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Theoretical consequences of the correlation of the Cooper pairs in the narrow band electron systems
Physica B 165&166 (1990) 1087-1088 North-Holland
THEORETICAL CONSEQUENCES OF THE CORRELATION OF THE COOPER PAIRS IN THE NARROW BAND ELECTRON SYSTEMS
Shun-ichiro KOH Physics Division,Faculty of Education,Kochi University,2-5-1,Akebono-cho,Kochi City 780, Japan
Interference and consequent breakdown of the electron-hole symmtry due to the correlation of the Cooper pairs in the narrow band electron systems are investigated.
1. INTRODUCTION Recently superconductive phases are found in the somewhat exotic materials (1)(2](3], where it's band width is so narrow and the band electron has so large density of states that, in it's superconductive phase, Cooper pairs are colliding to each other bitterly through the Coulomb repulsive interaction. The standard treatment (4](5) of the Coulomb effect on the superconductivity until now is the reduction scheme due to the scale difference between the phonon-mediated attractive frequency and the band width. In the above extreme situation with the high density of states, however, it would be worthwhile extending the theory to include , in addition to the normal electron process, the Coulomb interaction mediated by the Cooper pair theirself. Of course, simple order estimation tells us that, to realize such a situation, the following condition should be fulfilled, g-U~
---- U 2 ( D(£F»)2(U)2 2V "F
2. THE CORRELATION OF COOPER PAIRS The natural extension of the simple bubble graph to the superconducting electron is expressed as follows,
q,.,)-~f ~3G( k+q,.,+", i) r3G( k,., i )dk
(2)
where G is a 2x2 Green function matrix in the Nambu representation (6]. To classify X(q,,,,) according to it's symmetry, we will expand it with the Pauli matrixes as follows, (3)
Each coefficient x(i)(q,,,,) have the forms obtained by using the corresponding formula of the normal electron, setting a zero energy level to the Fermi level, and replacing the free electron energy E k to the Bcgolon energy E . k
0921-4526/90/$03.50
© 1990 -
X 10'; l
k +q
or
>1 2 + I1 k k+q
2
(4a)
+2
I
+q
>1 2 + I1 k k+q
2
(4b)
-2 X 111; !:
k +q
>
>+
k +q -
k k +q
>
U
k k+q
>
- +
The purpose of this paper is to investigate, once such a condition is fulfilled, what 'will happen to it's superconductivity?
X(
One essential difference to the normal electron case is existance of prefactors in each T process. These prefactors have the following forms,
or
k +q
>
k k+q
>-
k k+q
>
k k+q
>
(5)
(6)
21 12_ 2! 12 k k q
(7a)
21 12_ 21 1 k q
(7b)
2
where denotes,ukv + ' and so on. k In the above expresslons, ~he form of (a)'s correspond to the Fermi level crossing processes and (b)'s to the non crossing processes. In Eqs. (4),(5) and (6), there appear various types of interference terms between the occupied pairoccupied pair excitation, the empty pair-empty pair one, and the occupied pair-empty pair one. (See Fig.l.) In the normal phase, these interferences will disappear. T and T process therefore, reduce to the ngrmal etectron excitation, while, since
~
and \
process
consist
of the interferences oflly, th~se processes theirself disappear. '1 and, process are direct"reflection of the quantum n~ture of the superconductivity, and there exists no analogue in the normal electron case.
3. THE EXPRESSIONS of X("') AT THE ZERO TEMPERATURE To obtain the frequency dependence of 6(",) in Sec.4, we should get the shape of the frequency
Elsevier Science Publishers B.V. (North-Holland)
S.-i. Koh'
1088
(S}(2)-CS>(2) (Uk'U!t<')
I X
.J
(S}CZ}--CS)qQ)
(b)
Fig.l Possible combinations of the interferences between various excitations of Cooper pairs
Fig.3 The gap spectrum by Eqs.(9) and (10), setting (D/2V)'U'/(g-U) is 18.7. (a) the case of X'''=O, and (b) x"'"O.
