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Superconductivity due to coulomb interactions
Physica B 165&166 (1990) 1033-1034 North-Holland
SUPERCONDUCTIVITY DUE TO COULOMB INTERACTIONS S. Kiichenhoff Physik-Department T30, Technische Universitli.t Miinchen, D-8046 Garching, FRG S. Schiller Department of Applied Physics, Stanford University, Stanford, Ca 94305, USA We solve the two coupled, frequency independent Bethe-Salpeter equations for the two particle vertex function and determine the effective interaction and quasiparticle scattering amplitude for a degenerate electron gas for 1 < r. < 70. A phenomenological ansatz is employed for the direct interaction that selfconsistently reproduces spin and charge susceptibilities and the local field factor known from microscopic theories. We find p-wave superconductivity for 10 < r. < 40 and s-wave superconductivity for r. > 40, at r. 70 the transition temperature being Tc l'::l 2.5 K and the effective mass m· /m 15. The transport parameters decrease by several orders of magnitude with decreasing density.
=
=
A longstanding problem is the role of the Coulomb potential to superconductivity. While it is commonly accepted that the Coulomb potential yields a repulsive contribution to the pairing at metallic densities (r. = 4), the situation at lower densities is unclear and received new interest since one of the intriguing properties of the high T c superconductors is the low carrier density [I]. At low densities the electron liquid undergoes two phase transitions: at r. :::l 75 from the paramagnetic to the ferromagnetic state and at r. :::l 120 from the liquid to a solid [2], indicating that correlations are most important in that density regime and may also influence superconductivity. Sham, Rietschel and Grabowski have investigated the possibility of Cooper pair. formation by exchange of plasmons [3]. Solving the Eliashberg equations with a dynamically screening Coulomb interaction they find sizable transition temperatures even in the metallic range, however these are found to be suppressed upon inclusion of lowest-order vertex corrections, a conclusion also drawn by Shirron and Ruvalds [4]. More recently Takada [5] employed a variational method to study Cooper pair formation using purely repulsive potentials. He finds that for a special form of the potential T c can be as high as 200 K for s-wave pairing, the important features being a low carrier density and a long range tail of the interaction. However the total effect of vertex corrections, the behaviour at r. > 10 and the pairing in angular momenta I", 0 remained unclear. We will approach the problem by concentrating on the effective interaction at the Fermi surface and calculate the Cooper pair interaction from the scattering amplitude. The only interaction present is the Coulomb potential and the electron liquid will be embedded in a rigid, neutralizing positive charge background. To ensure that the Pauli principle is not violated we solve the two frequency independent BetheSalpeter equations for particle-hole scattering (we allow for 0921-4526/90/$03.50
© 1990 -
scattering with momentum transfer up to 2kJ) in both p-h channels simultaneously, using the direct interaction model (for details of that model see [6]). The two equations are given as:
Akk,(q) = Fkk,(q) + LFkk,,(q);n,,(q)A~'k.(q), v = s,a, k"
Akk,(q) = Ekk'(q) + (F}kk,(q) - Ikk'(q)·
(1)
(2)
Here A is the scattering amplitude, F the generalized Landau function, I the direct interaction (irreducible in both p-h channels) and q the momentum transfer in the p-h channel. The overbar on F in (2) indicates the exchanged quantities, i.e. the variables are corresponding to the interchange of the two in-going (or out-going) particles (s,a refers to the spinsymmetric/antisymmetric components). The Landaufunction F~ contains long range and short range contributions and it is necessary for the calculations to split off the long range part: F~ = Vc(q) + F~. Vc(q) is the Coulomb potential. F~(q) is related to the local field factor G(q) in the usual way F~(q) = v,,-l(q)G(q) and accounts for the correlations in the liquid. If the direct interaction is chosen to be exchange symmetric, equations (1) and (2) conserve the two particle exchange symmetry and a newly formulated q-dependent generalized Landau sum rule which is a direct consequence ofthe Pauli principle [7,8] holds. Since I is not known accurateley in the whole density regime we choose a phenomenological ansatz for I which reproduces spin and charge susceptibilities and the local field factor. These quantities are known from Green's-functions Monte Carlo calculations [2] and microscopic theories [9]. However the spin susceptibility is not known accurately in the regime 5 < r. < 60 and we therefore used a simple formula which interpolates smoothly between the known regions [8]: at r. < 5 the RPA give correct results and
Elsevier Science Publishers B.V. (North-Holland)
1034
S. Kiichenhoff, S. Schiller
75 po: -1. According to that interpolation -0.20 (-0.42, -0.51, -0.68, -0.76, -0.81) for r.= 1 (5,10,30,50,70). The values for the effective mass for the calculation of the spin susceptibility are the calculated ones of table 1. Our results are shown in table 1. (For the calculation of T c from the scattering amplitude we used the at r.
