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Flux structures in two-dimensional superconductors for nearly plane–parallel magnetic field
Physica B 284}288 (2000) 713}714
Flux structures in two-dimensional superconductors for nearly plane}parallel magnetic "eld U. Klein Institute for Theoretical Physics, University of Linz, Altenbergerstrasse 69, A-4040 Linz, Austria
Abstract The recent discovery of several classes of layered superconducting compounds such as cuprate compounds, organic superconductors, and ruthenate}cuprate compounds has renewed the interest in an experimental veri"cation of the Fulde}Ferrell}Larkin}Ovchinnikov (FFLO) state. For an external magnetic "eld nearly parallel to the layers one has competiton between spin and orbital pair-breaking e!ects (between FFLO and vortex state). This leads to superconducting states with pairing in Landau levels n'0. Recently, a nonlinear expansion of the quasiclassical equations has been performed in order to "nd the stable #ux structures below the upper critical "eld. Here, the states with Landau quantum numbers n"1, 3, the so-called `quasi-one-dimensionala states, are discussed in detail. 2000 Elsevier Science B.V. All rights reserved. Keywords: FFLO state; Landau levels; Orbital magnetism
Pair-breaking e!ects due to the electrons spin magnetic moment are much smaller than those due to the magnetic moment stemming from the electrons angular momentum. If an external magnetic "eld is applied at small angle H relative to the plane of a two-dimensional superconductor, both mechanisms become of comparable magnitude, since the perpendicular component, responsible for the orbital e!ect, is much smaller than the parallel component responsible for the spin e!ect. If the perpendicular component is exactly zero, a spatially modulated inhomogeneous state (FFLO state) has been predicted by Fulde and Ferrell [1] and by Larkin and Ovchinnikov [2] as stable state near the second-order transition to the normal conducting state. The interesting geometric situation where both mechanisms become competitive, has "rst been studied by Bulaevskii [3]. Later, his treatment was generalized to arbitrary temperatures and d-wave pairing by Shimahara and Rainer [4]. The upper critical "eld curve H ob tained by these workers showed that, for su$ciently
E-mail address: [email protected] (U. Klein)
small values of H, the superconducting pair-wave function must be characterized by Landau quantum numbers n larger than zero. Very recently, the structure of these stable states below H has been calculated [5] by min imizing the microscopic free energy. The H curve [3,4] for an external "eld at "nite angles H consists of several pieces belonging to di!erent n. For HP0 one has nPR and H approaches the FFLO result. To "nd the stable state below H the microscopic free energy has to be expanded to fourth-order terms in the order parameter amplitude, and the two-dimensional lattice structure of lowest free energy has to be found. This has been done in Ref. [5], for a general two-dimensional state, using Eilenberger's quasi-classical equations } properly generalized for spin paramagnetic e!ects. Note that the quasi-classical theory is valid for the present situation. To obtain the stable states near H a long analytical calculation, generalizing previous work by Eilenberger [6], must be performed, followed by a numerical minimization, for the most favourable lattice, of the fourth-order free energy contribution [5,7]. The nontrivial (n'0) pairing states occur for angles H smaller than a critical angle which varies considerably but is generally of the order of 13. A variety of di!erent order parameter structures has been found depending on
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 2 4 0 7 - 2
714
U. Klein / Physica B 284 }288 (2000) 713}714
Fig. 1. Square of modulus of order parameter for n"1 as a function of spatial coordinates measured in units of the length a of the unit cell vector.
to de"ne the angle H with arbitrary precision. Of particular interest are the states for n"1, 3; the corresponding order parameter structures are shown in Figs. 1 and 2. They consist of a mixture of vortex chains and onedimensional FFLO-like oscillations. For these states one "nds numerically a quasi-continuum of unit cells, which all give nearly identical free energy values. This fact is a consequence of the small coupling between vortices due to the intercalated FFLO domains. For these `quasione-dimensionala states, even small irregularities will destroy the (macroscopic) periodicity in the direction parallel to the chains; the energy barrier against motion of the vortex chains as a whole is extremely small. This should lead to sudden changes in transport properties at the transition from n"0 to 1.
Acknowledgements Stimulating and helpful discussions with my collaborators D. Rainer and H. Shimahara are gratefully acknowledged.
References
Fig. 2. Square of modulus of order parameter for n"3 as a function of spatial coordinates measured in units of the length a of the unit cell vector.
n. For higher n the unit cells contain several zeros of the order parameter with di!erent vorticities. In experiments on thin "lms and layered compounds the n'0 states may `hidea the FFLO state, since it is impossible
[1] P. Fulde, R.A. Ferrell, Phys. Rev. 135 (1964) A550. [2] A.I. Larkin, Y.N. Ovchinnikov, Sov. Phys. JETP 28 (1969) 1200. [3] L.N. Bulaevskii, Sov. Phys. JETP 38 (1974) 634. [4] H. Shimahara, D. Rainer, J. Phys. Soc. Japan 66 (1997) 3591. [5] U. Klein, D. Rainer, H. Shimahara, J. Low Temp. Phys. 118 (2000) in press. [6] G. Eilenberger, Phys. Rev. 153 (1967) 584. [7] U. Klein, in preparation.
