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Thermodynamic critical fields in high Tc superconductivity
Physica C 153-155 (1988) 699-700
North-Holland, Amsterdam
THERMODYNAMIC CRITICAL FIELDS IN HIGH Tc SUPERCONDUCTIVITY Dennis P. CLOUGHERTY and Keith H. JOHNSON Department of Physics and Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA Using the free electron approximation, a real space mechanism of how magnetic fields of sufficient strength destroy the superconducting state is outlined. Using the resultant equation together with the molecular orbital model of superconductivity 1,2, the thermodynamic critical magnetic field is calculated and compared to experiment for type I elemental superconductors and type II superconductors, including high-Tc superconductors. The expression for critical field compares favorably with an expression derived by Schrieffer. 1. INTRODUCT/ON The superconducting state of a material results from the pair formation of electrons at the Fermi energy. These Cooper pairs in a superconducting material can be thought of as quasi-molecular entities having a binding energy, A. If one of the electron valence bonds is broken, the density of superconducting electrons is decreased while the density of normal electrons is increased. A superconducting material reverts to normal when all the bonds are broken. These pair bonds can be broken thermally or magnetically. When a magnetic field, H, is applied to a superconductor, the superconducting pairs within the penetration depth of the surface acquire additional kinetic energy from the field 3. As this additional energy equals the binding energy, the pairs dissociate and the material reverts to the normal phase.
2. DIAMAGNETIC ENERGY SHIFT The highly correlated pairs of conduction electrons form a rigid shell on top of the Fermi distribution of electrons in momentum space. The many body wavefunction resists changing with the application of a magnetic field until the field is of adequate strength to cause a phase transformation and destroy superconducting order. In a free electron approximation, although the momentum of the conduction states is unperturbed by H, the velocity shifts: Pi e vi A m* m*c This is a manifestation of the perfect diamagnetism of a superconductor. The change in the kinetic energy is given by AEkinetic =
E
X e2
e pi.A + ~ A2 i m*c • 2m*c 2 The first term is a paramagnetic energy, and the second term is a diamagnetic energy. Choosing the London gauge, the sum of all the momenta is zero, leaving just the diamagnetic shift. The magnitude of A can be written in terms of H by solving the magnetic diffusion equation in London's gauge. At the surface of the superconductor,
0921-4534/88/$03.50 © ElsevierSciencePublishersB.V. (North-HollandPhysicsPublishingDivision)
A : H" ~'L, where ~.L : ~ ~
m* C2 / 1/2 )
Therefore, the kinetic energy increase per unit volume is equal to the energy density of the magnetic field, H2/8x. This is also known as the stabilization energy. 3. THERMODYNAMIC RELATIONS 3.1 Thermodynamic Critical Field The Cooper pairs on the surface of the superconductor acquire a boost in energy from the vector potential, A. On average, when this volume kinetic energy exceeds the volume binding energy, then the material reverts to the normal phase. This can be expressed as 8---~"= nc A where nc is the density of Cooper pairs in the superconductor. 3.2 Cooper Pair Density For conduction electrons which pair in a relative S-state, the distance of closest approach is less than or equal to a wavelength. This forms an approximate transverse measure of the extent of the pair wavefunction. The longitudinal extent is measured by the Pippard coherence length, ~. On average, one Cooper pair can be said to occupy a cylinder of radius k/2 and length ~. For calculational purposes, 1 n c = ~ (X/2) -2 t 0
It is possible to experimentally measure the density of Cooper pairs by microwave techniques. The resonant frequency shift resulting from a superconductor in a cavity is a measure of the inductance of the sample which is inversely proportional to the Cooper pair density.4 4. RESULTS 4.1 BCS Comparison For a pure material, the intrinsic coherence length is given by hVF t0 =
700
D.P. CIougherty and K.H. Johnson / Thermodynamic criticalfields
Combining terms shows that the thermodynamic critical field is proportional to A. This compares favorably with Schrieffer's expression 5
ACKNOWLEDGEMENTS The authors would like to thank IBM and Martin Marietta for supporting the research and NATO for a travel grant. One of us (DPC) would also like to thank Dr. M.E. Eberhart and the Center for Materials Science at Los Alamos National Lab for their hospitality during preparation of this paper.
