Academia.eduAcademia.edu

Outline

Title
Abstract
Figures
Introduction
References
All Topics
Physics
Condensed Matter Physics

Nonlinear σ model for disordered superconductors

Profile image of Igor YurkevichIgor Yurkevich

2001, Physical Review B

https://doi.org/10.1103/PHYSREVB.63.064522
visibility

…

description

8 pages

link

1 file

Abstract

We suggest a variant of the nonlinear σ model for the description of disordered superconductors. The main distinction from existing models lies in the fact that the saddle point equation is solved nonperturbatively in the superconducting pairing field. It allows one to use the model both in the vicinity of the metal-superconductor transition and well below its critical temperature with full account for the self-consistency conditions. We show that the model reproduces a set of known results in different limiting cases, and apply it for a self-consistent description of the proximity effect at the superconductor-metal interface.

Figures (5)
Averaging over u with the help of the standard replica trick gives the quartic in w term in the action. Using the Hubbard-Stratonovich transformation, one decouples both this term and the BCS term, Eq. (1c), the former with the help of a matrix field 6 = G(r;7,7’) and the latter with the help of a pairing field A = A(r;7) which will eventually play the role of the order parameter. This results in the following effective action:
Averaging over u with the help of the standard replica trick gives the quartic in w term in the action. Using the Hubbard-Stratonovich transformation, one decouples both this term and the BCS term, Eq. (1c), the former with the help of a matrix field 6 = G(r;7,7’) and the latter with the help of a pairing field A = A(r;7) which will eventually play the role of the order parameter. This results in the following effective action:
Here 7,(x) is a Grassmannian field!”> anti-periodic in p imaginary time 7 with period 1/T, x = (r,7), s = (f, |) is the spin index, Xo is the BCS coupling constant, and from now on we set h = 1. We consider the standard BCS Hamiltonian in the presence of a random potential, u(r). For complete- ness, we start with outlining the standard procedure? of a field theoretic representation in the temperature tech- nique for this Hamiltonian. The corresponding action has the form
Here 7,(x) is a Grassmannian field!”> anti-periodic in p imaginary time 7 with period 1/T, x = (r,7), s = (f, |) is the spin index, Xo is the BCS coupling constant, and from now on we set h = 1. We consider the standard BCS Hamiltonian in the presence of a random potential, u(r). For complete- ness, we start with outlining the standard procedure? of a field theoretic representation in the temperature tech- nique for this Hamiltonian. The corresponding action has the form
For w = 0 the coefficient ap = a=0(T) can be simplified in the two limits: where Tp = T.9(B =0) is the transition temperature of the clean superconductor in the absence of a magnetic field. The weak localisation correction is proportional to the coefficient a, given by
For w = 0 the coefficient ap = a=0(T) can be simplified in the two limits: where Tp = T.9(B =0) is the transition temperature of the clean superconductor in the absence of a magnetic field. The weak localisation correction is proportional to the coefficient a, given by
Here yn = x+ — x-— is the phase difference between two superconducting banks and L, = ,/D/2e is the coherence length for two particles with the energy dif- ference € propagating in the normal metal. For a long normal bridge between the two superconducting banks, L> Lr = VD/2rT, one may consider separately three regions: those close to the N/S boundaries (with the width of order Ly) and the bulk. Matching the solutions for all the regions, we find the following expression which well approximates the solution for the entire normal re- gion: FIG. 1. A spatial dependence of the phase x- across the SNS contact for quasi-1D geometry.
Here yn = x+ — x-— is the phase difference between two superconducting banks and L, = ,/D/2e is the coherence length for two particles with the energy dif- ference € propagating in the normal metal. For a long normal bridge between the two superconducting banks, L> Lr = VD/2rT, one may consider separately three regions: those close to the N/S boundaries (with the width of order Ly) and the bulk. Matching the solutions for all the regions, we find the following expression which well approximates the solution for the entire normal re- gion: FIG. 1. A spatial dependence of the phase x- across the SNS contact for quasi-1D geometry.
where the upper indices a, (3 refer to the time-reversal sector and tr refers only to the matrix indices which are not indicated explicitly. The matrix 7 in the Eqs.(A1) and (A2) has the following structure in the time-reversal sector:
where the upper indices a, (3 refer to the time-reversal sector and tr refers only to the matrix indices which are not indicated explicitly. The matrix 7 in the Eqs.(A1) and (A2) has the following structure in the time-reversal sector:

Related papers

Theory of Disordered Unconventional Superconductors
Ahmet Keles

Журнал Экспериментальной и Теоретической Физики, 2014

This article is dedicated to A. F. Andreev on the occasion of his 75th birthday. In contrast to conventional s-wave superconductivity, unconventional (e.g. p or d-wave) superconductivity is strongly suppressed even by relatively weak disorder. Upon approaching the superconductor-metal transition, the order parameter amplitude becomes increasingly inhomogeneous leading to effective granularity and a phase ordering transition described by the Mattis model of spin glasses. One consequence of this is that at low enough temperatures, between the clean unconventional superconducting and the diffusive metallic phases, there is necessarily an intermediate superconducting phase which exhibits s-wave symmetry on macroscopic scales.

View PDFchevron_right
The conductor-superconductor transition in disordered superconducting materials
Stephane Roux

Physica C: Superconductivity, 1990

We consider a network consisting of elements which are normal down to a threshold current, below which they become superconducting.

View PDFchevron_right
Nonlinear effects in the theory of superconductivity
Nicolae Enaki

Journal of Physics: Conference Series, 2012

It is proposed the non-linear cooperative interaction between the electrons through the vibration states of thermostat (the large system with big degree of freedom). As an example is studied the cooperative interaction between the cooper pairs through the non-linear lattice vibration in single and two-phonon exchange processes between the electrons. In the case of superconductivity phase transition, the cooperative interaction between the carriers in two and single phonon exchange processes depends on the temperature. This temperature dependence of exchange integral between the quasi-particles drastically changes the second order diagram as this is demonstrated in paper . Unlike the approach proposed in papers , in this report it is takes into account the non-linear vibration of thermostat modes (phonon, optical modes etc.). It is observed that this effect influence the temperature dependence of the order parameter in the second order phase transition like superconductivity, super-radiance, ferromagnetism, etc.

View PDFchevron_right
Granular superconductors: From the nonlinear σ model to the Bose-Hubbard description
Igor Yurkevich

Physical Review B, 2001

We modify a nonlinear σ model (NLσM) for the description of a granulated disordered system in the presence of both the Coulomb repulsion and the Cooper pairing. We show that under certain controlled approximations this model is reduced to the Bose-Hubbard (or "dirty-boson") model with renormalized coupling constants. We obtain a more general effective action (which is still simpler than the full NLσM action) which can be applied in the region of parameters where the reduction to the Bose-Hubbard model is not justified. This action may lead to a different picture of the superconductor-insulator transition in 2D systems.

View PDFchevron_right
Augmented space recursion study of the effect of disorder on superconductivity
Kartick Tarafder

Physical Review B, 2009

We present a real-space approach to study the effect of disorder on superconductivity. The method is based on augmented space formalism that goes beyond mean-field approximations for configuration averaging and effectively deals with the influence of configuration fluctuations of the neighborhood of an atom. In the regime of validity of Anderson's theorem our results for s-and d-wave dirty superconductors have excellent agreement with existing results. The formalism is extended and tested for random negative U Hubbard model. Having verified the reliability of our method we use it to study environment-dependent inhomogeneous randomness in disordered superconducting systems.

View PDFchevron_right
Granular superconductors: From the nonlinear sigma model to the Bose-Hubbard description
Igor Yurkevich

Phys Rev B, 2001

We modify a nonlinear σ model (NLσM) for the description of a granular disordered system in the presence of both the Coulomb repulsion and the Cooper pairing. We show that under certain controlled approximations the action of this model is reduced to the Ambegaokar-Eckern-Schön (AES) action, which is further reduced to the Bose-Hubbard (or ``dirty-boson'') model with renormalized coupling constants. We obtain an effective action which is more general than the AES one but still simpler than the full NLσM action. This action can be applied in the region of parameters where the reduction to the AES or the Bose-Hubbard model is not justified. This action may lead to a different picture of the superconductor-insulator transition in two-dimensional systems.

