Bennemann K.H., Ketterson J.B. Superconductivity: Volume 1: Conventional and Unconventional Superconductors; Volume 2: Novel Superconductors
Подождите немного. Документ загружается.
4 Coexistence of Singlet Superconductivity and Magnetic Order 197
-contact, oscillation of T
c
with the thickness of the
F-layer, etc. This provides the unique possibility to
study the properties of superconducting electrons
under the influence of a huge exchange field acting
on the electron spins.
Acknowledgments
We would like to thank L.N. Bulaevskii for fruitful
collaboration in formulating and solving a number
of problems in the physics of magnetic superconduc-
tors. We are thankful to Vitalii Lazarevich Ginzburg
and Aleksei Alekseevich Abrikosov for a number of
inspiring and useful discussions as well for sup-
port. M.L.K. thanks Karl Bennemann for support
and fruitful collaboration in solving some interest-
ing problems in the field, and Ivan Boˇzovi´c for sup-
port. This work was supported in part by DOE (Con-
tract No. DE-AC02-98CH10886) Brookhaven Na-
tional Laboratory, Department of Condensed Matter
Physics and Material Science.
Appendix
The Hamiltonian of magnetic superconductors is
given by Eqs. (4.2)–(4.3). The coexistence of SC and
magnetic order with a wave vector Q can be stud-
ied in principle by using Gorkov equations for any
Q. However, in systems where the atomic length
a Q
−1
is fulfilled the quasiclassical Eilenberger–
Larkin–Ovchinikov (ELO) equations are more suit-
able and efficient. We first start with Gorkov’s equa-
tions and then introduce the ELO method.
A.1 Gor’kov Equations for Magnetic Superconductors
These equations contain nor mal and anomalous
Green’s functions (
ˆ
G)
˛ˇ
(x, y)=−<
ˆ
T
˛
(x)
†
ˇ
(y) >
and (
ˆ
F
†
)
˛ˇ
(x, y)=<
ˆ
T
†
˛
(x)
†
ˇ
(y) >,wherex ≡
(r, )and˛, ˇ =↑, ↓. The superconducting order
parameter is defined by
∗
(r
i
− r
j
)=−
1
2
V(r
i
− r
j
)[<
†
↑
(r
i
)
†
↓
(r
j
)>
−<
†
↓
(r
i
)
†
↑
(r
j
)>] , (4.82)
where the pairing interaction V(r
i
− r
j
)isre-
sponsible for the singlet pairing in absence of
magnetic order. Note that in the presence of any
(in)commensurate magnetic order should be non-
uniform,i.e.(r
1
, r
2
)=(r
1
− r
2
, r), r =(r
i
+ r
j
)/2.
However, it was shown in [10] that when h v
F
Q,
which is realized in the rare earth ternary supercon-
ductors, the non-uniform part of is small and of
the order (h/v
F
Q). This is the reason for our as-
sumption
∗
(r
i
, r
j
)=
∗
(r
i
− r
j
). In the absence of
magnetic field (A = 0) the set of equations for the
Green’s functions is given by
i!
n
−
ˆ
0
(
ˆ
p)−
ˆ
V
ex
(r)
ˆ
G
!
n
(r, r
)
+
d
3
x
ˆ
(r, x)
ˆ
F
†
!
n
(x, r
)=ı(r − r
) , (4.83)
i!
n
+
ˆ
0
(
ˆ
p)+
ˆ
V
tr
ex
(r)
ˆ
F
†
!
n
(r, r
)
−
d
3
x
ˆ
∗
(r − x)
ˆ
G
!
n
(x, r
)=0, (4.84)
where
ˆ
=
01
−1 0
,
ˆ
V
ex
=
h
z
(r) h
⊥
e
−i(Qr+)
h
⊥
e
i(Qr+)
−h
z
(r)
. (4.85)
In the case of the spiral magnetic ordering (with
spatial rotation of magnetization) besides the sin-
glet pairing amplitude F
†
↓↑
∼ there is also a triplet
pairing amplitude F
†
↑↑
∼ · h
∗
(and F
†
↓↓
), which is
due to the rotating magnetization. The latter mixes
spin up and down, [10]. F
†
↓↑
is responsible for singlet
pairing in the rare earth magnetic superconductors
giving rise to the order parameter
↑↓
(r), while the
triplet amplitude F
†
↑↑
doesnotgiverisetothetriplet
pairing since it is assumed from the very beginning
that
↑↑
=
↓↓
=0.
A.2 Eilenberger–Larkin–Ovchinikov Equations
for Magnetic Superconductors
In problems where the exchange field h(r)=h(r)e
z
changes slowly on the atomic length a generaliza-
tion of the standard quasiclassical ELO equations is
possible [8, 34, 93]. In this case the Gor’kov equa-
tions contain 4 × 4 Green’s functions
ˆ
G(x
1
, x
2
)=
−
ˆ
T
ˆ
¦ (x
1
)
ˆ
¦
†
(x
2
) with the four-component spinor
198 M.L. Kuli´candA.I.Buzdin
ˆ
¦
†
(x
1
)=(ˆ
†
↑
ˆ
†
↓
↑
↓
). Here ˆ
i
are Nambu matrices
in the electron-holespace and ˆ
i
are Pauli matrices in
the spin-space and x ≡ (r, ). The SC order parame-
ter is
ˆ
(r)=iˆ
+
ˆ
2
(r)−i ˆ
−
ˆ
2
∗
(r)with
(r)=iVTr
ˆ
+
ˆ
2
ˆ
G(x, x)
/2 . (4.86)
By integrating out unimportant degrees of freedom,
i.e. changes at small distances, one defines the quasi-
classical Green’s function ˆg
p
F
,!
n
(R, )
ˆg
p
F
,!
n
(R)=
d
p
d(r
1
− r
2
)
× e
−ip((r
1
−r
2
)
ˆ
3
ˆ
G(r
1
, r
2
, !
n
).