H c = 2[nN(0)] 1/2 A 4.2 Molecular Orbital Theory of Superconductivity Within the context of a real space molecular orbital description of the superconducting state, two calculable parameters are introduced 1,2: the Coulomb screening length, d, and the dynamic Jahn-Teller coupling, 1~.The pair binding energy becomes a function of d and 1~.The half wavelength Z/2 is shown to be the Coulomb screening distance resulting from the orbital charges at the Fermi energy. The tables below summarize the result of the calculations using the molecular orbital parameters.
REFERENCES (1) K.H. Johnson and R.P. Messmer, Synth. Metals 5 (1983) 151. (2) K.H. Johnson et al., Molecular Orbital Basis for Superconductivity with Applications to High-Tc Materials, in: Novel Superconductivity (Plenum, 1987). K.H. Johnson et al., Quantum Chemistry and High Tc Superconductivity, this volume. (3) V.F. Weisskopf, Contemp. Phys. 22 (1981) 375. (4) M.Poirier, G. Quirion, K.R. Poeppelmeier, and J.P. Thiel, Phys. Rev. B 36 (1987) 3906. (5) J.R. Schrieffer, Theory of Superconductivity (Benjamin, 1964). (6) M.E. McHenry, private communication. (7) B.W. Roberts, J. Phys.Chem. Ref. Data 5 (1976), No. 3, 581. (8) S. Senoussi, M. Oussena, M. Ribault, G. Collin, Phys. Rev. B 36 (1987) 4003. (9) T.R. McGuire, T.R. Dinger, P.J.P. Freitas, W.J. Gallagher, T.S. Plaskett, R.L. Sandstrom, and T.M. Shaw, Phys. Rev B 36 (1987) 4032. (10) J.S. Moodera, R. Meservey, J.E. Tkaczyk, C.X. Hao, G.A. Gibson, and P.M. Tedrow, Phys. Rev. B (in print).
5. CONCLUSIONS The breaking of superconducting order by magnetic fields can be thought of as being the result of kinetic energy transfer from the magnetic field to the paired conduction electrons at the surface of the superconductor. Within the context of the molecular orbital description of superconductivity, the thermodynamic critical field can be calculated efficiently. Since Hc is approximately the geometric mean of Hcl and Hc2, one can use the above equations to take experimentally measured values of Hcl and estimate Hc2. As evidenced by the tables below, this formalism has general applicability to all known superconductors.
Table of Hc for selected elemental superconductors Element AI Be Mo Os Re Ru Ta Ti Zr
d ([k) 1,6 1.7 0.92 1.4 2.0 2.5 2.0 2.0 2.1 2.3
1] 1,6 0.50 0.50 0.37 0.049 0.48 0.051 0.33 0.050 0.042
Tc (exp) 7 K 1.2 0.026 0.92 0.66 1.7 0.49 4.5 0.40 0.61
Tc (calc.) K 1.1 0.03 0.86 0.55 1.7 0.51 6.36 0.42 0.41
Hc (exp) 7 G 105 96 70 200 69 829 56 47
Hc (calc) G 144 6 135 72 200 66 828 54 50
Table of Hc for high Tc superconductors T]cpe 2 La2CuO4- doped YBa2Cu307
d (~) 2 2.7 3.9
1~ 2 0.25 - 0.30 0.14 - .25
Tc (exp) 2 K ~ 40 ~ 90
Tc (calc.) K 34 - 50 87 - 149
* Reference 8. ** Reference 9. t Reference 10. t t Reference 4.
