Bardeen-Cooper-Schrieffer Model Research Articles

Operation of kinetic-inductance travelling wave parametric amplifiers at millimetre wavelengths

It is expected that the operation of microwave kinetic inductance travelling wave parametric amplifiers (KITWPAs) can be extended to the millimetre (mm) and the sub-mm wavelength range as long as the frequency is below the gap frequency of the superconducting film. This paper presents possible mm-wave designs for KITWPAs based on microstrip transmission lines. Our device is designed based on the BCS (Bardeen–Cooper–Schrieffer) model which successfully reproduces the measured transmission profile, gain, bandwidth, and nonlinear response of a fabricated KITWPA operating in the Ka-band, and includes the millimetre-wave dielectric loss determined by fitting the quality factor of a W-band microstrip resonator. We suggest a layout for a KITWPA operating near 220 GHz that can be fabricated using the same superconducting properties as the Ka-band device and can be coupled to a waveguide system. We conclude the paper by extending the 220 GHz design to higher frequency regimes approaching 1 THz.

Ta181 nuclear quadrupole resonance study of the noncentrosymmetric superconductor PbTaSe2

We report on a pure $^{181}$Ta-nuclear quadrupole resonance (NQR) measurement of PbTaSe$_2$ at zero magnetic field, which has the advantage of directly probing the intrinsic superconducting phase and electronic states of the TaSe$_2$ layer. We observed the $^{181}$Ta-NQR spectrum of the intrinsic structure with space group $P6$-$m2$, which agrees well with density functional theory (DFT) calculations. The nuclear spin relaxation rate ($1/T_1$) shows an exponential decrease well below $T_{\rm_c}$, indicating that the superconducting state is fully gapped in the framework of Bardeen-Cooper-Schrieffer (BCS) theory. The gap size obtained by $^{181}$Ta-NQR was smaller than the value in previous reports, which may imply that the Fermi surfaces composed of Ta-5$d$ orbitals, where the average pairing interactions are expected to be weaker than in BCS model, are primarily probed. The temperature dependence of $1/T_1$ below $T_{\rm_c}$ can be reproduced well by the superposition of quadrupole and magnetic relaxation mechanisms, together with the distribution of superconducting gap size inherent to multiple Fermi surfaces theoretically proposed in PbTaSe$_2$.

Boundary states with elevated critical temperatures in Bardeen-Cooper-Schrieffer superconductors

Bardeen-Cooper-Schrieffer (BCS) theory describes a superconducting transition as a single critical point where the gap function or, equivalently, the order parameter vanishes uniformly in the entire system. We demonstrate that in superconductors described by standard BCS models, the superconducting gap survives near the sample boundaries at higher temperatures than superconductivity in the bulk. Therefore, conventional superconductors have multiple critical points associated with separate phase transitions at the boundary and in the bulk. We show this by revising the Caroli-De Gennes-Matricon theory of a superconductor-vacuum boundary and finding inhomogeneous solutions of the BCS gap equation near the boundary, which asymptotically decay in the bulk. This is demonstrated for a BCS model of almost free fermions and for lattice fermions in a tight-binding approximation. The analytical results are confirmed by numerical solutions of the microscopic model. The existence of these boundary states can manifest itself as discrepancies between the critical temperatures observed in calorimetry and transport probes.

Postquench gap dynamics of two-band superconductors

Recent experimental progress in the fields of cold quantum gases and ultrafast optical spectroscopy of quantum materials allows to controllably induce and probe non-adiabatic dynamics of superconductors and superfluids. The time-evolution of the gap function before relaxation with the lattice is determined by the superposition of coherently evolving individual Cooper pairs within the manifold of the Bardeen-Cooper-Schrieffer (BCS) wavefunction. While dynamics following an abrupt quench of the pairing interaction strength in the single-band BCS model has been exactly solved due to the integrability of the model, the dynamics of post-quench multi-band superconductors remain under scrutiny. Here, we develop a generalization of the Volkov-Kogan Laplace-space perturbative method that allows us to determine the non-adiabatic gap dynamics of two-band fully gapped superconductors for a wide range of quench amplitudes. Our approach expands the long-time dynamics around the steady-state asymptotic value of the gap, which is self-consistently determined, rather than around the equilibrium value of the gap. We explicitly demonstrate that this method recovers the exact solution of the long-time gap dynamics in the single-band case and perfectly agrees with a numerical solution of the two-band model. We discover that dephasing of Cooper pairs from different bands leads to faster collisionless relaxation of the gap oscillation with a power-law of $t^{-3/2}$ instead of the well-known $t^{-1/2}$ behavior found in the single-band case. Furthermore, the gap oscillations display beating patterns arising from the existence of two different asymptotic gap values. Our results have important implications to a variety of two-band superconductors driven out of equilibrium, such as iron-based superconductors, MgB$_{2}$, and SrTiO$_{3}$.