(4.87)
In the presence of non-magnetic impurities the ELO
equations for the quasiclassical Green’s functions
read [34]
iv
F
∇
R
ˆg
!
n
=
i!
n
ˆ
3
+ ˆ
3
ˆ
(R)+h(R) ˆ
3
− ˆ
3
£
imp
p
F
,!
n
(R), ˆg
!
n
,
(4.88)
where the impurity self-energy £
imp
!
n
in the Born ap-
proximation is given by
£
imp
p
F
!,
n
(R)=c
i
N(0)
9
ˆu
p
F
−p
F
ˆg
p
F
,!
n
(R)ˆu
p
F
−p
F
:
p
F
.
(4.89)
If the impurity potential (matrix) contains both the
non-magnetic and spin-orbit scattering then
ˆu
p
F
−p
F
= uˆ
3
+ iu
so
[ˆp
F
׈p
F
] ˆ˛ , (4.90)
with the vector matrix ˆ˛ = [(1 + ˆ
3
) +(1−ˆ
3
)
2
2
]/2and ˆp
F
= p
F
/ | p
F
|.Thematrixˆg
p
F
!
n
(R)
is given by [8,34,93]
ˆg
p
F
,!
n
(R)=−i
⎛
⎜
⎜
⎝
g
+
00if
+
0 g
−
if
−
o
0−if
†
−
−g
−
0
−if
†
+
00−g
+
⎞
⎟
⎟
⎠
. (4.91)
By including also orbital effects of the magnetic field
B(R)=∇×A(R) the ELO equations for the compo-
nents g
±
and f
±
read
[ ˜!
n,±
+ iev
F
· A(R) ± ih(R)−
1
2
v
F
·∇
R
]f
±p
F
(R, !
n
)
=
˜
±p
F
(R, !
n
)g
±p
F
(R, !
n
) , (4.92)
[ ˜!
n,±
+ iev
F
· A(R) ± ih(R)+
1
2
v
F
·∇
R
]f
†
±p
F
(R, !
n
)
=
∗
±p
F
(R,!
n
)g
±p
F
(R, !
n
) . (4.93)
The normalization condition and the self-consist-
ency equation are given by
g
2
±
+ f
†
±
f
±
=1, (4.94)
(p
F
, R)=
T
2
!
n
,p
F
V(p
F
, p
F
) (4.95)
×
=
f
+
(R, p
F
, !
n
)+f
−
(R, p
F
, !
n
)
>
.
˜!
n,±
and
˜
±
(p
F
, R, !
n
) are defined by
˜!
n,±
= !
n
+
1
2
g
±p
F
(R, !
n
)
p
F
(4.96)
+
3
2
so
9
g
∓p
F
(R, !
n
)sin
2
( −
)
:
p
F
,
˜
±p
F
(R, !
n
)=
p
F
(R)+
1
2
f
±p
F
(R, !
n
)
p
F
(4.97)
+
1
so
9
f
∓p
F
(R, !
n
)sin
2
( −
)
:
p
F
,
where 1/ = c
i
N(0) | u |
2
and 1/
so
= c
i
N(0) | u
so
|
2
.
The microscopic theory of magnetic supercon-
ductors, which takes into account spin (exchange)
and orbital (electromagnetic) effects of magnetic or-
der, as well as of non-magnetic impurity scattering,
has been developed by using the above generalized
ELO equations [8,34].