Hc (calc) G 13500 - 16300 21400 - 27500
Hcl (exp) G ~650 *
~400"*
Hc2 (exp) KG ~380 t? 600-2000 t
North-Holland, Amsterdam
THERMODYNAMIC CRITICAL FIELDS IN HIGH Tc SUPERCONDUCTIVITY Dennis P. CLOUGHERTY and Keith H. JOHNSON Department of Physics and Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA Using the free electron approximation, a real space mechanism of how magnetic fields of sufficient strength destroy the superconducting state is outlined. Using the resultant equation together with the molecular orbital model of superconductivity 1,2, the thermodynamic critical magnetic field is calculated and compared to experiment for type I elemental superconductors and type II superconductors, including high-Tc superconductors. The expression for critical field compares favorably with an expression derived by Schrieffer. 1. INTRODUCT/ON The superconducting state of a material results from the pair formation of electrons at the Fermi energy. These Cooper pairs in a superconducting material can be thought of as quasi-molecular entities having a binding energy, A. If one of the electron valence bonds is broken, the density of superconducting electrons is decreased while the density of normal electrons is increased. A superconducting material reverts to normal when all the bonds are broken. These pair bonds can be broken thermally or magnetically. When a magnetic field, H, is applied to a superconductor, the superconducting pairs within the penetration depth of the surface acquire additional kinetic energy from the field 3. As this additional energy equals the binding energy, the pairs dissociate and the material reverts to the normal phase.
2. DIAMAGNETIC ENERGY SHIFT The highly correlated pairs of conduction electrons form a rigid shell on top of the Fermi distribution of electrons in momentum space. The many body wavefunction resists changing with the application of a magnetic field until the field is of adequate strength to cause a phase transformation and destroy superconducting order. In a free electron approximation, although the momentum of the conduction states is unperturbed by H, the velocity shifts: Pi e vi A m* m*c This is a manifestation of the perfect diamagnetism of a superconductor. The change in the kinetic energy is given by AEkinetic =
E
X e2
e pi.A + ~ A2 i m*c • 2m*c 2 The first term is a paramagnetic energy, and the second term is a diamagnetic energy. Choosing the London gauge, the sum of all the momenta is zero, leaving just the diamagnetic shift. The magnitude of A can be written in terms of H by solving the magnetic diffusion equation in London's gauge. At the surface of the superconductor,
0921-4534/88/$03.50 © ElsevierSciencePublishersB.V. (North-HollandPhysicsPublishingDivision)
A : H" ~'L, where ~.L : ~ ~
m* C2 / 1/2 )
Therefore, the kinetic energy increase per unit volume is equal to the energy density of the magnetic field, H2/8x. This is also known as the stabilization energy. 3. THERMODYNAMIC RELATIONS 3.1 Thermodynamic Critical Field The Cooper pairs on the surface of the superconductor acquire a boost in energy from the vector potential, A. On average, when this volume kinetic energy exceeds the volume binding energy, then the material reverts to the normal phase. This can be expressed as 8---~"= nc A where nc is the density of Cooper pairs in the superconductor. 3.2 Cooper Pair Density For conduction electrons which pair in a relative S-state, the distance of closest approach is less than or equal to a wavelength. This forms an approximate transverse measure of the extent of the pair wavefunction. The longitudinal extent is measured by the Pippard coherence length, ~. On average, one Cooper pair can be said to occupy a cylinder of radius k/2 and length ~. For calculational purposes, 1 n c = ~ (X/2) -2 t 0
It is possible to experimentally measure the density of Cooper pairs by microwave techniques. The resonant frequency shift resulting from a superconductor in a cavity is a measure of the inductance of the sample which is inversely proportional to the Cooper pair density.4 4. RESULTS 4.1 BCS Comparison For a pure material, the intrinsic coherence length is given by hVF t0 =
700
D.P. CIougherty and K.H. Johnson / Thermodynamic criticalfields
Combining terms shows that the thermodynamic critical field is proportional to A. This compares favorably with Schrieffer's expression 5
ACKNOWLEDGEMENTS The authors would like to thank IBM and Martin Marietta for supporting the research and NATO for a travel grant. One of us (DPC) would also like to thank Dr. M.E. Eberhart and the Center for Materials Science at Los Alamos National Lab for their hospitality during preparation of this paper.