Real and imaginary energy gaps: a comparison between single excitation Superradiance and Superconductivity and robustness to disorder

A comparison between the single particle spectrum of the discrete Bardeen-Cooper-Schrieffer (BCS) model, used for small superconducting grains, and the spectrum of a paradigmatic model of Single Excitation Superradiance (SES) is presented. They are both characterized by an equally spaced energy spectrum (Picket Fence) where all the levels are coupled between each other by a constant coupling which is real for the BCS model and purely imaginary for the SES model. While the former corresponds to the discrete BCS-model describing the coupling of Cooper pairs in momentum space and it induces a Superconductive regime, the latter describes the coupling of single particle energy levels to a common decay channel and it induces a Superradiant transition. We show that the transition to a Superradiant regime can be connected to the emergence of an imaginary energy gap, similarly to the transition to a Superconductive regime where a real energy gap emerges. Despite their different physical origin, it is possible to show that both the Superradiant and the Superconducting gaps have the same magnitude in the large gap limit. Nevertheless, some differences appear: while the critical coupling at which the Superradiant gap appears is independent of the system size $N$, for the Superconductivity gap it scales as $(\ln N)^{-1}$. The presence of a gap in the imaginary energy axis between the Superradiant and the Subradiant states shares many similarities with the "standard" gap on the real energy axis: the superradiant state is protected against disorder from the imaginary gap as well as the superconducting ground state is protected by the real energy gap. Moreover we connect the origin of the gapped phase to the long-range nature of the coupling between the energy levels.

Heavy Electron Superconductors in Terms of Block Spin Phenomenology

We investigate the relationship between superconductivity and spin glasses, which were observed to be coexistent in heavy fermion superconductors (HFSs). We begin by explaining the phase of spin glass using concepts of finite-sized block spin. We then introduce the phase of superconductivity in HFSs as collective Cooper pairing, that is, the pairing of block spins with net spin = 1/2, each of which is comprised of a large number of random spins that together produce a majority spin direction. The superconducting Bardeen-Cooper-Schrieffer (BCS)-type phonon-mediated electron-electron interaction is substituted for the BCS-type electron-electron interaction mediated by phonon-enhanced spin flips previously suggested by us. An effective charge of any value stemming from an electron attached to the flux of an electric field can be referred to as a composite charge. The distinguished difference between ours and the original BCS model is the following: (i) another BCS-type interaction, (ii) bare electrons that are substituted for block spins with net spin = 1/2 and net charge = e in the presence of electric fields, and (iii) Fermi-Dirac distributions that are replaced by a new distribution called as Brillouin distribution.

Studying temperature dependence of pairing gap parameter in a nucleus as a small superconducting system

In this paper, we have taken the effect of small size of nucleus and static fluctuations into account in the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity calculations of [Formula: see text]Ti nucleus. Thermodynamic quantities of [Formula: see text]Ti have been extracted within the BCS model with the inclusion of the average value of the pairing gap square, extracted by the modified Ginzburg–Landau (MGL) method for small systems. Calculated values of the excitation energy and entropy within the MGL+BCS method improve the extracted results within the usual BCS model and show a smooth behavior around the critical temperature with a very good agreement with the semi-empirical values. The result of using MGL+BCS method for the heat capacity of [Formula: see text]Ti is compared with the corresponding semi-empirical values and the calculated values within the BCS, static path approximation (SPA) and Modified Pairing gap BCS (MPBCS) which is a method that was proposed in our previous publications. Both MGL+BCS and MPBCS avoid the discontinuity of the heat capacity curve, which is observed in the usual BCS method, and lead to an S-shaped curve with a good agreement with the semi-empirical results.