4 Coexistence of Singlet Superconductivity and Magnetic Order 199
References
1. V.L. Ginzburg, Zh. Eksp. Teor. Fiz.31, 202 (1958)
2. V.L.Ginzburg,G.F.Zharkov,Zh.Eksp.Teor.Fiz.34, 412 (1956); ibid, 37, 1784 (1958)
3. J. Bardeen, L.N. Cooper, J.R. Schrieffer, Phys. Rev. 108, 1175-11204(1957)
4. B.T.Matthias,H. Suhl and E. Corenzwit, Phys.Rev. Lett. 1, 92 (1958)
5. A.A.Abrikosov and L.P. Gor’kov, Sov. Phys. JETP, 12, 1243 (1961)
6. M.B. Maple and O . Fischer (Eds.), In: Superconductivity and Magnetism (Springer, Berlin, 1982)
7. P. Fulde, J. Keller, In: Superconductivity and Magnetism, Eds. M.B. Maple and O. Fischer (Springer, Berlin, 1982)
Chap.9,p.249
8. L.N. Bulaevskii, A.I. Buzdin, M.L. Kuli´candS.V.Panyukov,Adv.Phys.34,176 (1985)
9. P.W.Anderson,H.Suhl,Phys.Rev.116, 898 (1959)
10. L.N. Bulaevskii, M.L. Kuli´c and A.I. Rusinov, Solid State Comm. 30, 59 (1979); J.Low T. Phys. 39, 255 (1980)
11. S.K.Sinha, G.W.Crabtree, D.G. Hinks and H. Mook,Phys. Rev. Lett. 48, 950-953 (1982)
12. J.W.Lynn, G. Shirane, W. Thomlinson, R.N. Shelton and D.E. Moncton, Phys.Rev. B24, 3817 (1981)
13. W.A.Fertig,J.K.Bachtacharjee,A.Bagchi,Phys.Rev.Lett.38, 987 (1977)
14. S.Rehmann, T. Herrmannsd¨orfer and F.Pobell, Phys. Rev. Lett. 78, 1122 (1997)
15. M.L.Kuli´c, L.N. Bulaevskii and A.I. Buzdin, Phys. Rev. B56, R11 415 (1997)
16. M.L.Kuli´c, I.M. Kuli´c, Phys. Rev. B63, 104503 (2001)
17. M. L. Kuli´c, CR Phys. 7, 4 (2006)
18. F.S. Bergeret, A.F.Volkov and K.B. Efetov, Rev. Mod. Phys. 77, 1321 (2005)
19. Yu.A. Izyumov,Yu.N. Proshin, M.G. Khusainov, Phys. Uspekhi 45, 109 (2002)
20. A.I.Buzdin,L.N.Bulaevskii,S.V.Panyukov,JETPLett.35, 147 (1982)
21. A.I.Buzdin,Rev.Mod.Phys.77, 935 (2005)
22. V.V. Ryazanov et al., Phys. Rev. Lett. 86, 2427 (2001)
23. J.Flouquet,G.Knebel,D.Braithwaite,D.Aoki,J.-P.Brison,F.Hary,A.Huxley,S.Raymond,B.Salce,I.Sheikin,CRPhys.
7, 22 (2006)
24. M.L.Kuli´c, Physics Reports 338, 1 (2000); M.L. Kuli´c,O.V.Dolgov,Phys.Stat.Sol.B242, 151(2005)
25. J.W.Lynn, In: High Temperature Superconductivity,J.W. Lynn, (Ed.) (Springer, Berlin,1990) pp.268–302
26. M.L.Kuli´c, I.M. Kuli´c, Physica C 391, 42 (2003)
27. D. Pines, Physica B 163, 78 (1990); P. Montoux, D. Pines, Phys. Rev. B47, 6069 (1993); M. Eschrig, M. Norman, Phys.
Rev. Lett. 85,
3261 (2000)
28. I.W.Sumarlin,S.Skanthakumar,J.W.Lynn,J.L.Peng,Z.Y.Li,W.Jiang,andR.L.Greene,Phys.Rev.Lett.68,2228 (1992)
29. T.Nachtrab,C.Bernhard,C.Lin,D.Koelle,R.Kleiner,CR Phys.7,68 (2006)
30. J.W.Lynn,B.Keimer,C.Ulrich,C.BernhardandJ.L.Tallon,Phys.Rev.B61, R14964–R14967 (2000)
31. S.S. Saxena, P. Agarwal, K. Ahilan, F.M. Grosche, R.K.W. Haselwimmer, M.J. Steiner, E. Pugh, I.R. Walker, S.R. Julian,
P.Monthoux,G.G.Lonzarich,A.Huxley,I.Sheikin,D.Braithwaite,J.Flouquet,Nature406, 587 (2000)
32. D. Aoki, A. Huxley, E. Ressouche, D. Braithwaite, J. Flouquet, J.-P. Brison, E. Lhotel and C. Paulsen, Nature 413, 613
(2000)
33. U.Krey,Int.J.Magnetism3, 65-73 (1972); ibid 4, 153 (1973)
34. L.N. Bulaevskii, A.I. Buzdin, M.L. Kuli´c, S.V. Panyukov, Phys. Rev. B28, 1370 (1983)
35. M. Kaufman, O. Entin-Wohlman, Physica 84, 77 (1976)
36. K.B. Blagoev et al., Phys.Rev. Lett. 82, 133 (1999); ibid Phil. Mag. 83, 3247 (1998); N.I. Karchev et al., Phys. Rev. Lett.