View PDFchevron_right
Disordered two-dimensional superconductors: Roles of temperature and interaction strength
Raimundo dos Santos

Physical Review B, 2008

View PDFchevron_right
Effect of disorder on superconductivity in the boson-fermion model
K. Wysokinski

Physical Review B, 2002

We study how a randomness of either boson or fermion site energies affects the superconducting phase of the boson fermion model. We find that, contrary to what is expected for s-wave superconductors, the non-magnetic disorder is detrimental to the s-wave superconductivity. However, depending in which subsystem the disorder is located, we can observe different channels being affected. Weak disorder of the fermion subsystem is responsible mainly for renormalization of the single particle density of states while disorder in the boson subsystem directly leads to fluctuation of the strength of the effective pairing between fermions.

View PDFchevron_right
On the relation between two models for superconductors with a structural distortion
Jordan Brankov

Physica A: Statistical Mechanics and its Applications, 1976

Two three-dimensional models exhibiting coexistence of superconductivity and lattice distortion are discussed. One of the models, studied in detail by Rusinov, Do Chart Kat and Kopayev, takes into account the superconductivity inter-and intra-band pairing phenomenologically. The other has been derived by Mattis and Langer from a Fr6hlich-type hamiltonian by eliminating the linear electron-phonon coupling in the presence of a "condensing" phonon mode. Both approaches are shown to agree in the limit ~oo ~ It, where ~oo is the cut-off energy and/~ is the deviation of the chemical potential from the band centre.

View PDFchevron_right
Nature of the superconductor–insulator transition in disordered superconductors
Yigal Meir

Nature, 2007

The interplay of superconductivity and disorder has intrigued scientists for several decades. Disorder is expected to enhance the electrical resistance of a system, whereas superconductivity is associated with a zero-resistance state. Although, superconductivity has been predicted to persist even in the presence of disorder 1 , experiments performed on thin films have demonstrated a transition from a superconducting to an insulating state with increasing disorder or magnetic field 2 . The nature of this transition is still under debate, and the subject has become even more relevant with the realization that high-

View PDFchevron_right

References (29)

  1. F. Wegner, Z. Phys. B 35, 207 (1979).
  2. A. M. Finkelshtein, JETP Letters 45, 46 (1987).
  3. A. M. Finkelstein, Physica B 197, 636 (1994).
  4. D. Belitz and T. R. Kirkpatrick, Rev. Mod. Phys. 66, 261 (1994).
  5. K. B. Efetov, A. I. Larkin, and D. E. Khmel'nitskii, Sov. Phys. JETP 52, 568 (1980).
  6. A. M. Finkelshtein, Sov. Phys. JETP 57, 97 (1983).
  7. R. Oppermann, Physica A 167, 301 (1990).
  8. V. E. Kravtsov and R. Oppermann, Phys. Rev. B 43, 10865 (1991).
  9. V. E. Kravtsov, Z. Phys. B 82, 205 (1991).
  10. A. Altland, B. D. Simons, and D. Taras-Semchuk, JETP Letters 67, 22 (1998);
  11. Adv. Phys. 49, 321 (2000).
  12. K. B. Efetov, Adv. Phys. 32, 53 (1983).
  13. A. Kamenev and M. Mezard, Phys. Rev. B 60, 3944 (1999).
  14. I. V. Yurkevich and I. V. Lerner, Phys. Rev. B 60, 3955 (1999).
  15. A. Altland and A. Kamenev, cond-mat/0006363 (2000).
  16. A. Kamenev and A. Andreev, Phys. Rev. B 60, 3944 (1999).
  17. M. V. Feigel'man, A. I. Larkin, and M. A. Skvortsov, Phys. Rev. B 61, 1622 (2000).
  18. F. A. Berezin, Introduction to Superanalysis, Reidel, Dor- drecht (1987).
  19. L. P. Gor'kov, A. I. Larkin, and D. E. Khmel'nitskii, JETP Letters 30, 228 (1979).
  20. B. L. Altshuler, V. E. Kravtsov, and I. V. Lerner, Sov. Phys. JETP 64, 1352 (1986).
  21. K. D. Usadel, Phys. Rev. Lett. 25, 507 (1970).
  22. R. A. Riedel, L.-F. Chang, and P. Bagwell, Phys. Rev. B 54, 16082 (1996).
  23. I. V. Yurkevich and I. V. Lerner, unpublished.
  24. J. S. Langer and V. Ambegaokar, Phys. Rev. 164, 498 (1967);
  25. D. E. McCumber and B. I. Halperin, Phys. Rev. B 1, 1054 (1970).
  26. J.-M. Duan, Phys. Rev. Lett. 74, 5128 (1995);
  27. A. D. Zaikin, D. S. Golubev, A. van Otterlo, and G. T. Zimanyi, Phys. Rev. Lett. 78, 1552 (1997).
  28. A. Altland and M. R. Zirnbauer, Phys. Rev. B 55, 1142 (1997).
  29. I. V. Lerner, Phys. Rev. B 48, 9462 (1993).

