z)TO*oO -fkTO’o,-J -Z(~,z)A(~,z)~,~u,]-‘,
(6)
with the so far unknown renormalization functions Z(a, z) and A(a, z). The self-energy contribution of the paramagnetic impurities is described by the Shiba f-matrix element [lo] which can easily be shown to be identical to Rusinov’s [ 111 results if one restricts the calculation to s-wave scattering only Q$,z)
(10)
Equation (9) together with the self consistency equation (4) determines completely the physical behaviour of the system. Close to the critical temperature Tz A(!d, z) becomes infinitesimally small and only terms linear in A(a, z) are kept in the equations (9) and (4). In their linear form these equations imply the following ansatz for the gap function
z>= A,(z) + 4WLtz),
(11)
where the isotropic [A,(z)] and anisotropic [A,(z)] parts of the anisotropy function are now independent of the angle a. Impurity scattering affects the degree of anisotropy through A but not the anisotropy function a(n) itself. The combination of equation (11) with the linearized equations (9) and (4) result in two equations which determine A, and A, : A,(z)
= So
1 -isig:Imzrl(l
A,(z)
= a0
1-
i
1 (12a
+eo)
sign Im z
(W
(r, + r2)
Z
I
Schachinger and Carbotte [4] introduced the ratio Re [A,(z)]/Re [A,(z)] as a measure of gap anisotropy and the equation (12) give for the SR-treatment the interesting result: lim r2=o;r1-t-
= M(z)
Vol. 52, No. 6
r2 = n7rN(0)V2.
1 +a2
A(& Here Ti Oj denotes the direct product of two Pauli spin matrices, where the different symbols 7 and u were chosen to make the notation clearer, V is the momentum scattering potential which is assumed to be a contact potential for matters of simplicity, and N(0) is the quasiparticle density of states at the Fermi energy. The following standard ansatz was used for the Green’s function:
ALLOYS
z
(13)
= (1 -eo)2. 0
=--n----
da ZT~.U~ + eoA(Q z)~~*u~ P 1 +cw2 I 47T e;A2(a2, z) -z2
x dA2(Q,z)-z2
,
(7)
with p = N(O)n
f’;
0
(Y = N(O)+;
1 --(IL2 Eg = 1+oJ2’
(8)
J denotes the exchange integral, S is the classical spin of the impurity atom and n is the impurity concentration. Solving equation (1) finally gives one equation which determines the so far arbitrary function A(Sl, z) which obviously plays the role of the gap function in the superconducting alloy : A(a, Z) = [ 1 + ~(a)] 60 + r2 j g
- r2 j
2
The reported AG-limit of Re A,(z)/Re A,(z) = 4 is found for e. = 1 which corresponds to the AG-limit of the SR-theory. Thus we find as a first result that the anisotropy of the gap function is less affected by impurities of the SR kind. Using the ansatz (11) in the nonlinear equations (9) and (4) leads to: A(u) = A, +aA, @qq=-s Ao
=
60 -
rl(l
+
EO)&
j
j
daaWz)
EoA2ca>_z2
E2A2ca)_z2 0
A’~$~,vz~f;f’ ,
[W-22, z) -co A@, z>l
daP(a)
dE@=s -rlEoA,
A1
=
4-j
&o---IA,
(9) with
(14a)
--F2Al
j
j
d@(a)
d-gp-+ 3
e2A2~u~_z2 0
(14c)
GAP ANISOTROPY
Vol. 52, No. 6
IN SUPERCONDUCTING
ALLOYS
625
1
1 -
Re6o.t) --
-1miiiJ
1
F&. 1. Isotropic (Ae = Ae /6 e) and anisotropic part (A, = A,/&,) of the gap function for different values of yp and for yN = 0 as a function of the reduced frequencies G = w/6. The anisotropy parameter (a’) = 0.04 and the localized state is described by e. = 0.7. where we introduced the anisotropy distribution P(u) following Clem [ 161. P(u) is defined as the fraction of the Fermi surface at which a
= 1;
j- d&‘(a)
= (a?).