826, 846 (2001)
37. K.H. Bennemann, M.L.Kuli´c, P. Stampfli, Sol. State. Comm. 64, 1359 (1987)
38. L.N. Bulaevskii, A.I. Buzdin, M.L. Kuli´c, Phys. Rev. B34, 4928 (1986)
39. H.S.Greenside,E.I.Blount,C.M.Varma,Phys.Rev.Lett.46, 49 (1981); M. Tachiki, Physica B 109-110, 1699 (1982)
40. J.W. Lynn, J.A. Gotaas, R.W. Erwin, R.A. Ferrell, J.K. Bhattacharjee, R.N. Shelton, P. Klavins, Phys. Rev. Lett. 52,
133–136(1984)
41. L.N. Bulaevskii, A.I. Buzdin, M.L. Kuli´c,S.V.Panyukov,J.LowTemp.Phys.59, 487(1985)
42. F.S. Bergeret, K.B. Efetov, A.I. Larkin, Phys. Rev. B62, 11872 (2000)
200 M.L. Kuli´candA.I.Buzdin
43. L.N. Bulaevskii, A.I. Buzdin, M.L. Kuli´c, Solid State Comm. 56, 351(1985)
44. Yu.V. Kopaev, Fizika tverd. Tela B7, 2907 (1965) (in Russian)
45. M.L.Kuli´c, Phys. Lett.A83, 4928 (1981)
46. A.I.Buzdin,L.N.Bulaevskii,S.V.Panyukov,JETP87, 299 (1984)
47. M. Baltensperger, S. Str¨assler, Phys. Kondens. Mat.1, 20 (1963)
48. G.Zwicknagel,P.Fulde,Z.Phys.43, 23 (1981)
49. A.I.Buzdin,L.N.Bulaevskii,Sov.Phys.Uspekhi,29, 412 (1986)
50. D.Rainer,Z.Phys.252, 174 (1972)
51. M.I. Nass,K. Levin, G.S. Grest, Phys.Rev. Lett. 46, 614 (1981)
52. A.I. Morozov, Fiz. Tverd. Tela (in Russian), 22, 3372 (1980)
53. A.I.Buzdin,L.N.Bulaevskii,S.S.Krotov,Sol.StateComm.48, 719 (1983)
54. N.R.Werthamer, E. Helfland, P.C.Hohenberg, Phys. Rev. 147, 295 (1966)
55. L.W.Gruenberg,L.Gˇunther, Phys. Rev. Lett. 16, 996 (1966)
56. P. Fulde, R.A. Ferrell,Phys.Rev. 135, A550 (1964)
57. A.I. Larkin,Yu.N. Ovchinikov, Zh. Eksp. Teor. Fiz. 47, 1136 (1964) (Sov. Phys.JETP 20, 762 (1965))
58. G.W.Crabtree,F.Behroozi,S.A.Campbell,D.G.Hinks,Phys.Rev.Lett.49, 1342 (1982)
59. L.N. Bulaevskii, A.I. Buzdin, M.L. Kuli´c, Solid State Comm. 41, 309 (1982); Phys.Lett. 85A, 161(1981)
60. H.R. Ott,W.A.Fertig, D.C.Johnston, M.B. Maple, B.T. Matthias,J.Low Temp.Phys.33, 159 (1978)
61. S. Kirkpatrick, Rev. Mod. Phys. 45, 574(1973)
62. K.H. Bennemann,A.I.Buzdin, M.L. Kuli´c, J. Low Temp. Phys. 66, 1(1987)
63. L.N. Bulaevskii,V.V. Kuzii, A.A. Sobyanin, JETP Letters 25, 290 (1977) ; ibid, Solid State Commun. 25, 1053 (1978)
64. F.S. Nogueira, K.-H. Bennemann,Eur. Phys. Lett. 67, 620 (2004), 67 (2005)
65. I.F. Lyuksyutov,V.L. Pokrovsky,Adv. Phys.54, 67 (2005)
66. L.G.Aslamazov, Sov. Phys. JETP 28, 773–775 (1969)
67. A.I.Buzdin,M.Y.Kuprianov,JETPLett.52, 487–491 (1990)
68. K. Nakanishi,K. Maki, Prog. Theor. Phys. 48, 143 (1972)
69. A.I. Buzdin, H. Kachkachi, Phys. Lett. A 225, 341 (1997); M. Houzet, A.I.Buzdin, Phys.Rev. B63, 184521 (2001)
70. A.I.Buzdin,M.L.Kuli´c, J. Low Temp. Phys. 54, 203 (1984)
71. A.I. Buzdin, Phys. Rev. B62
, 11377 (2000)
7
2. T.Kontos,M.Aprili,J.Lesueur,XGrison,X.Phys.Rev.Lett.86, 3043 (2001)
73. Z. Radovi´c,M.Ledvij,L.Dobrosaljevi´c-Gruji´c,A.I.Buzdin,J.R.Clem,Phys.Rev.B44, 759-764 (1991)
74. J.S. Jiang, D. Davidovic,D.H. Reich, C.L. Chien, Phys.Rev. Lett. 74, 314 (1995)
75. V.V. Ryazanov,V.A. Oboznov, A.S. Prokof’ev,V.V. Bolginov,A.K. Feofanov, J. Low Temp. Phys. 136, 385 (2004)
76. Ya.V.Fominov,N.M.Chtchelkatchev,A.A.Golubov,Phys.Rev.66, 014507 (2002)
77. V.V.Ryazanov,V.A. Oboznov, A.Yu. Rusanov,A.V.Veretennikov,A.A. Golubov, J.Aarts, Phys. Rev.Lett. 86, 2427 (2001)
78. T.Kontos, M.Aprili, J. Lesueur, F. Genet, B. Stephanidis, R. Boursier, Phys. Rev. Lett. 89, 137007 (2002)
79. Y.Blum,A.Tsukernik,M.Karpovski,A.Palevski,Phys.Rev.Lett.89, 187004 (2002)
8 0. V. A . O boz n ov, V. V. B ol g in ov, A .K. Feo fan ov, V. V. Ryazanov, A . I. Buzd in, P hy s . Rev. Lett. 96, 197003(2006)
81. A.A.Golubov,M.Yu.Kupriyanov,E.Il’ichev,Rev.Mod.Phys.76,411 (2004)
82. L.N. Bulaevskii,V.V. Kuzii, A.A. Sobyanin, JETP Lett. 25, 290–294 (1977)
83. P.G. de Gennes, Phys. Lett. 23, 10 (1966)
84. J.J. Hauser, Phys. Rev. Lett. 23, 374 (1969)
85. G. Deutscher, F. Meunier, F. Phys. Rev. Lett. 22, 395 (1969)
86. L.R.Tagirov, Phys. Rev. Lett. 83, 2058 (1999)
87. A.I. Buzdin, A.V.Vedyaev, N.V. Ryzhanova, Europhys. Lett. 48, 686 (1999)
88. J.Y. Gu, C.-Y.You,Y.S. Jiang, J. Pearson,Ya.B. Bazaliy, S.D. Bader, Phys. Rev. Lett. 89, 267001 (2002)
89. I.C. Moraru,W.P. Jr.Pratt, N.O. Birge N.O., Phys. Rev. Lett. 96, 037004 (2006)
90. M.L.Kuli´c, M. Endres, Phys. Rev. B62, 11846 (2000)
91. Z. Pajovi´c, M. Boˇzovi´c, Z. Radovi´c, J. Cayssol and A.I. Buzdin, Phys.Rev. B74, 184509(2006)
92. A.I. Buzdin,A.S. Melnikov, Phys. Rev. B67, R020503 (2003)
93. E.A.Demler, G.B.Arnold, M.R. Beasley, Phys. Rev. B55, 15174 (1997)
5 Theory of Superconducting Alloys
L. P. Gor’kov Florida State University, National High Magnetic Field Laboratory, Tallahassee, USA
5.1 Introduction ..............................................................................201
5.2 Averages of Green Functions over Impurities .................................................202
5.2.1“Cross-Technique”inNormalStateMetals................................................202
5.2.2GeneralizationtoSuperconductors......................................................205