dependence of the space averaged X(q,w) by using Eq.(2). Although there is no space to explain it's proceedure in details, we can obtain their shape by using analytic and numerical integration of k and q variables in Eq.(2). 4. THE SELF CONSISTENT EQUATION AT THE ZERO TEMPERATURE To elucidate qualitative feature of.this system as simply as possible, we assume a simple electronhole symmetric band in the normal phase. To investigate the gap spectrum ,under. the condition of Eq.(l) , we should include all possible X process in Eq. (3) as follows (Fig.2),
xJ(s -
XCk, .. ) = i
U -
xCk-q,,,-zl)r3G(q,zlr3~' (~)
(Since, in the symmetric band, the prefactor of x12 ' at the zero temperature becomes zero, X'" is dropped in Eq.(8).) On the other hand, in t and G (not in X), " term is dropped, while, since the order parameter A=A ±A 1 2 is the quantity to be determined by Eq.(8), '1 and' term should be included 2 The algebra of pauli matrix enebles us to connect the all possible interaction channels to the order parameter automatically and produce the following coup lied equation of A and A ' l 2 A, (w)= O(tlJdz(g-U-U'xlO'(w-z) 2V +
)Rer~l ,If(Z)
O(tliJdZU'X"'(w-z)Re[~ J 2V
(9)
~
+
/!Q;\
,
,
hUllllunna 0'
Fig.2 The diagram of the self consistent equation. G is nonlinear in the third term of the right side.
+ D(E)
2V
f dzU'x"'(w-z)Re[-z--] If(z)
A,(w)=
where
and the imaginary signs represent the phase angle of the order parameter A. Using the non perturbative value (A =ReA ,A =0) l l 2 we can obtain the solution by iterative procedure. The absence of inhomogenious term for ImA in Eq. l (9) , and for ReA in Eq. (10) ensure us to get the 2 following real, but asymmetric type of solution, Redlrl+iIm62r2
(
0
,Redl-rm02)
Re~ l+rm~ 2,
0
(11)
5. CONCLUSION We explored the highly self correlated superconducting electron system by the Coulomb interaction, and concluded that the symmetry breaking of the electron-hole symmetry can be possible due to the cooperation effect in the self energy, that is, the nonlinearity of the Green function matrix in the self consistent equation depicted in Fig.2. References (1) For example,S.V.Vonsovsky,Y.A.Izyumov and E.Z.Kuramaev, Superconductivity of Transition Metals (Springer,1982) [2) For example,G.R.Stewart, Rev.Mod.Phys.56 755(1984) (3) For example,S.Kagoshima,N.Nagasawa and T.Sambongi, One-Oimmensional Condactor (Springer,1988) (4) N.N.BogoliuboVi Nuovo Cimento,7,6,794(1958) [5] P.Morel and P.W.Anderson, Phys.Rev.125,1263 (1962) [6J Y.Nambu, Phys.Rev,117,648(1960)
THEORETICAL CONSEQUENCES OF THE CORRELATION OF THE COOPER PAIRS IN THE NARROW BAND ELECTRON SYSTEMS
Shun-ichiro KOH Physics Division,Faculty of Education,Kochi University,2-5-1,Akebono-cho,Kochi City 780, Japan
Interference and consequent breakdown of the electron-hole symmtry due to the correlation of the Cooper pairs in the narrow band electron systems are investigated.
1. INTRODUCTION Recently superconductive phases are found in the somewhat exotic materials (1)(2](3], where it's band width is so narrow and the band electron has so large density of states that, in it's superconductive phase, Cooper pairs are colliding to each other bitterly through the Coulomb repulsive interaction. The standard treatment (4](5) of the Coulomb effect on the superconductivity until now is the reduction scheme due to the scale difference between the phonon-mediated attractive frequency and the band width. In the above extreme situation with the high density of states, however, it would be worthwhile extending the theory to include , in addition to the normal electron process, the Coulomb interaction mediated by the Cooper pair theirself. Of course, simple order estimation tells us that, to realize such a situation, the following condition should be fulfilled, g-U~
---- U 2 ( D(£F»)2(U)2 2V "F
2. THE CORRELATION OF COOPER PAIRS The natural extension of the simple bubble graph to the superconducting electron is expressed as follows,
q,.,)-~f ~3G( k+q,.,+", i) r3G( k,., i )dk
(2)
where G is a 2x2 Green function matrix in the Nambu representation (6]. To classify X(q,,,,) according to it's symmetry, we will expand it with the Pauli matrixes as follows, (3)
Each coefficient x(i)(q,,,,) have the forms obtained by using the corresponding formula of the normal electron, setting a zero energy level to the Fermi level, and replacing the free electron energy E k to the Bcgolon energy E . k
0921-4526/90/$03.50
© 1990 -
X 10'; l
k +q
or
>1 2 + I
2
(4a)
+2
I
+q
>1 2 + I
2
(4b)
-2
k +q
>
>+
k +q -
k k +q
>
U
k k+q
>
The purpose of this paper is to investigate, once such a condition is fulfilled, what 'will happen to it's superconductivity?