Related papers

Nonlinear sigma model for disordered superconductors
Igor Yurkevich

Phys Rev B, 2001

We suggest a variant of the nonlinear σ model for the description of disordered superconductors. The main distinction from existing models lies in the fact that the saddle point equation is solved nonperturbatively in the superconducting pairing field. It allows one to use the model both in the vicinity of the metal-superconductor transition and well below its critical temperature with full account for the self-consistency conditions. We show that the model reproduces a set of known results in different limiting cases, and apply it for a self-consistent description of the proximity effect at the superconductor-metal interface.

View PDFchevron_right
Superconductivity in the (2+1)-dimensional nonlinearσmodel
Adriaan Schakel

Physical Review D, 1991

A (2+1)-dimensional field-theoretical model is constructed that shows magnetic-induced supercon- ductivity. The model consists of the O(3) nonlinear o. model interacting via a Yukawa coupling with Dirac fermions. where o. are the Pauli matrices. Since n is a unit vector, ~z~= 1; i.e. , z is constrained to the manifold CP' -SU(2)/U(1). Written in terms of the z fields model (1) becomes , (a"z'a~z b"b~), b, = i-z'a"z . As is well known [1], the CP' version of the o model is invariant under local gauge transformations of the form z~e' "z . A special feature of the Cu-0 superconductors is the presence of antiferromagnetism in the undoped case, where these systems are Mott-Hubbard insulators. Upon doping the Neel temperature decreases until the long- range magnetic order is destroyed and, eventually, these systems become superconducting. These phenomena motivate us to study a (2+ 1)-dimensional field- theoretical model describing Dirac fermions coupled to the O(3) nonlinear cr model. In this paper we point out that in the disordered phase of the nonlinear o. model

View PDFchevron_right
Theoretical studies of the interplay between superconductivity and disorder
Anirban Gangopadhyay

2012

Title of dissertation: THEORETICAL STUDIES OF THE INTERPLAY BETWEEN SUPERCONDUCTIVITY AND DISORDER Anirban Gangopadhyay, Doctor of Philosophy, 2012 Dissertation directed by: Dr. V. M. Galitski Department of Physics In this thesis, I explore a variety of disordered condensed matter systems and investigate questions pertinent to transport in such systems. In the first part of the thesis, I seek explanations for the strange feature of a giant magnetoresistance peak seen in the vicinity of superconductor-insulator transitions. To this end, I propose a semiclassical two-component Coulomb glass model for 2D insulators close to such transitions. I show that a local pairing attraction in Coulomb glasses can lead to crucial modification of the low-energy density of states which may affect transport. In another explanation for the peak, I consider an Anderson insulator of localized pairs and develop a theory of their transport. I study the change in localization length of the pairs (treated a...

View PDFchevron_right
Inhomogeneous superconductivity in disordered metals
Sergey Panyukov

A parameter rD is introduced to describe the temperature region near T,, in which the statistical spatial fluctuations of the order parameter are strong. It is shown on the basis of the Ginzburg-Landau functional, with the aid of the replica method, that two temperature superconductivity regimes are realized, depending on the degree of the disorder. At T, > rg = 2 . 4 9~~~ where T, is the Ginzburg parameter that characterizes the size of the region of strong thermodynamic fluctuations, the superconductivity is produced in spatially inhomogeneous fashion with droplike seeds. The drop density and their contribution to the free energy and to the diamagnetic susceptibility are obtained in a model of non-interacting drops. If rD < rg, superconductivity sets in below Tc simultaneously in the entire volume, i.e., the usual second-order transition is realized.