5.2.3PropertiesofHomogeneousSuperconductingAlloys......................................207
5.3 Superconducting Alloys with a Small Gap ....................................................208
5.3.1TheSuperconductingTransitionTemperature............................................208
5.3.2OrdinaryImpurities...................................................................209
5.3.3Spin-OrbitScattering..................................................................210
5.3.4ParamagneticImpurities...............................................................211
5.3.5ImpuritiesandCrystallineAnisotropy...................................................212
5.3.6Ginzburg–LandauEquationsforSuperconductingAlloys..................................213
5.3.7UpperCriticalFieldfor“DirtyAlloys”...................................................215
5.4 Paramagnetic Alloys and Gapless Superconductivity ..........................................216
5.5 Eilenberger Equations ......................................................................218
5.5.1Quasi-ClassicalApproximation.........................................................218
5.5.2DerivationoftheEilenbergerEquations.................................................219
5.5.3“Dirty”Alloys:UsadelEquations........................................................223
5.6 Final Remark ..............................................................................223
References................................................................................224
5.1 Introduction
Various lattice defects (dissolved foreign atoms, va-
cancies, atomic substitutions, etc.) in normal metals
determine the so-called residual resistance. The lat-
ter is characterized by the corresponding mean free
path,,and for many physical phenomena studied in
thephysics of normal metalsthe longer themean free
path is, the more pronounced are most of the physi-
cal effects. For instance, to observe galvanomagnetic
phenomena in normal metals very pure and perfect
crystal samples are necessary.In other words,defects
are important for studying normal phase properties,
buttheir role is oftena negative one.The roleof static
defects in superconductors can be quite diverse. On
the one hand, electron scattering by randomly posi-
tioned defects breaks down the spatial correlation of
electrons forming a Cooper pair. The characteristic
length scale for the latter is
0
v
F
/2T
c
1
.Thus
one expects that the role of defects becomes impor-
tant for ≤
0
. On the other hand, such scatter-
ing does not always have a destructive character and
1
Mostlywe use units with =1, c =1( is Planck constant,c is speed of light). However,in some cases, especially when
it is desirable to emphasize the gauge invariance of a result obtained, c and sometimes appear explicitly in formulae.
202 L.P. Gor’kov
may result in important changes of superconducting
characteristics and even in the appearance of qual-
itatively new phenomena which may not be present
in the pure superconductor.Forexample,an increase
of the critical magnetic fields may occur in alloys.
Also one may have the so-called“gapless”state in su-
perconductors with paramagnetic impurities.There-
fore, the theory of superconductors with impurities,
called “superconducting alloys”, or “dirty supercon-
ductors”, constitutes an essential part of our under-
standing of the phenomenon of superconductivity.
The theoretical methods and results of this chap-
ter were mainly developedfor superconductors of the
“old generation”, the so-called “ordinary, or classical
superconductors”. In other words, they utilize ideas
and approaches of the microscopic weak coupling
BCS theory (for the phonon mechanism, this can be
extended to the strong coupling regime by making
use of the Migdal–Eliashberg approximation). Since
the discovery of high temperature superconductiv-
ity in cuprates by Bednorz and M¨uller in 1986, other
newsuperconductors havebeen foundsomeof which
may also not belong to the same class as the “old-
fashioned” superconductors. Undoubtedly cuprates
are strongly correlated electronic systems; although
there is no consensus yet regarding a theory explain-
ing their superconducting properties. In spite of this,
the treatment of defects in such superconductors re-
mains based on the use of the classical techniques.
While various mechanisms of superconductivity in
the new materials are still being debated, the basic
idea of forming the Cooper pair condensate appar-
ently remains correct in most of the recently discov-
ered superconductors. For this reason many theoret-
ical predictions and results known for ordinary su-
perconductors continue to be of interest for studying
the physics of High-T
c
and other new materials.
This chapter is not intended to provide an exhaus-
tive coverage of all the results accumulated since the
publication of the microscopic BCS-theory in 1957.