The Higgs mechanism and superconductivity: A case study of formal analogies

Following the experimental discovery of the Higgs boson, physicists explained the discovery to the public by appealing to analogies with condensed matter physics. The historical root of these analogies is the analogies to models of superconductivity that inspired the introduction of spontaneous symmetry breaking (SSB) into particle physics in the early 1960s. We offer a historical and philosophical analysis of the analogies between the Higgs model of the electroweak (EW) interaction and the Ginsburg–Landau (GL) and Bardeen–Cooper–Schrieffer (BCS) models of superconductivity, respectively. The conclusion of our analysis is that both sets of analogies are purely formal in virtue of the fact that they are accompanied by substantial physical disanalogies. In particular, the formal analogies do not map the temporal, causal, or modal structures of SSB in superconductivity to temporal, causal, or modal structures in the Higgs model. These substantial physical disanalogies mean that analogies to models of superconductivity cannot supply the basis for the physical interpretation of EW SSB; however, an appreciation of the contrast between the physical interpretations of SSB in superconductivity and the Higgs model does help to clarify some foundational issues. Unlike SSB in superconductivity, SSB in the Higgs sector of the Standard Model (without the addition of new physics) is neither a temporal nor a causal process. We discuss the implications for the ‘eating’ metaphor for mass gain in the Higgs model. Furthermore, the distinction between the phenomenological GL model and the dynamical BCS model does not carry over to EW models, which clarifies the desiderata for the so-called ‘dynamical’ models of EW SSB (e.g., minimal technicolor). Finally, the development of the Higgs model is an illuminating case study for philosophers of science because it illustrates how purely formal analogies can play a fruitful heuristic role in physics.

The entropy excess and moment of inertia excess ratio with inclusion of statistical pairing fluctuations

The entropy excess of 163 Dy compared to 162 Dy as a function of nuclear temperature have been investigated using the mean value Bardeen–Cooper–Schrieffer (BCS) method based on application of the isothermal probability distribution function to take into account the statistical fluctuations. Then, the spin cut-off excess ratio (moment of inertia excess ratio) introduced by Razavi [Phys. Rev. C88 (2013) 014316] for proton and neutron system have been obtained and are compared with their corresponding data on the BCS model. The results show that the overall agreement between the BCS model and mean value BCS method is satisfactory and the mean value BCS model reduces fluctuations and washes out singularities. However, the expected constant value in the entropy excess is not reproduced by the mean value BCS method.

A study of self-consistent Hartree–Fock plus Bardeen–Cooper–Schrieffer calculations with finite-range interactions

In this work we test the validity of a Hartree–Fock plus Bardeen–Cooper–Schrieffer model in which a finite-range interaction is used in the two steps of the calculation by comparing the results obtained to those found in fully self-consistent Hartree–Fock–Bogoliubov calculations using the same interaction. Specifically, we consider the Gogny-type D1S and D1M forces. We study a wide range of spherical nuclei, far from the stability line, in various regions of the nuclear chart, from oxygen to tin isotopes. We calculate various quantities related to the ground state properties of these nuclei, such as binding energies, radii, charge and density distributions, and elastic electron scattering cross sections. The pairing effects are studied by direct comparison with the Hartree–Fock results. Despite its relative simplicity, in most cases, our model provides results very close to those of the Hartree–Fock–Bogoliubov calculations, and it reproduces the empirical evidence of pairing effects rather well in the nuclei investigated.

A Possible Fundamental Theory of Cooper Pair Formation in Unconventional Superconductivity

The successful application of the electron-phonon interaction (EPI) mechanism in formulating the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity is among the most outstanding intellectual achievements in theoretical physics because of the successful application of the theory to the conventional superconducting materials. Therefore, its unsuccessful application to the nonconventional superconducting materials has led to the search for new theories which generalized it, include an interplay with other mechanisms or are formulated from non – EPI mechanisms. We observe in this current study that to achieve a generalized theory of superconductivity, there is need to first developed a quantitative structure model of the Cooper pair formation (CPF) in line with the formulation of the molecules in nature which has given birth to hadronic mechanics. This generalized formulation is the iso-superonductivity model which is based on the observation that the Cooper pair of the standard BCS model may have a nonlocal-nonhamiltonian structure equivalent to the strong interaction (hadronic mechanics (HM)) structure of the neutral pion, as compressed positronium atom at short distances ( 1 F) is by the quantum mechanical superexchange interaction. This generalized CPF has been used to successfully explain the high-Tc superconductivity in the cuprates as well as to account for the transition temperatures of these materials. It is also used to account for the high-Tc iron based superconducting materials. PACS: 74.20._z, 74.62.Bf; 75.30.Et,

