We consider the single particle spectral function for a two-dimensional clean superconductor in a regime of strong critical thermal phase fluctuations. In the limit where the maximum of the superconducting gap is much smaller than the Fermi energy we obtain an exact expression for the spectral function integrated over the momentum component perpendicular to the Fermi surface.
We studied the influence of the amplitude fluctuations of a non-Fermi superconductor on the energy spectrum of the 2D Anderson non-Fermi system. The classical fluctuations give a temperature dependence in the pseudogap induced in the fermionic excitations.
It is shown that the specific heat jump at the superconducting transition temperature in pnictide and cuprate superconductors is produced by ferroelastic fluctuations just below the transition temperature. The amplitude of the corresponding lattice distortion is estimated. It is shown that a contribution from ferroelastic fluctuations to the specific heat jump is also present in low-temperature superconductors.
A theory of fluctuations in two-band superconductor MgB$_{2}$ is developed. Since the standard Ginzburg-Landau (GL) approach fails in description of its properties, we generalize it basing on the microscopic theory of a two-band superconductor. Calculating the microscopic fluctuation propagator, we build up the nonlocal two-band GL functional and the corresponding time-dependent GL equations. This allows us to calculate the main fluctuation observables such as fluctuation specific heat and conductivity.
Based on recent experimental results for electron-doped cuprate oxides and ferromagnetic superconductors, it is shown that antiferromagnetic fluctuations always develop in the superconducting phase of both low- and high-temperature superconductors. The relation between the magnitude of the antiferromagnetic pseudogap and the characteristic temperature of the antiferromagnetic pseudogap opening is obtained. The characteristic temperature of the antiferromagnetic pseudogap opening for metal superconductors is estimated.
Iron-based superconductors can exhibit different magnetic ground states and are in a critical magnetic region where frustrated magnetic interactions strongly compete with each other. Here we investigate the longitudinal modes of spin fluctuations in an unified effective magnetic model for iron-based superconductors. We focus on the collinear antiferromagnetic phase (CAF) and calculate the behavior of the longitudinal modes when different phase boundaries are approached. The results can help to determine the nature of the magnetic fluctuations in iron-based superconductors.
Magnetic field fluctuations are inevitable in practical applications of superconductors and it is often necessary to estimate the AC losses these fluctuations induce. If the fluctuation wavelength is greater than the size of a superconductor, known estimates for an alternating uniform external magnetic field can be employed. Here we consider the opposite case and analyze, using a model critical-state problem, penetration of spatially nonuniform fluctuations into type-II superconductors. Numerical simulation is based on a variational formulation of the Bean model. The analytical solutions, foun…
We study the effects of low-energy nodal quasiparticles on the classical phase fluctuations in a two-dimensional d-wave superconductor. The singularities of the phase-only action at T\to 0 are removed in the presence of disorder, which justifies using an extended classical XY-model to describe phase fluctuations at low temperatures.
We study the effects of quantum vortex fluctuations in two-dimensional superconductors using a dual theory of vortices, and investigate the relevance to underdoped cuprates where the superconductor-insulator transition (SIT) is possibly driven by quantum vortex proliferation. We find that a broad enough phase fluctuation regime may exist for experimental observation of the quantum vortex fluctuations near SIT in underdoped cuprates. We propose that this scenario can be tested via pair-tunneling experiments which measure the characteristic resonances in the zero-temperature pair-field susceptib…
Spin fluctuations are widely believed to play an important role in the superconducting mechanisms of unconventional high-temperature superconductors. Spin fluctuations have been observed in iron-based superconductors as well. However, in some iron-based superconductors such as LaFePO$_{0.9}$, they have not been observed by inelastic neutron scattering (INS). LaFePO$_{0.9}$ is an iron-based superconductor with a low superconducting transition temperature ($T_{\rm c}$= 5 K), where line nodes are observed in the superconducting gap function. The line-node symmetry typically originates from sign r…
Thermal fluctuation conductivity for a layered superconductor in perpendicular magnetic field is treated in the frame of the self-consistent Hartree approximation for an arbitrarily strong in-plane electric field. The simultaneous application of the two fields results in a slightly stronger suppression of the superconducting fluctuations, compared to the case when the fields are applied individually.
