- Email: [email protected]
Interplay between the Anderson model of localization and BCS superconductivity: The finite band width transition
PHYSICA
Physica B 194-196 (1994) 1463-1464 North-Holland
Interplay B e t w e e n T h e Anderson M o d e l o f Localization and B C S Superconductivity" The Finite B a n d width Transition R. Rangel a and F. P. Marin b aCentro de Ffsica, Instituto Venezolano de Investigaciones Cientfficas, Apartado 21827, Caracas 1020 A, Venezuela. bFacultad de Ciencias, Universidad Central de Venezuela. We combine the tight binding Anderson model of disorder with a mean field BCS interaction. The transition temperature is obtained as a general relation which depends on the disorder averaged spectral density associated with the electron propagator between neighboring lattice sites. This relation is obtained using the locator expansion technique. As a main result we predict superconductivity in the disorder induced localized state. There is a maximun bandwidth where superconductivity is destroyed. This is consequence of the local pairing assumed in the model.. In disordered superconductors a strong reduction of the transition temperature has been observed [ 1]. This is not in contradiction with Anderson theorem valid for the case when KFI >> 1. Recently Kirkpatrick [2] used renormalization group arguments to study the disorder dependence of the mean field superconducting T c within a BCS model. We present in this paper an extension of the Wiecko and Allup model [3] to the case of a finite bandwidth. A general relation between T c and the electron propagation betweeen neighboring sites is found. This permits us to investigate the case of finite bandwidth. This is achieved by using the locator expansion technique. A very detailed exposition of extensive analytical calculations will be given elsewhere [4]. We consider a Hamiltonian that combines the Anderson model and a mean field BCS interaction H=HA + HS.
g ,c:oc,o + g tijc;ocio it
<1>
N is the number of lattice sites. We use a Cauchy distribution of {~i}
where W models disorder. H S is given by Hs = -gAZ(c~?c~'j. + Cil,Ci,[,)+gNA2 i
(3) A = 1 Z < c~Tc+~ > (4) N i A is the order parameter and g A plays the role of a BCS like gap, g is the coupling constant., <....> is the thermal average. Since the model emphasizes local pairing in configuration space it is a model for describing granular superconductors. [5]. By substituting tij as ~, tij we obtain the disorder averaged free energy as an integral over ~,: 1
i,j,o +
where cio(cio ) destroys (creates) an electron at site i with spin O and txj= t, i f i and j are nearest
P(Ei)=(W/X)/(e2+w2)
F =F0 + Etijfd~< i,j,o
CioC + jo > k
(5)
0
F00 = -2NT In 2 + NgA2
neighbors, otherwise tij=0. The on-sites electron energies { Ei} are random variables. Any function M(E1E 2 I~N) can be averaged as . . . . .
__
N
+oo
M= I-I ldeiP(ei)M(ele2...e N) i=l-,,o
(2)
T=~ -1 and h=KB=I. F00 is the zero bandwidth limit of the disorder averaged free energy [3]. We evaluate this integral and obtain
1 (tanh(13c /
='2g(.'
~
0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved
S S D I 0921-4526(93)E1312-A
2))
1464
1
--
1
(6)
2Ng Tc 5",.~.tii I d~R00 nl,j J0
I
g
i
I
ij
=
+
\
/
I
/ I \
\
+
/
I
I
i
\
/
~
I
/ll~\
I \ I
+
't
\
I
/
1
",
I I
~" I
x %
where +
R (~')= limA__>0{1 3--~Tr{crzGij (iOn)k} )
Fig. 1. Feynman Diagrams expansion disorder averaged Green's Functions.
of the
The trace in R (~,) depends on A only through the IO i
combination (o 2 +g2A2) as demonstrated in full detail [4]. Thus we find
I
\
I(0.224)
0.8
1 (tanh(13¢e l=~g~ ~ / 2)'/ J
t -~*
i~.~\
+" a tanh([3cE) + g ~ toi ! d~_j" dEpio (E'~ ~E'E{ ~'} '
w
h e r e
po(e)~
'(0.3~2)-
0
0.6
0.370) 0.386)
.............
\
Jr
From this expresion we obtain T c (A=0) which depends in a functional form from the density Pi0(E)~. We now solve the Dyson equation for the Matsubara Green's function using the locator expansion technique and then summing the class of Feynman diagrams depicted in Fig.1. We obtain 8ij + t gij = o) + iW (co+ iW) 2 - t 2
(8)
~'-0" 04
02
0.01
0.0
We use this solution to evaluate eq.(7) and obtain
1= 2g(l(O)+ 2z(I(t)-I(O)))
(9)
z is the number of nearest neighbors of any lattice site, and I(t) can be written in terms of the digamma function and this is our main result, I [1 W + i t ] ] i(t)=___W ~ ( 1 / 2 ) + 2 R e ~ l 2 + 2rcTc J~] W2+t 2 W+it |
J
(10)
,
20
i
40
l
60
W/Too Fig. 2. Variation of Tc/Tco as a function of WFFco" Tco=g/4. (W=t=0). We use Eq. 10.
References. 1. Superconductivity in d and f metals. Edited by H. Suhl and M. Mapel. (Academic, New York, 1980). 2. T. Kirkpatrick, and D.Belitz. Phys. Rev. Lett.68, 3232 (1992) 3. R. Allup, A. Caro, and C. Wiecko. J. Low Temp. Phys. 75, 27 (1989). 4. F. P. Matin, and R. Rangel ,unpublished. 5. F.P. Matin and R.Rangel, in preparation.