PO :
r:;
IOrmula of Patton and Zaringhalam [10] and the cutoff energy is estimated to be the Fermi energy.)
r.
m"/m
go
gl
1 5 10 20 30 40 50 60 70
0.88 1.02 1.15 1.46 2.06 2.99 4.55 6.52 14.8
0.45 0.59 0.63 0.50 0.24 -0.02 -0.29 -0.49 -0.85
0.05 0.03 0.00 -0.03 -0.06 -0.06 -0.06 -0.05 -0.02
TF exp(l/g/)
K
8 X 10-11 , I : 1 5 X 10- 6 , I : 1 2 X 10- 5 , I : 1 1.7,1=0 3.3,1= 0 2.5, I : 0
The transport parameters may also be calculated from the scattering amplitudes. They are found [8] to change by more than 10 orders of magnitude, a behaviour which is expected for a liquid which becomes ferromagnetic and/or localised. The values for the spin diffusion constant (in K2 m 2 / s) are: DT2 : 2.0 X 107 (4.4 X 103 , 1.0 X 102 ,6.9 X 10- 2 ,3.6 x 10-4, 1.4 X 10- 6 for r.:1 (5,10,30,50,70). Part of the decrease of the transport parameters is of course due to the decrease of the Fermi velocity. In concluding we have shown that the dilute electron gas is unstable against formation of Cooper pairs, the purely repulsive Coulomb interaction providing an attractive interaction. If an additional attractive mec1Ianism is present fairly ~high transition temperatures may be reached as they have been found in the high T c superconductors. Our treatment of the Coulomb interaction includes all vertex corrections and it is seen that vertex corrections are very important for Coulombic systems - Migdal's theorem does not hold - and even at metallic densities may not be neglected. Our model may also be directly applied to systems were electronic correlations are important; if spin and c1Iarge response are known, the superconducting coupling constants may be calculated. Acknoledgements We are indebted to ·Prof. Peter Wolfle for continuous support during all stages of this work.
References Table 1: Effective mass, pair interaction constants g/ and superconductivity temperature scale TJ!) as a function of the electron density
[1] T. M. Rice, Z. Phys. B 67, 141 (1987) [2] D.M. Ceperley and B.J. Alder, Phys. Rev. Lett. 45, 566 (1980) [3] 1. J. Sham, Physica B 135,451 (1985)
The quasiparticle become very heavy as the density is lowered, in accordance with the approac1I to the transition point of solidification. Note the dip in the effective mass (m"/m: 1 for r...... 0) whic1I is also found in other calculations [11], but which is somewhat stronger in ours. At metallic densities due to the lack of a low energy mode the mass is not strongly renormalized. A~( q : 2kf) is found to increase as the densities is lowered indicating also a transition to a solid. (Due to the long range of the Coulomb interaction AMO) : 1 irrespectively of the density.) The p-wave coupling constant is negative for r. > 10 with a minnimum value of gl : -0.06. The transition temperatures are fairly small, the attractive potential is provided by spin density fluctuations as can be seen from the enhanced spin susceptibility. The s-wave coupling constant is decreasing rapidly from r. : 10 on and goes negative at r. : 40. The highest positive values of go are in the metallic density range, where a strong attractive potential has to be present in superconducting materials to overcome that repulsion due to the Coulomb interaction. The transition temperatures at lowest densities are in the range of a few Kelvins. Looking at the different contributions to go it can be shown, that the attractive mechanism is provided by transverse current fluctuations [12].
[4] J. J. Shirron and J. Ruvalds, Phys. Rev. B 34, 7596 (1986) [5] Y. Takada, Phys. Rev. B 39, 11575 (1989) [6] M. Pfitzner and P. Wolfle, Phys. Rev. B 35, 4699, (1987) [7] S. Schiller, Diplomarbeit, Technisc1Ie Universitat Miinchen, 1987 [8] S. Kiichenhoff and S. Schiller, to be published [9] KS. Singwi and M.P. Tosi, in Solid Sta.te Physics, H. Ehrenreich, F. Seitz and D. Turnbull, eds., Vol. 36, p.l77, Academic, 1981 [10] B.R. Patton and A. Zaringhalam, Phys. Lett. 45, 566 (1980) [11] T.M. Rice, Ann. Phys. 100, 100 (1965); T.K Ng and KS. Singwi, Phys.Rev. B 34, 7743 (1986) [12] S. Kiichenhoff and P. Wolfle, Phys. Rev. B 38, 935 (1988)
SUPERCONDUCTIVITY DUE TO COULOMB INTERACTIONS S. Kiichenhoff Physik-Department T30, Technische Universitli.t Miinchen, D-8046 Garching, FRG S. Schiller Department of Applied Physics, Stanford University, Stanford, Ca 94305, USA We solve the two coupled, frequency independent Bethe-Salpeter equations for the two particle vertex function and determine the effective interaction and quasiparticle scattering amplitude for a degenerate electron gas for 1 < r. < 70. A phenomenological ansatz is employed for the direct interaction that selfconsistently reproduces spin and charge susceptibilities and the local field factor known from microscopic theories. We find p-wave superconductivity for 10 < r. < 40 and s-wave superconductivity for r. > 40, at r. 70 the transition temperature being Tc l'::l 2.5 K and the effective mass m· /m 15. The transport parameters decrease by several orders of magnitude with decreasing density.