We derive Ginzburg-Landau-like action for two-dimensional disordered superconductor under far-from-equilibrium conditions in a fluctuational regime. Then, utilizing it, we calculate fluctuation induced density of states, Maki-Thomson and Aslamazov-Larkin type contributions to the in-plane electrical conductivity. We apply our approach to thin superconducting film sandwiched between a gate and a substrate that have different temperatures and different electrochemical potentials.
The review is devoted to the detailed description of the fluctuation phenomena in superconductors. Developing phenomenological methods through the first five sections we deal with the direct fluctuation pair contributions only. The indirect fluctuation effects are discussed in the last two sections, devoted to the microscopic justification of the time dependent Ginzburg-Landau equation, the description of the microscopic theory of fluctuations, and discussion of their manifestations in various physical properties of superconductors. The first six sections are written in detail, so they can ser…
The effect of fluctuation induced weakly first order phase transition known for three dimensional (3D) type I superconductors appears in a modified and strongly enhanced variant in thin (quasi-2D) superconducting films. The unusual thermodynamic properties of this new type of first order phase transitions and the possibility for an experimental verification of the effect are established and discussed.
We study the superconducting order parameter fluctuations near the phase transition into the Larkin-Ovchinnikov-Fulde-Ferrell state in the clean limit at zero temperature. In contrast to the usual normal metal-to-uniform superconductor phase transition, the fluctuation corrections are dominated by the modes with the wave vectors away from the origin. We find that the superconducting fluctuations lead to a divergent spin susceptibility and a breakdown of the Fermi-liquid behavior at the quantum critical point.
It is shown that a ferroelastic lattice distortion is associated with the superconducting transition both in low- and high-temperature superconductors. In layered tetragonal superconductors with a sufficiently high transition temperature, a ferroelastic distortion is improper and is caused by an out-of-plane charge ordering. The relation between the out-of-plane charge gap and the maximum superconducting transition temperature for these superconductors is obtained. It is shown also that the specific heat jump at the superconducting transition temperature in pnictide and cuprate superconductors…
Spin subsystem of the copper oxides within the polar Jahn-Teller centers model corresponds to a singlet-triplet magnet where the local boson movement accompanied by the induced longitudinal spin fluctuations. These fluctuations determine the features of the magnetic inelastic scattering in the high-Tc cuprates.
We study the order parameter phase fluctuation effects in cuprate superconductors near T=0, using a quasi-two-dimensional d-wave BCS model. An effective phason theory is obtained which is used to estimate the strength of the fluctuations, the fluctuation correction to the in-plane penetration depth, and the pair-field susceptibility. We find that while the phase fluctuation effects are difficult to observe in the renormalization of the superfluid phase stiffness, they may be observed in a pair tunneling experiment which measures the pair-field susceptibility.
Within the phase fluctuation picture for the pseudogap state of a high-$T_{c}$ superconductor, we incorporate the phase fluctuations generated by the classical XY model with the Bogoliubov-de Gennes formalism utilizing a field-theoretical method. This picture delineates the inhomogeneous characteristics of local order parameters observed in high-$T_{c}$ superconductors above $T_{c}$. We also compute the local density of states near a non-magnetic impurity with a strong scattering potential. The resonance peak smoothly evolves as temperature increases through $T_{c}$ without showing any sudden …
Recent measurements of fluctuation diamagnetism in high temperature superconductors show distinct features above and below Tc, which can not be explained by simple gaussian fluctuation theory. Self consistent calculation of magnetization in layered high temperature superconductors, based on the Ginzburg-Landau-Lawrence-Doniach model and including all Landau levels is presented. The results agree well with the experimental data in wide region around Tc, including both the vortex liquid below Tc and the normal state above Tc. The gaussian fluctuation theory significantly over estimates the diama…
The observation of a strong nonlinear diamagnetic signal well above T_c in several cuprate superconductors constitutes clear evidence of the persistence of local superconducting correlations.