= 0; (15)
It is also obvious from the structure of the equations (14) that they can be renormalized by dividing by ho. It is rather easy to derive an approximate solution of the equations (14) assuming the anisotropy of the system to be small. In this case it is possible to develop the square root in a power series in a keeping only terms of maximum order (u2). The resulting system of equations can be solved numerically using either a Newton algorithm or an algorithm of successive approximation. Figure 1 presents the results for a system with (0’) = 0.04 and e. = 0.7 for various values of the parameter yp = F2/60 and yN = F1/ho set equal to zero. Thus only paramagnetic scattering is regarded in this figure. The result differs distinctively from the one found for the AG-theory [4]. Around o/Se = ee the
Fig, 2. Same as Fig, 1 but for an almost isotropic system with YN = 10.
imaginary part of both A, and A, becomes finite and this region corresponds to the localized state within the energy gap and a comparison with Fig. 2 which shows the results for the identical but isotropic system (&,, L 10) reveals immediately the influence of anisotropy. The localized state is distinctively broader in the anisotropic case and there is an additional structure found for both the A, and the A, curve. This structure is a result of our separable model and should not be observable by experiment. These results are found to be even more pronounced in the quasiparticle density of states which is determined from the gap function by N(w) N(0)
=
da 5 z
= I
0 Re d/w’ - A2(s2, w)
d&‘(u) Re
w dw”
- A2(Q, o) ’
In Figs. 3 and 4 the tunneling characteristic according to equation (16) is presented for the anisotropic system (YN = 0) and the isotropic one (UN = 10) respectively. A comparison of the two figures depicts the main influence of anisotropy on the quasiparticle density of states: (i) The localized state within the energy gap is
(16)
626
GAP ANISOTROPY
IN SUPERCONDUCTING
ALLOYS
Vol. 52, No. 6
n
Fig. 3. Quasiparticle density of states for the system with ee = 0.7 and (a’> 0 0.04 with no normal scattering impurities (yN = 0.0) as a function of the reduced frequencies z = o/S. significantly broader in the anisotropic system. Nevertheless this broadening becomes less and less pronounced when the parameter EOgoes to zero. (ii) The maximum in the quasiparticle density of states in the region of the localized state is larger in the anisotropic system. (iii) The onset of the gapless regime is observed for smaller impurity concentrations in an anisotropic system. Acknowledgements - This research was supported by Fonds zur Forderung der wlssenschaftlichen Forschung, contract Nr. 4440. All numerical calculations were performed on the UNIVAC 1 lOO/Sl computer of EDVZentrum der Technischen Universitft Graz, Austria. REFERENCES 1. 2. 3.
P. Fulde,Phys. Rev. 139, A726 (1965). D. Markowitz & L.P. Kadanoff, Phys. Rev. 131, 563 (1963). A.A. Abrikosov & L.P. Gor’kov, Sov. Phys. -JETP 9,220 (1959).
Fig. 4. Same as Fig. 3 but for the almost isotropic system (ye = 10).
15.
E. Schachinger & J.P. Carbotte, J. Low Temp. Phys. 42,81 (1981). J.W. Belzius & J.P. Carbotte, Can. J. Phys. 62, 158 (1984). J.M. Daams, E. Schachinger & J.P. Carbotte, J. Low Temp. Phys. 42,69 (1981). M.D. Whitmore & J.P. Carbotte, Phys. Rev. B23, 5782 (1981). C.R. Leavens, A.H. Macdonald & D.J.W. Geldar, Phys. Rev. B26,3960 (1982). M.D. Whitmore, J.P. Carbotte & E. Schachinger, Phys. Rev. B29,2510 (1984). H. Shiba, Prop. Theor. Phys. 40,435 (1968). A.I. Rusinov, Zh. Eksp. Teor. Fiz. 56,2047 (1969); [Sov. Phys.-JETP 29,llOl (1969)]. K.B. Wanier & A.D.S. Nag&J. Low Temp. Phys. 4597 (1981). Y. Okabe & A.D.S. Nagi, Phys. Rev. B28, 1323 (1983). A.A. Abrikosov, L.P. Gor’kov & I.E. Dzyaloskinskii,Methods of Quantum Field Theory in Statistical Physics, Dover Publications, New York (1975). Y. Okabe & A.D.S. Nagi, - Phys. - Rev. B28,1320
16.
f!?%m,
4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
Phys. Rev. 148,392
(1966).