=
=
A longstanding problem is the role of the Coulomb potential to superconductivity. While it is commonly accepted that the Coulomb potential yields a repulsive contribution to the pairing at metallic densities (r. = 4), the situation at lower densities is unclear and received new interest since one of the intriguing properties of the high T c superconductors is the low carrier density [I]. At low densities the electron liquid undergoes two phase transitions: at r. :::l 75 from the paramagnetic to the ferromagnetic state and at r. :::l 120 from the liquid to a solid [2], indicating that correlations are most important in that density regime and may also influence superconductivity. Sham, Rietschel and Grabowski have investigated the possibility of Cooper pair. formation by exchange of plasmons [3]. Solving the Eliashberg equations with a dynamically screening Coulomb interaction they find sizable transition temperatures even in the metallic range, however these are found to be suppressed upon inclusion of lowest-order vertex corrections, a conclusion also drawn by Shirron and Ruvalds [4]. More recently Takada [5] employed a variational method to study Cooper pair formation using purely repulsive potentials. He finds that for a special form of the potential T c can be as high as 200 K for s-wave pairing, the important features being a low carrier density and a long range tail of the interaction. However the total effect of vertex corrections, the behaviour at r. > 10 and the pairing in angular momenta I", 0 remained unclear. We will approach the problem by concentrating on the effective interaction at the Fermi surface and calculate the Cooper pair interaction from the scattering amplitude. The only interaction present is the Coulomb potential and the electron liquid will be embedded in a rigid, neutralizing positive charge background. To ensure that the Pauli principle is not violated we solve the two frequency independent BetheSalpeter equations for particle-hole scattering (we allow for 0921-4526/90/$03.50
© 1990 -
scattering with momentum transfer up to 2kJ) in both p-h channels simultaneously, using the direct interaction model (for details of that model see [6]). The two equations are given as:
Akk,(q) = Fkk,(q) + LFkk,,(q);n,,(q)A~'k.(q), v = s,a, k"
Akk,(q) = Ekk'(q) + (F}kk,(q) - Ikk'(q)·
(1)
(2)
Here A is the scattering amplitude, F the generalized Landau function, I the direct interaction (irreducible in both p-h channels) and q the momentum transfer in the p-h channel. The overbar on F in (2) indicates the exchanged quantities, i.e. the variables are corresponding to the interchange of the two in-going (or out-going) particles (s,a refers to the spinsymmetric/antisymmetric components). The Landaufunction F~ contains long range and short range contributions and it is necessary for the calculations to split off the long range part: F~ = Vc(q) + F~. Vc(q) is the Coulomb potential. F~(q) is related to the local field factor G(q) in the usual way F~(q) = v,,-l(q)G(q) and accounts for the correlations in the liquid. If the direct interaction is chosen to be exchange symmetric, equations (1) and (2) conserve the two particle exchange symmetry and a newly formulated q-dependent generalized Landau sum rule which is a direct consequence ofthe Pauli principle [7,8] holds. Since I is not known accurateley in the whole density regime we choose a phenomenological ansatz for I which reproduces spin and charge susceptibilities and the local field factor. These quantities are known from Green's-functions Monte Carlo calculations [2] and microscopic theories [9]. However the spin susceptibility is not known accurately in the regime 5 < r. < 60 and we therefore used a simple formula which interpolates smoothly between the known regions [8]: at r. < 5 the RPA give correct results and
Elsevier Science Publishers B.V. (North-Holland)
1034
S. Kiichenhoff, S. Schiller
75 po: -1. According to that interpolation -0.20 (-0.42, -0.51, -0.68, -0.76, -0.81) for r.= 1 (5,10,30,50,70). The values for the effective mass for the calculation of the spin susceptibility are the calculated ones of table 1. Our results are shown in table 1. (For the calculation of T c from the scattering amplitude we used the at r.