X(
One essential difference to the normal electron case is existance of prefactors in each T process. These prefactors have the following forms,
or
k +q
>
k k+q
>-
k k+q
>
k k+q
>
(5)
(6)
21
(7a)
21
(7b)
2
where
~
and \
process
consist
of the interferences oflly, th~se processes theirself disappear. '1 and, process are direct"reflection of the quantum n~ture of the superconductivity, and there exists no analogue in the normal electron case.
3. THE EXPRESSIONS of X("') AT THE ZERO TEMPERATURE To obtain the frequency dependence of 6(",) in Sec.4, we should get the shape of the frequency
Elsevier Science Publishers B.V. (North-Holland)
S.-i. Koh'
1088
(S}(2)-CS>(2) (Uk'U!t<')
I X
.J
(S}CZ}--CS)qQ)
(b)
Fig.l Possible combinations of the interferences between various excitations of Cooper pairs
Fig.3 The gap spectrum by Eqs.(9) and (10), setting (D/2V)'U'/(g-U) is 18.7. (a) the case of X'''=O, and (b) x"'"O.
dependence of the space averaged X(q,w) by using Eq.(2). Although there is no space to explain it's proceedure in details, we can obtain their shape by using analytic and numerical integration of k and q variables in Eq.(2). 4. THE SELF CONSISTENT EQUATION AT THE ZERO TEMPERATURE To elucidate qualitative feature of.this system as simply as possible, we assume a simple electronhole symmetric band in the normal phase. To investigate the gap spectrum ,under. the condition of Eq.(l) , we should include all possible X process in Eq. (3) as follows (Fig.2),
xJ(s -
XCk, .. ) = i
U -
xCk-q,,,-zl)r3G(q,zlr3~' (~)
(Since, in the symmetric band, the prefactor of x12 ' at the zero temperature becomes zero, X'" is dropped in Eq.(8).) On the other hand, in t and G (not in X), " term is dropped, while, since the order parameter A=A ±A 1 2 is the quantity to be determined by Eq.(8), '1 and' term should be included 2 The algebra of pauli matrix enebles us to connect the all possible interaction channels to the order parameter automatically and produce the following coup lied equation of A and A ' l 2 A, (w)= O(tlJdz(g-U-U'xlO'(w-z) 2V +
)Rer~l ,If(Z)
O(tliJdZU'X"'(w-z)Re[~ J 2V
(9)
~
+
/!Q;\
,
,
hUllllunna 0'
Fig.2 The diagram of the self consistent equation. G is nonlinear in the third term of the right side.
+ D(E)
2V
f dzU'x"'(w-z)Re[-z--] If(z)
A,(w)=
where
and the imaginary signs represent the phase angle of the order parameter A. Using the non perturbative value (A =ReA ,A =0) l l 2 we can obtain the solution by iterative procedure. The absence of inhomogenious term for ImA in Eq. l (9) , and for ReA in Eq. (10) ensure us to get the 2 following real, but asymmetric type of solution, Redlrl+iIm62r2
(
0
,Redl-rm02)
Re~ l+rm~ 2,
0
(11)
5. CONCLUSION We explored the highly self correlated superconducting electron system by the Coulomb interaction, and concluded that the symmetry breaking of the electron-hole symmetry can be possible due to the cooperation effect in the self energy, that is, the nonlinearity of the Green function matrix in the self consistent equation depicted in Fig.2. References (1) For example,S.V.Vonsovsky,Y.A.Izyumov and E.Z.Kuramaev, Superconductivity of Transition Metals (Springer,1982) [2) For example,G.R.Stewart, Rev.Mod.Phys.56 755(1984) (3) For example,S.Kagoshima,N.Nagasawa and T.Sambongi, One-Oimmensional Condactor (Springer,1988) (4) N.N.BogoliuboVi Nuovo Cimento,7,6,794(1958) [5] P.Morel and P.W.Anderson, Phys.Rev.125,1263 (1962) [6J Y.Nambu, Phys.Rev,117,648(1960)