Physica B 194-196 (1994) 1463-1464 North-Holland
Interplay B e t w e e n T h e Anderson M o d e l o f Localization and B C S Superconductivity" The Finite B a n d width Transition R. Rangel a and F. P. Marin b aCentro de Ffsica, Instituto Venezolano de Investigaciones Cientfficas, Apartado 21827, Caracas 1020 A, Venezuela. bFacultad de Ciencias, Universidad Central de Venezuela. We combine the tight binding Anderson model of disorder with a mean field BCS interaction. The transition temperature is obtained as a general relation which depends on the disorder averaged spectral density associated with the electron propagator between neighboring lattice sites. This relation is obtained using the locator expansion technique. As a main result we predict superconductivity in the disorder induced localized state. There is a maximun bandwidth where superconductivity is destroyed. This is consequence of the local pairing assumed in the model.. In disordered superconductors a strong reduction of the transition temperature has been observed [ 1]. This is not in contradiction with Anderson theorem valid for the case when KFI >> 1. Recently Kirkpatrick [2] used renormalization group arguments to study the disorder dependence of the mean field superconducting T c within a BCS model. We present in this paper an extension of the Wiecko and Allup model [3] to the case of a finite bandwidth. A general relation between T c and the electron propagation betweeen neighboring sites is found. This permits us to investigate the case of finite bandwidth. This is achieved by using the locator expansion technique. A very detailed exposition of extensive analytical calculations will be given elsewhere [4]. We consider a Hamiltonian that combines the Anderson model and a mean field BCS interaction H=HA + HS.
g ,c:oc,o + g tijc;ocio it
<1>
N is the number of lattice sites. We use a Cauchy distribution of {~i}
where W models disorder. H S is given by Hs = -gAZ(c~?c~'j. + Cil,Ci,[,)+gNA2 i
(3) A = 1 Z < c~Tc+~ > (4) N i A is the order parameter and g A plays the role of a BCS like gap, g is the coupling constant., <....> is the thermal average. Since the model emphasizes local pairing in configuration space it is a model for describing granular superconductors. [5]. By substituting tij as ~, tij we obtain the disorder averaged free energy as an integral over ~,: 1
i,j,o +
where cio(cio ) destroys (creates) an electron at site i with spin O and txj= t, i f i and j are nearest
P(Ei)=(W/X)/(e2+w2)
F =F0 + Etijfd~< i,j,o
CioC + jo > k
(5)
0
F00 = -2NT In 2 + NgA2
neighbors, otherwise tij=0. The on-sites electron energies { Ei} are random variables. Any function M(E1E 2 I~N) can be averaged as . . . . .
__
N
+oo
M= I-I ldeiP(ei)M(ele2...e N) i=l-,,o
(2)
T=~ -1 and h=KB=I. F00 is the zero bandwidth limit of the disorder averaged free energy [3]. We evaluate this integral and obtain
1 (tanh(13c /
='2g(.'
~
0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved
S S D I 0921-4526(93)E1312-A
2))
1464
1
--
1
(6)
2Ng Tc 5",.~.tii I d~R00 nl,j J0
I
g
i
I
ij
=
+
\
/
I
/ I \
\
+
/
I
I
i
\
/
~
I
/ll~\
I \ I
+
't
\
I
/
1
",
I I
~" I
x %
where +
R (~')= limA__>0{1 3--~Tr{crzGij (iOn)k} )
Fig. 1. Feynman Diagrams expansion disorder averaged Green's Functions.
of the
The trace in R (~,) depends on A only through the IO i
combination (o 2 +g2A2) as demonstrated in full detail [4]. Thus we find
I
\
I(0.224)
0.8
1 (tanh(13¢e l=~g~ ~ / 2)'/ J
t -~*
i~.~\
+" a tanh([3cE) + g ~ toi ! d~_j" dEpio (E'~ ~E'E{ ~'} '
w
h e r e
po(e)~
'(0.3~2)-
0
0.6
0.370) 0.386)
.............
\
Jr
From this expresion we obtain T c (A=0) which depends in a functional form from the density Pi0(E)~. We now solve the Dyson equation for the Matsubara Green's function using the locator expansion technique and then summing the class of Feynman diagrams depicted in Fig.1. We obtain 8ij + t gij = o) + iW (co+ iW) 2 - t 2
(8)
~'-0" 04
02
0.01
0.0
We use this solution to evaluate eq.(7) and obtain
1= 2g(l(O)+ 2z(I(t)-I(O)))
(9)
z is the number of nearest neighbors of any lattice site, and I(t) can be written in terms of the digamma function and this is our main result, I [1 W + i t ] ] i(t)=___W ~ ( 1 / 2 ) + 2 R e ~ l 2 + 2rcTc J~] W2+t 2 W+it |
J
(10)
,
20
i
40
l
60
W/Too Fig. 2. Variation of Tc/Tco as a function of WFFco" Tco=g/4. (W=t=0). We use Eq. 10.
References. 1. Superconductivity in d and f metals. Edited by H. Suhl and M. Mapel. (Academic, New York, 1980). 2. T. Kirkpatrick, and D.Belitz. Phys. Rev. Lett.68, 3232 (1992) 3. R. Allup, A. Caro, and C. Wiecko. J. Low Temp. Phys. 75, 27 (1989). 4. F. P. Matin, and R. Rangel ,unpublished. 5. F.P. Matin and R.Rangel, in preparation.