Flux structures in two-dimensional superconductors for nearly plane}parallel magnetic "eld U. Klein Institute for Theoretical Physics, University of Linz, Altenbergerstrasse 69, A-4040 Linz, Austria
Abstract The recent discovery of several classes of layered superconducting compounds such as cuprate compounds, organic superconductors, and ruthenate}cuprate compounds has renewed the interest in an experimental veri"cation of the Fulde}Ferrell}Larkin}Ovchinnikov (FFLO) state. For an external magnetic "eld nearly parallel to the layers one has competiton between spin and orbital pair-breaking e!ects (between FFLO and vortex state). This leads to superconducting states with pairing in Landau levels n'0. Recently, a nonlinear expansion of the quasiclassical equations has been performed in order to "nd the stable #ux structures below the upper critical "eld. Here, the states with Landau quantum numbers n"1, 3, the so-called `quasi-one-dimensionala states, are discussed in detail. 2000 Elsevier Science B.V. All rights reserved. Keywords: FFLO state; Landau levels; Orbital magnetism
Pair-breaking e!ects due to the electrons spin magnetic moment are much smaller than those due to the magnetic moment stemming from the electrons angular momentum. If an external magnetic "eld is applied at small angle H relative to the plane of a two-dimensional superconductor, both mechanisms become of comparable magnitude, since the perpendicular component, responsible for the orbital e!ect, is much smaller than the parallel component responsible for the spin e!ect. If the perpendicular component is exactly zero, a spatially modulated inhomogeneous state (FFLO state) has been predicted by Fulde and Ferrell [1] and by Larkin and Ovchinnikov [2] as stable state near the second-order transition to the normal conducting state. The interesting geometric situation where both mechanisms become competitive, has "rst been studied by Bulaevskii [3]. Later, his treatment was generalized to arbitrary temperatures and d-wave pairing by Shimahara and Rainer [4]. The upper critical "eld curve H ob tained by these workers showed that, for su$ciently
E-mail address: [email protected] (U. Klein)
small values of H, the superconducting pair-wave function must be characterized by Landau quantum numbers n larger than zero. Very recently, the structure of these stable states below H has been calculated [5] by min imizing the microscopic free energy. The H curve [3,4] for an external "eld at "nite angles H consists of several pieces belonging to di!erent n. For HP0 one has nPR and H approaches the FFLO result. To "nd the stable state below H the microscopic free energy has to be expanded to fourth-order terms in the order parameter amplitude, and the two-dimensional lattice structure of lowest free energy has to be found. This has been done in Ref. [5], for a general two-dimensional state, using Eilenberger's quasi-classical equations } properly generalized for spin paramagnetic e!ects. Note that the quasi-classical theory is valid for the present situation. To obtain the stable states near H a long analytical calculation, generalizing previous work by Eilenberger [6], must be performed, followed by a numerical minimization, for the most favourable lattice, of the fourth-order free energy contribution [5,7]. The nontrivial (n'0) pairing states occur for angles H smaller than a critical angle which varies considerably but is generally of the order of 13. A variety of di!erent order parameter structures has been found depending on
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 2 4 0 7 - 2
714
U. Klein / Physica B 284 }288 (2000) 713}714
Fig. 1. Square of modulus of order parameter for n"1 as a function of spatial coordinates measured in units of the length a of the unit cell vector.
to de"ne the angle H with arbitrary precision. Of particular interest are the states for n"1, 3; the corresponding order parameter structures are shown in Figs. 1 and 2. They consist of a mixture of vortex chains and onedimensional FFLO-like oscillations. For these states one "nds numerically a quasi-continuum of unit cells, which all give nearly identical free energy values. This fact is a consequence of the small coupling between vortices due to the intercalated FFLO domains. For these `quasione-dimensionala states, even small irregularities will destroy the (macroscopic) periodicity in the direction parallel to the chains; the energy barrier against motion of the vortex chains as a whole is extremely small. This should lead to sudden changes in transport properties at the transition from n"0 to 1.
Acknowledgements Stimulating and helpful discussions with my collaborators D. Rainer and H. Shimahara are gratefully acknowledged.
References
Fig. 2. Square of modulus of order parameter for n"3 as a function of spatial coordinates measured in units of the length a of the unit cell vector.
n. For higher n the unit cells contain several zeros of the order parameter with di!erent vorticities. In experiments on thin "lms and layered compounds the n'0 states may `hidea the FFLO state, since it is impossible
[1] P. Fulde, R.A. Ferrell, Phys. Rev. 135 (1964) A550. [2] A.I. Larkin, Y.N. Ovchinnikov, Sov. Phys. JETP 28 (1969) 1200. [3] L.N. Bulaevskii, Sov. Phys. JETP 38 (1974) 634. [4] H. Shimahara, D. Rainer, J. Phys. Soc. Japan 66 (1997) 3591. [5] U. Klein, D. Rainer, H. Shimahara, J. Low Temp. Phys. 118 (2000) in press. [6] G. Eilenberger, Phys. Rev. 153 (1967) 584. [7] U. Klein, in preparation.