The disorder inevitably present in the doped High-
T
c
cuprates, has resulted in an enormous amount
of theoretical activity on the role played by defects
in anisotropic superconductivity such as the d-wave
superconducting order parameter now commonly
accepted for cuprates. In what follows we describe
the basic theoretical approaches and methods devel-
oped so far for treating defects.The classical theories
have a well defined range of applicability within the
framework of the old microscopic theory. Whether
it is still true for superconductivity in cuprates or in
heavy-fermion/mixed valence materials is an open
question. Nevertheless, it should be mentioned once
again that, so far, there is no alternative for the old
tools.
Below we begin with a discussion of the diagram-
matic“cross”technique. This is the method by which
most of the qualitative theoretical results were first
obtained. We then introduce the more powerful and
efficient approach of the so-called quasi-classical
Green functions followed by a summary of the prin-
cipal results.
5.2 Averages of Green Functions
Over Impurities
5.2.1 ``Cross-Technique´´ in Normal State Metals
In the case of normal metals one is mainly interested
in the role the impurities play for kinetic (transport)
properties. In the framework of the Landau Fermi
Liquid (FL) theory the familiar transport equations
for free quasi-particles provide an adequate math-
ematical description. The applicability of the latter
rests on the quasi-classical character of electronic
motion in metals for p
F
1, where p
F
is a scale for
the Fermi momentum and is the electronic mean
free path. In the superconducting phase excitations
have a more complex character. The Cooper channel
pairing introduces a new energy scale, T
c
,thetem-
perature of the phase transition, or equilivantly the
energy gap, (0), and the spatial coherence length,
0
= v
F
/2T
c
. Therefore, to account for the exis-
tence of these new scales different approaches be-
come necessary. Diagrammatic methods turn out to
be the most convenient tool to include scattering by
impurities into the theory of superconductivity. To
begin with, let us first consider the so-called “cross-
technique” for static defects in normal metals (A.A.
Abrikosov and L.P. Gor’kov, 1958 [1]; S.F. Edwards,
1958 [2]. It goes without saying that both methods
are completely equivalent). The diagrammatic tech-
5 Theory of Superconducting Alloys 203
nique reproduces all the well-known results obtained
by the standard kinetic equation approach.
Let U (r−r
a
) be the potential energy for an electron
in the field of a defect positioned at the point r
a
.The
Hamiltonian of electrons interacting with impurity
atoms in the second-quantized form is
ˆ
H
int
=
a
U(r − r
a
) ˆ
†
˛
(r) ˆ
˛
(r)d
3
r . (5.1)
The summation in Eq. (5.1) runs over all defects. For
the positions r
a
of the defects we assume the random
(Poisson) distribution, i.e. there are no correlations
between different impurity positions.
In what follows we employ the diagrammatic
methods associated with the thermodynamic Green
function
G
˛ˇ
(x, x
)=−
ˆ
T
ˆ
˛
(x), ˆ
†
ˇ
(x
)
(5.2)
expressed in terms of the field operators, ˆ
˛
(x),
ˆ
†
ˇ
(x
), written in the so-called Matsubara represen-
tation:
ˆ
˛
(x)=exp
ˆ
H
ˆ
˛
(r)exp
−
ˆ
H
,
ˆ
†
ˇ
(x)=exp
ˆ
H
ˆ
†
ˇ
(r)exp
−
ˆ
H
, (5.3)
where (x = r, )and, the “imaginary time”, varies
in the interval ∈ (0, 1/T). Notice that the ˆ (x),
ˆ
†
(x)-operators are no longer Hermitian conjugates.
However, the notation Eq. (5.3) causes no misunder-
standing below.The symbol
ˆ
T implies the chronolog-
icalordering ofoperatorswithrespectto (,
),while
... denotes the thermodynamic (Gibbs) average.
The Hamiltonian,
ˆ
H, is routinely sub-divided into
the free-electron term,
ˆ
H
0
− N with and N stand-
ing for the chemical potential and the total number
of particles, respectively, and an interaction Hamil-
tonian
ˆ
H
int
.Wehave
ˆ
H =
ˆ
H
0
− N +
ˆ
H
int
. (5.4)
In our case
ˆ
H
int
is given by Eq. (5.1).
The diagrammatic methods, including the ther-
modynamic technique, are described in a number of
good textbooks [3,4]). It is assumed that the reader
is familiar with them. In this section, we recount only
some details relevant for the impurity problem.
The Green function Eq. (5.2) in the momentum
and frequency representation is taken in the form
G
˛ˇ
(x, x
)=(2)
−3
T
!
n
G
˛ˇ
(p, p
; !
n
) (5.5)
× e
ip·r−ip
·r
e
−i!
n
d
3
pd
3
p
with !
n
=(2n +1)T, ≡ −
. G
˛ˇ
(x, x
)de-
pends both on r and r
in the presence of impuri-
ties. The methods developed provide recipes for cal-
culating the physical characteristics averaged over
random positions of impurities. The Green function
G
˛ˇ
(p, p
; !
n
)canbeformally expanded in powers
of
ˆ
H
int
Eq. (5.1), as shown in Fig. 5.1. Each cross
in Fig. 5.1 symbolizes the Fourier component of the
impurities potential Eq. (5.1). The diagrammatic ex-
pansioninFig.5.1canberewrittenasanintegral
equation for G
˛ˇ
(p, p
; !
n
)=ı
˛ˇ
G(p, p
; !
n
):
G(p, p
; !
n
)=ı(p − p
)G
0
(p, !
n
)+
1
(2)
3
a
G
0
(p; !
n
)
×
U(p − p
)e
i(p−p
)r
a
G(p
, p
; !
n
)d
3
p
(5.6)
with
G
0
(p, !
n
)=(i!
n
− )
−1
(5.7)
( = v
F
(p − p
F
)). We consider here the simplest case
of scattering potentials without spin dependence. To
average the Green function
G(p, p
; !
n
)=G(p; !
n
)ı(p − p
) , (5.8)
consider a few diagrams in Fig. 5.1. For the single
cross in Fig. 5.1 the matrix element is
Fig. 5.1. Formal expansion of the Green
function in powers of the impurity po-
tential
204 L.P. Gor’kov
1
(2)
3
a
U(p − p
)e
i(p−p
)r
a
.