PO :
r:;
IOrmula of Patton and Zaringhalam [10] and the cutoff energy is estimated to be the Fermi energy.)
r.
m"/m
go
gl
1 5 10 20 30 40 50 60 70
0.88 1.02 1.15 1.46 2.06 2.99 4.55 6.52 14.8
0.45 0.59 0.63 0.50 0.24 -0.02 -0.29 -0.49 -0.85
0.05 0.03 0.00 -0.03 -0.06 -0.06 -0.06 -0.05 -0.02
TF exp(l/g/)
K
8 X 10-11 , I : 1 5 X 10- 6 , I : 1 2 X 10- 5 , I : 1 1.7,1=0 3.3,1= 0 2.5, I : 0
The transport parameters may also be calculated from the scattering amplitudes. They are found [8] to change by more than 10 orders of magnitude, a behaviour which is expected for a liquid which becomes ferromagnetic and/or localised. The values for the spin diffusion constant (in K2 m 2 / s) are: DT2 : 2.0 X 107 (4.4 X 103 , 1.0 X 102 ,6.9 X 10- 2 ,3.6 x 10-4, 1.4 X 10- 6 for r.:1 (5,10,30,50,70). Part of the decrease of the transport parameters is of course due to the decrease of the Fermi velocity. In concluding we have shown that the dilute electron gas is unstable against formation of Cooper pairs, the purely repulsive Coulomb interaction providing an attractive interaction. If an additional attractive mec1Ianism is present fairly ~high transition temperatures may be reached as they have been found in the high T c superconductors. Our treatment of the Coulomb interaction includes all vertex corrections and it is seen that vertex corrections are very important for Coulombic systems - Migdal's theorem does not hold - and even at metallic densities may not be neglected. Our model may also be directly applied to systems were electronic correlations are important; if spin and c1Iarge response are known, the superconducting coupling constants may be calculated. Acknoledgements We are indebted to ·Prof. Peter Wolfle for continuous support during all stages of this work.
References Table 1: Effective mass, pair interaction constants g/ and superconductivity temperature scale TJ!) as a function of the electron density
[1] T. M. Rice, Z. Phys. B 67, 141 (1987) [2] D.M. Ceperley and B.J. Alder, Phys. Rev. Lett. 45, 566 (1980) [3] 1. J. Sham, Physica B 135,451 (1985)
The quasiparticle become very heavy as the density is lowered, in accordance with the approac1I to the transition point of solidification. Note the dip in the effective mass (m"/m: 1 for r...... 0) whic1I is also found in other calculations [11], but which is somewhat stronger in ours. At metallic densities due to the lack of a low energy mode the mass is not strongly renormalized. A~( q : 2kf) is found to increase as the densities is lowered indicating also a transition to a solid. (Due to the long range of the Coulomb interaction AMO) : 1 irrespectively of the density.) The p-wave coupling constant is negative for r. > 10 with a minnimum value of gl : -0.06. The transition temperatures are fairly small, the attractive potential is provided by spin density fluctuations as can be seen from the enhanced spin susceptibility. The s-wave coupling constant is decreasing rapidly from r. : 10 on and goes negative at r. : 40. The highest positive values of go are in the metallic density range, where a strong attractive potential has to be present in superconducting materials to overcome that repulsion due to the Coulomb interaction. The transition temperatures at lowest densities are in the range of a few Kelvins. Looking at the different contributions to go it can be shown, that the attractive mechanism is provided by transverse current fluctuations [12].
[4] J. J. Shirron and J. Ruvalds, Phys. Rev. B 34, 7596 (1986) [5] Y. Takada, Phys. Rev. B 39, 11575 (1989) [6] M. Pfitzner and P. Wolfle, Phys. Rev. B 35, 4699, (1987) [7] S. Schiller, Diplomarbeit, Technisc1Ie Universitat Miinchen, 1987 [8] S. Kiichenhoff and S. Schiller, to be published [9] KS. Singwi and M.P. Tosi, in Solid Sta.te Physics, H. Ehrenreich, F. Seitz and D. Turnbull, eds., Vol. 36, p.l77, Academic, 1981 [10] B.R. Patton and A. Zaringhalam, Phys. Lett. 45, 566 (1980) [11] T.M. Rice, Ann. Phys. 100, 100 (1965); T.K Ng and KS. Singwi, Phys.Rev. B 34, 7743 (1986) [12] S. Kiichenhoff and P. Wolfle, Phys. Rev. B 38, 935 (1988)