View PDFchevron_right
Emergence of nanoscale inhomogeneity in the superconducting state of a homogeneously disordered conventional superconductor
Jayesh Parmar

Scientific Reports, 2013

The notion of spontaneous formation of an inhomogeneous superconducting state is at the heart of most theories attempting to understand the superconducting state in the presence of strong disorder. Using scanning tunneling spectroscopy and high resolution scanning transmission electron microscopy, we experimentally demonstrate that under the competing effects of strong homogeneous disorder and superconducting correlations, the superconducting state of a conventional superconductor, NbN, spontaneously segregates into domains. Tracking these domains as a function of temperature we observe that the superconducting domains persist across the bulk superconducting transition, T c , and disappear close to the pseudogap temperature, T*, where signatures of superconducting correlations disappear from the tunneling spectrum and the superfluid response of the system. T he emergence of an inhomogeneous superconducting state in the presence of strong disorder is a recurring theme in the study of strongly disordered superconductors. When disorder is homogeneously distributed over atomic length scales, one would expect that its effect on the superconducting state will get averaged over length scales of the order of the superconducting coherence length (j), leaving a spatially uniform superconducting ground state. Quite early on, it was conjectured that for such a state the pairing energy scale, and consequently the superconducting transition temperature, T c , will remain finite even in the limit of very strong disorder 1,2 . However, more recent numerical simulations 3-5 indicate that even in the presence of homogeneous disorder the superconducting state can segregate into superconducting and non-superconducting regions both made of Cooper pairs, where global phase coherence is achieved below T c through Josephson coupling between the superconducting domains. From a theoretical standpoint it is now understood that although the pairing energy scale can remain finite even at strong disorder 6,7 , phase fluctuations between the superconducting domains can drive the transition from superconducting to non-superconducting state. Indeed, recent observation of a pseudogap state in several strongly disordered s-wave superconductors 8-10 (NbN, TiN and InO x ), characterized by a soft gap in the electronic energy spectrum which persists up to a temperature T*, well above T c , is consistent with this scenario.

View PDFchevron_right

Related topics

  • Proximity Effect

    • Cited by

    Granular superconductors: From the nonlinear σ model to the Bose-Hubbard description
    Igor Yurkevich

    Physical Review B, 2001

    We modify a nonlinear σ model (NLσM) for the description of a granulated disordered system in the presence of both the Coulomb repulsion and the Cooper pairing. We show that under certain controlled approximations this model is reduced to the Bose-Hubbard (or "dirty-boson") model with renormalized coupling constants. We obtain a more general effective action (which is still simpler than the full NLσM action) which can be applied in the region of parameters where the reduction to the Bose-Hubbard model is not justified. This action may lead to a different picture of the superconductor-insulator transition in 2D systems.

    View PDFchevron_right
    Superconducting edge states in a topological insulator
    V. Kagalovsky

    Scientific Reports

    We study the stability of multiple conducting edge states in a topological insulator against perturbations allowed by the time-reversal symmetry. A system is modeled as a multi-channel Luttinger liquid, with the number of channels equal to the number of Kramers doublets at the edge. Assuming strong interactions and weak disorder, we first formulate a low-energy effective theory for a clean translation invariant system and then include the disorder terms allowed by the time-reversal symmetry. In a clean system with N Kramers doublets, N − 1 edge states are gapped by Josephson couplings and the single remaining gapless mode describes collective motion of Cooper pairs synchronous across the channels. Disorder perturbation in this regime, allowed by the time reversal symmetry is a simultaneous backscattering of particles in all N channels. Its relevance depends strongly on the parity if the number of channel N is not very large. Our main result is that disorder becomes irrelevant with t...

    View PDFchevron_right
    580 California St., Suite 400
    San Francisco, CA, 94104
    © 2025 Academia. All rights reserved