Since the impurity atoms are distributed randomly,
one has to average any expression over the positions
of each impurity site. We assume that the sizes of
defects are of atomic dimensions and that the con-
centration of defects is low enough to allow averaging
to be carried out over distances much greater than
inter-atomic distances. In metals the characteristic
values of momenta in Eqs. (5.6)–(5.8) are given by
the Fermi momentum,i.e. electronic wavelengths are
of the order of an atomic distance. Averaging gives
then the constant
nU(0) ,
where n is the impurity concentration.Thisterm can
be interpreted as a correctionto the chemical poten-
tial. The last diagram in Fig. 5.1 contains two matrix
elements of the same form with the sum over (r
a
, r
b
).
If a = b, the same averaging procedure over r
a
and r
b
results in the product (nU(0))
2
. However, for a = b
the resulting average is
n
|U(p − p
)|
2
G
0
(p, !
n
)
d
3
p
(2)
3
=
nmp
F
(2)
3
|U(p − p
)|
2
dd§
p
i!
n
−
. (5.9)
Convergence of the integration over the energy vari-
able, , is of course guaranteed by the momentum
dependence of U(p − p
). However, the contribution
to the integral which comes from an integration over
the values of momenta far from the Fermi surface
gives a constant. This, like the expression nU(0), can
be included in the definition of the chemical po-
tential.A nontrivial (imaginary) contribution arises
only in the vicinity of the Fermi surface in spite of a
formally divergent term of the integral in Eq. (5.9).
Thus, one gets
−isign !
n
1
2
,
where
1
=
nmp
F
(2)
2
|U()|
2
d§
is the inverse collision time resulting for the impurity
potential in the Born approximation.
Diagrams with more crosses do not produce any
structurally new terms. Besides the contributions
whichcome from different impuritiesfor each“cross”
involving the matrix elements, and all possible pairs.
There are of course higher order corrections to the
Born scattering amplitudes. Also interference terms
occur. The latter ones represent contributionsdue to
scattering by two or more different defects.As an ex-
ample, consider a few of the fourth order diagrams
Fig. 5.2. Different contributions to the“self-energy” parts resulting from an averaging of the fourth order terms in Fig. 5.1
5 Theory of Superconducting Alloys 205
shown in Fig. 5.2. The dashed lines connecting dif-
ferent crosses denote possible ways of selecting pairs
of atoms in the product of the four matrix elements.
Thus, Fig. 5.2(a,b) give nothing but corrections to
the Born expressions. The contribution to the ma-
trix element of the form shown in Fig. 5.2(c) is pro-
portional to n
2
and describes mixed (interference)
scattering processes by two different (a = b)de-
fects. Diagrammatically the latter can also be viewed
as a “vertex” correction to the “self-energy” part of
Fig. 5.3. Although Fig. 5.2(c) describes the matrix el-
ement which contains three G
0
lines integrated over
two independent momenta q
1
, q
2
,itcanbeneglected
because thedenominators of thethree internal Green
functions are only small when the values of all mo-
menta lie close to the Fermi surface. This may hap-
pen only inside a narrow fraction of the phase space.
At the same time contributions such as shown in
Fig. 5.2(d,e) formally provide an expansion in pow-
ers of 1/T and must be accounted for in the aver-
aged Green function for T ∼ 1/.A summation over
all “non-intersecting” diagrams is given by the fol-
lowing nonlinear equation for the averaged function
G(p; !
n
):
G(p; !
n
)=G
0
(p; !
n
)+G
0
(p; !
n
)
n
(2)
3
×
|U(p − p
)|
2
G(p
; !
n
)d
3
p
× G(p; !
n
) . (5.10)
The same arguments as we have already used for
the evaluation of the “self-energy” allow one to write
down for G(p; !
n
)thesolution
G(p; !
n
)=[i!
n
− + isign!
n
/2]
−1
.
It is easy to verify that for instance the “self-energy”
part shown in Fig. 5.2(d) is in our approximation
identically zero. Equation (5.10) in its diagrammatic
form is shown in Fig.5.4. The dotted line connecting
Fig. 5.3. “Self-energy” part in the Dyson equation for the
average electronic Green function
two crosses and carrying the momentum q corre-
sponds to the factor n|U (q)|
2
in Eq. (5.10).
Having developed the above diagrammatic tech-
nique for calculating the averaged Green functions,
one may apply it to other correlators as well. We will
not describe here its applications to physical phe-
nomena in normal metals. As was emphasized from
the very beginning the technique is applicable for
quasi-classical electrons (i.e. p
F
1), but it does
not produce any results which could not be obtained
by the standard methods.