Superconductivity in cuprates: Details of electron phonon coupling

AbstractThe Bardeen‐Cooper‐Schrieffer (BCS) model explains superconductivity (SC) as due to correlation between electronic momentum and nuclear momentum (phonons) in a free electron gas. The BCS model lacks chemical specificity, however, since the coupling mechanism is left unspecified. After the discovery of high TC superconductivity in 1986 it was concluded that electron–phonon interactions are insufficient to explain electron pairing. A large part of theoretical research has since been aiming at finding another mechanism that would allow us to consider the superconducting system as a gas of charged free bosons. However, there appears to be no reason to assume free electrons in oxides. In this article the free‐electron criterion is therefore replaced by the criterion that a pair of electrons can move freely between sites without resistance, i.e., without activation energy. Electron pair transfer is treated in a many‐electron real space approach using standard mixed‐valence theories. Mott‐Hubbard‐U is strictly defined, its dependence on breathing mode coordinates analyzed, and the connection between U and the energy gap for superconductivity clarified. d‐wave gap anisotropy is found to be consistent with the general atomic level model presented here. Softening of phonon half‐breathing modes in inelastic neutron scattering (INS) is connected to mixed‐valency. The fundamental vibronic interaction between spin density wave (SDW) and charge density wave (CDW) states leads to a new phase with energy gap and electron pair carriers that can only be the superconducting phase. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010

Magnetic enhancement of superconductivity from electron spin domains.

Since the discovery of superconductivity, there has been a drive to understand the mechanisms by which it occurs. The BCS (Bardeen-Cooper-Schrieffer) model successfully treats the electrons in conventional superconductors as pairs coupled by phonons (vibrational modes of oscillation) moving through the material, but there is as yet no accepted model for high-transition-temperature, organic or 'heavy fermion' superconductivity. Experiments that reveal unusual properties of those superconductors could therefore point the way to a deeper understanding of the underlying physics. In particular, the response of a material to a magnetic field can be revealing, because this usually reduces or quenches superconductivity. Here we report measurements of the heat capacity and magnetization that show that, for particular orientations of an external magnetic field, superconductivity in the heavy-fermion material CeCoIn(5) is enhanced through the magnetic moments (spins) of individual electrons. This enhancement occurs by fundamentally altering how the superconducting state forms, resulting in regions of superconductivity alternating with walls of spin-polarized unpaired electrons; this configuration lowers the free energy and allows superconductivity to remain stable. The large magnetic susceptibility of this material leads to an unusually strong coupling of the field to the electron spins, which dominates over the coupling to the electron orbits.

Pair-kinetic-levels sum and difference in BCS model for generic systems with non-Fermi liquid states-I

The Bardeen–Cooper–Schrieffer (BCS) singlet-pairing model, now with spin-dependent electronic kinetic levels, is analysed in mean-field approximation. Outside the usual equikinetic-pair (EP) superconductivity, new pair-regimes are revealed as in some correlated electron systems (CES), which it now generally models. The essential new feature of this pair scheme is its conformity with Haldane's fractional exclusion principle. Application of the parametrized principle on the model, leads to novel realizations of non-Fermi liquid (NFL) states. The EP superconductivity, and the NFL regimes with non-equikinetic-pair (NEP) states, are governed by non-vanishing symmetry parameters: pair-kinetic-levels sum and difference, respectively. New NEP superconductivity arises from NFL normal state, the later exhibiting magnetism. The obtained pair-states are consistent with those experimentally reported for hole-doped cuprates and, more recently, for CePd 2Si 2, by the Cambridge–Hiroshima collaboration. Green's function analysis of the normal state NFL yields an energy spectrum leading to new semi-empirical formulae for the antiferromagnetic exchange and Neel temperature. Theory is in good qualitative and quantitative agreement with experiment on NFL YBa 2Cu 3O 6+ y and CePd 2(Al 1− x Ga x ) 3.

N-quantum approach to the BCS theory of superconductivity

A method of general applicability to the solution of second-quantized field theories at finite temperature is illustrated using the BCS (Bardeen–Cooper–Schrieffer) model of superconductivity. Finite-temperature field theory is treated using the thermo field-theory formalism of Umezawa and collaborators. The solution of the field theory uses an expansion in thermal modes analogous to the Haag expansion in asymptotic fields used in the N-quantum approximation at zero temperature. The lowest approximation gives the usual gap equation.