H c = 2[nN(0)] 1/2 A 4.2 Molecular Orbital Theory of Superconductivity Within the context of a real space molecular orbital description of the superconducting state, two calculable parameters are introduced 1,2: the Coulomb screening length, d, and the dynamic Jahn-Teller coupling, 1~.The pair binding energy becomes a function of d and 1~.The half wavelength Z/2 is shown to be the Coulomb screening distance resulting from the orbital charges at the Fermi energy. The tables below summarize the result of the calculations using the molecular orbital parameters.
REFERENCES (1) K.H. Johnson and R.P. Messmer, Synth. Metals 5 (1983) 151. (2) K.H. Johnson et al., Molecular Orbital Basis for Superconductivity with Applications to High-Tc Materials, in: Novel Superconductivity (Plenum, 1987). K.H. Johnson et al., Quantum Chemistry and High Tc Superconductivity, this volume. (3) V.F. Weisskopf, Contemp. Phys. 22 (1981) 375. (4) M.Poirier, G. Quirion, K.R. Poeppelmeier, and J.P. Thiel, Phys. Rev. B 36 (1987) 3906. (5) J.R. Schrieffer, Theory of Superconductivity (Benjamin, 1964). (6) M.E. McHenry, private communication. (7) B.W. Roberts, J. Phys.Chem. Ref. Data 5 (1976), No. 3, 581. (8) S. Senoussi, M. Oussena, M. Ribault, G. Collin, Phys. Rev. B 36 (1987) 4003. (9) T.R. McGuire, T.R. Dinger, P.J.P. Freitas, W.J. Gallagher, T.S. Plaskett, R.L. Sandstrom, and T.M. Shaw, Phys. Rev B 36 (1987) 4032. (10) J.S. Moodera, R. Meservey, J.E. Tkaczyk, C.X. Hao, G.A. Gibson, and P.M. Tedrow, Phys. Rev. B (in print).
5. CONCLUSIONS The breaking of superconducting order by magnetic fields can be thought of as being the result of kinetic energy transfer from the magnetic field to the paired conduction electrons at the surface of the superconductor. Within the context of the molecular orbital description of superconductivity, the thermodynamic critical field can be calculated efficiently. Since Hc is approximately the geometric mean of Hcl and Hc2, one can use the above equations to take experimentally measured values of Hcl and estimate Hc2. As evidenced by the tables below, this formalism has general applicability to all known superconductors.
Table of Hc for selected elemental superconductors Element AI Be Mo Os Re Ru Ta Ti Zr
d ([k) 1,6 1.7 0.92 1.4 2.0 2.5 2.0 2.0 2.1 2.3
1] 1,6 0.50 0.50 0.37 0.049 0.48 0.051 0.33 0.050 0.042
Tc (exp) 7 K 1.2 0.026 0.92 0.66 1.7 0.49 4.5 0.40 0.61
Tc (calc.) K 1.1 0.03 0.86 0.55 1.7 0.51 6.36 0.42 0.41
Hc (exp) 7 G 105 96 70 200 69 829 56 47
Hc (calc) G 144 6 135 72 200 66 828 54 50
Table of Hc for high Tc superconductors T]cpe 2 La2CuO4- doped YBa2Cu307
d (~) 2 2.7 3.9
1~ 2 0.25 - 0.30 0.14 - .25
Tc (exp) 2 K ~ 40 ~ 90
Tc (calc.) K 34 - 50 87 - 149
* Reference 8. ** Reference 9. t Reference 10. t t Reference 4.
Hc (calc) G 13500 - 16300 21400 - 27500
Hcl (exp) G ~650 *
~400"*
Hc2 (exp) KG ~380 t? 600-2000 t