5.2.2 Generalization to Superconductors
The diagrammatic cross technique can easily be ex-
tended to superconductors. As discussed in Chap-
ter 2, the superconducting phase is characterized by
the normal, G(x, x
) and the two anomalous Gor’kov
Green functions (L.P. Gor’kov, 1958 [5])
F
˛ˇ
(x, x
)=
ˆ
T
ˆ
˛
(x) ˆ
ˇ
(x
)
,
F
†
˛ˇ
(x, x
)=
ˆ
T
ˆ
†
˛
(x) ˆ
†
ˇ
(x
)
. (5.11)
In the absence of impurities (and any external fields)
we will assume that all the functions depend only
on the coordinate differences (x − x
) and can be ex-
pressed in the momentum and frequency represen-
tation in the standard manner. Scattering by static
defects breaks only the spatial homogeneity. For the
Green functions we again write
Fig. 5.4. Diagrammatic illustration of
the Dyson equation (5.10) for the av-
erage electronic Green function in the
normal phase
206 L.P. Gor’kov
G
˛ˇ
(x, x
)=T
!
n
G
˛ˇ
(p, p
; !
n
) (5.12)
× e
ip·r−ip
·r
e
−i!
n
(−
)
d
3
pd
3
p
(2)
3
(and similarly for the two anomalous functions). If
we neglect the effects of the spin-orbit interactions
and assume non-magnetic defects, the spin indices
can be omitted; the ordinary Green function is then
diagonal in (˛, ˇ), G
˛ˇ
∝ Gı
˛ˇ
,whilethefunctions
F
˛ˇ
,F
†
˛ˇ
areproportional to the spinor metric tensor
ˆ
I
˛ˇ
.Thus,
F
˛ˇ
=−F
ˆ
I
˛ˇ
; F
†
˛ˇ
= F
†
ˆ
I
˛ˇ
, (5.13)
where
ˆ
I
˛ˇ
≡ i(
y
)
˛ˇ
=
01
−1 0
˛ˇ
is nothing but the antisymmetric (singlet) spin wave
function of the two electrons forming the Cooper
pair (if not said otherwise,we assume s-wave pairing
below).
Let us consider again a random, homogeneous
distribution of impurities. The full equations for
G(r, r
; !
n
)andF(r, r
; !
n
) are of the form
1
i!
n
−
ˆ
H
0
+ −
a
U(r − r
a
)
2
G(r, r
; !
n
)
+ (r)F
†
(r, r
; !
n
)=ı(r − r
) ,
1
−i!
n
−
ˆ
H
0
+ −
a
U(r − r
a
)
2
F
†
(r, r
; !
n
)
−
∗
(r)G(r, r
; !
n
)=0. (5.14)
Here,
ˆ
H
0
is the free electron operator. As we plan to
comment below on the role of crystalline anisotropy,
ˆ
H
0
is left unspecified. In the weak coupling limit one
may write (r)and
∗
(r)as
(r)=
T
!
n
g(r − r
)F(r, r
; !
n
)d
3
r
,
∗
(r)=
T
!
n
g(r − r
)F
†
(r, r
; !
n
)d
3
r
. (5.15)
Here, g(r − r
) is a short-ranged interaction. For the
isotropic model
g(r − r
)=gı(r − r
) (5.16)
(if crystalline anisotropy is not specifically men-
tioned, we always assume g(r − r
) has the “contact”
form, Eq. (5.16)). Equations (5.14) must be supple-
mented by equations determining F(r, r
; !
n
). For
completeness we shall later write down explicitly the
whole set of Gor’kov equations. However, at least
in the absence of an external magnetic field, both
anomalous functions can be expressed in terms of
each other using their definitions and their symme-
try properties.
Before proceeding to discuss the averaging of
the Green functions G(r, r
; !
n
), F(r, r
; !
n
)and
F
†
(r, r
; !
n
) by means of the cross-technique, it is
important to discuss whether one can use the aver-
aged values
¯
and
¯
∗
for (r)and
∗
(r). When an
impurity is introduced into a superconductor, (r)
is expected to experience local changes. This change
of course depends on the strength of the impurity
potential. However, local variations of (r)rapidly
decrease away from an impurity center. Therefore,
we anticipate that the order parameter, (r), is a self-
averaging quantity, at least at low enough impurity
concentrations. It will be seen from the results below
that the assumption (r)
¯
indeed turns out to be
valid.
Taking again the impurity Hamiltonian Eq. (5.1)
as the perturbation for the field operators Eq. (5.3),
or merely expanding the equations in terms of
a
U(r − r
a
), one obtains an expansion for each of
the two Green functions which is formally similar
to the one shown in Fig. 5.1. The major difference is
that for the Green functions G
0
(r − r
; !
n
) connect-
ing the crosses in Fig. 5.1 can now alternate with the
bare functions F
0
(r − r
; !
n
)andF
†0
(r − r
; !
n
)of
the pure superconductor.Theabove lineof argument
(including theneglect ofcrossing diagrams suchasin
Fig. 5.2(c)) can be repeated in writing the equations
for the averaged G(p; !
n
), F(p; !
n
)andF
†
(p; !
n
).
One then obtains the two equations shown schemat-
ically in Fig. 5.5(b), where double lines stand for the
averaged (exact) functions G, F and F
†
(Fig. 5.5(a)).
Writing them in a more convenient and compact
form with the help of the Gor’kov equations in the
absence of impurities leads to