Analytic solution of the Bardeen–Cooper–Schrieffer gap equation in one, two, and three dimensions

Analytic expressions for the Bardeen–Cooper–Schrieffer (BCS) gap of a many-electron system with the BCS model interaction are obtained in one, two, and three dimensions in the weak coupling limit, but for arbitrary interaction width ν≡ℏωD/EF, 0<ν<∞. Here ℏωD is the maximum energy of a force-mediating boson, and EF is the Fermi energy (which is fixed by the electronic density). Our results address both phononic (ν≪1) as well as nonphononic (e.g., exciton, magnon, plasmon, etc.) pairing mechanisms where the mediating boson energies are not small compared with EF, provided weak electron–boson coupling prevails. The essential singularity in coupling, sometimes erroneously attributed to the two-dimensional character of the BCS model interaction with ν≪1, is shown to appear in one, two, and three dimensions before the limit ν→0 is taken.

Two-component Fermi systems: II. Superfluid coupled cluster theory

In this second paper of a series the coupled cluster method (CCM) or exp(S) formalism is applied to two-component Fermi superfluids using a Bardeen-Cooper-Schrieffer (BCS) ground state as a zeroth-order approximation. We concentrate on developing the formalism necessary for carrying out eventual numerical calculations on realistic superconducting systems. We do this by generalising the one-component formalism in an appropriate manner and by using the results in the first paper of this series, where we studied two-component Fermi fluids. We stress the previous successes of the CCM, both from the point of view of analytic and numerical results, and we further indicate its potential for studying superconductivity. We restrict ourselves here to a so-called ring plus single particle energy (RING+SPE) approximation for general potentials and show how it can be formulated as a set of four coupled, bilinear integral equations for the cluster-integrated amplitudes. These latter amplitudes are themselves derived from the four-point functions of the system which provide a measure of the two-particle/two-hole component in the true ground-state wavefunction with respect to the BCS model state. We indicate how to obtain possible analytic solutions.

Two-component Fermi systems. I. Fluid coupled cluster theory

In this second paper of a series the coupled cluster method (CCM) or exp(S) formalism is applied to two-component Fermi superfluids using a Bardeen-Cooper-Schrieffer (BCS) ground state as a zeroth-order approximation. We concentrate on developing the formalism necessary for carrying out eventual numerical calculations on realistic superconducting systems. We do this by generalising the one-component formalism in an appropriate manner and by using the results in the first paper of this series, where we studied two-component Fermi fluids. We stress the previous successes of the CCM, both from the point of view of analytic and numerical results, and we further indicate its potential for studying superconductivity. We restrict ourselves here to a so-called ring plus single particle energy (RING+SPE) approximation for general potentials and show how it can be formulated as a set of four coupled, bilinear integral equations for the cluster-integrated amplitudes. These latter amplitudes are themselves derived from the four-point functions of the system which provide a measure of the two-particle/two-hole component in the true ground-state wavefunction with respect to the BCS model state. We indicate how to obtain possible analytic solutions.

Common Interpretation of Phase Transitions in Various Models

A discussion is presented of the formal analogy which exists between the Bardeen-Cooper-Schrieffer (BCS) model for superconductivity and the several variations of the model suggested by Kac for the exact study of phase transitions. Some of the algebraic techniques, pioneered by Haag for his formulation of axiomatic field theory and his application of this theory to the BCS model, are shown to extend to all these cases. Some arguments are given in the beginning of the paper to point out the necessity of a new interpretation of the existence of various phases in the same physical system. The results obtained so far for equilibrium situations confirm the consistency of the proposed tentative description of the phenomena considered.

Ultrasonic Determination of the Superconducting Energy Gap in Thallium

The temperature-dependent energy gap $2\ensuremath{\Delta}(T)$ in single crystals of high-purity thallium has been determined by measurements of the attenuation in the normal and superconducting states of longitudinal acoustic waves. Determinations were made for sound-wave propagation along the [0002], [$10\overline{1}0$], and [$1\overline{2}10$] crystallographic axes. The energy gap $2\ensuremath{\Delta}(0)$ at 0\ifmmode^\circ\else\textdegree\fi{}K for propagation along these axes, respectively, was found to be $3.76k{T}_{c}\ifmmode\pm\else\textpm\fi{}0.03k{T}_{c}$, $4.10k{T}_{c}\ifmmode\pm\else\textpm\fi{}0.03k{T}_{c}$, and $4.00k{T}_{c}\ifmmode\pm\else\textpm\fi{}0.10k{T}_{c}$. Ultrasonic attenuation gives a larger value for this limiting energy gap than that observed by other methods. The decrease in attenuation below ${T}_{c}$ was found to be more abrupt than predicted by the Bardeen-Cooper-Schrieffer (BCS) theory. It is suggested that the simple BCS model does not give a complete picture of the ultrasonic attenuation in very pure superconductors.