If phase coherence determines the superconducting transition temperature Tc in the cuprate oxides, it is of great interest to understand the role that dynamics of the phase fluctuation plays in bringing about depletion of the superconducting condensate. We will show that a phase correlation function can be calculated that allows us to describe depletion of superconducting condensate as a result of the quantum fluctuation. In two dimension, dynamic phase fluctuation or pair fluctuation gives rise to a condensate depletion linear with temperature as T$\to O$ in superconductors with nodes at the …
We report neutron inelastic scattering measurements on polycrystalline LaFePO and Sr2ScO3FeP, two members of the iron phosphide families of superconductors. No evidence is found for any magnetic fluctuations in the spectrum of either material in the energy and wavevector ranges probed. Special attention is paid to the wavevector at which spin-density-wave-like fluctuations are seen in other iron-based superconductors. We estimate that the magnetic signal, if present, is at least a factor of four (Sr2ScO3FeP) or seven (LaFePO) smaller than in the related iron arsenide and chalcogenide supercond…
Here we consider the universal heat conduction and the angular dependent thermal conductivity in the vortex state for a few nodal superconductors. We present the thermal conductivity as a function of impurity concentration and the angular dependent thermal conductivity in a few nodal superconductors. This provides further insight in the gap symmetry of superconductivity in Sr$_2$RuO$_4$ and UPd$_2$Al$_3$.
We use large-scale Monte-Carlo simulations to study thermal fluctuations in chiral $p$-wave superconductors in an applied magnetic field in three dimensions. We consider the thermal stability of previously predicted unusual double-quanta flux-line lattice ground states in such superconductors. In previous works it was shown that, neglecting thermal fluctuations, a chiral $p$-wave superconductor forms an hexagonal lattice of doubly-quantized vortices, except extremely close to the vicinity of $H_{c2}$ where double-quanta vortices split apart. We find dissociation of double-quanta vortices drive…
As a part of my doctoral thesis, I studied the effect of phase fluctuations in two dimensional (2D) and three dimensional (3D) superconducting films of conventional s-wave superconductor, NbN, by measuring the electrodynamics response through magnetic penetration depth at low frequency and the complex conductivity at microwave frequencies. We show that the phase transition in 2D or quasi-2D superconductors can be reconciled with the standard Berezinskii-Kosterlitz-Thouless (BKT) transition when the small vortex core energy is taken into consideration. In 3D NbN films, we show that in presence …
We study the effect of site diagonal disorder on the pairing amplitude by a perturbation method. Using an extended Hubbard model with the intersite attraction we analyze fluctuations of order parameter in presence of non-magnetic disorder and discuss the instability of various solutions with p-wave paring.
The paraconductivity, originating from critical superconducting order-parameter fluctuations in the vicinity of the critical temperature in a layered superconductor is calculated in the frame of the self-consistent Hartree approximation, for an arbitrarily strong electric field and zero magnetic field. The paraconductivity diverges less steep towards the critical temperature in the Hartree approximation than in the Gaussian one and it shows a distinctly enhanced variation with the electric field. Our results indicate that high electric fields can be effectively used to suppress order-parameter…
We study the renormalization of sound attenuation and sound velocity by fluctuation Cooper pairs in layered superconductors. We consider the influence of s- and d-wave symmetry of the fluctuating order parameter, on both longitudinal and transverse phonon modes. We show that both unconventional order parameter symmetry and transverse sound polarization suppress the AL and MT terms, while the DOS contribution is the least affected. The combination of these effects can change the sign of the overall fluctuation corrections above Tc. We also compare the results obtained using the Ginzburg-Landau …
ARPES measurements on HiTc superconductors such as BiSrCaCuO reveal a quasiparticle peak, a broad background and a dip. The background may originate from inelastic spin fluctuation emission. Calculations of the spectral weight are presented incorporating these inelastic processes. Superconducting correlations have been included in the magnetic susceptibility. The dip feature in the spectral weight is evidence for a spin gap in the superconducting state.
We compare the phase-diagrams of an effective theory of a three-dimensional multi-band superconductor obtained within standard and cluster mean-field theories, and in large-scale Monte Carlo simulations. In three dimensions, mean field theory fails in locating correctly the positions of the phase transitions, as well as the character of the transitions between the different states. A cluster mean-field calculations taking into account order-parameter fluctuations in a local environment improves the results considerably for the case of extreme type-II superconductors where gauge-field fluctuati…
We calculate suppression of inter- and intralayer superconducting currents due to equilibrium phase fluctuations and find that, in contrast to a recent prediction, the effect of thermal fluctuations cannot account for linear temperature dependence of the superfluid density in high-Tc superconductors at low temperatures. Quantum fluctuations are found to dominate over thermal fluctuations at low temperatures due to hardening of their spectrum caused by the Josephson plasma resonance. Near Tc sizeable thermal fluctuations are found to suppress the critical current in the stack direction stronger…
The excess Hall conductivity, resulting from thermal fluctuations of the superconducting order parameter, is calculated for a layered superconductor for an arbitrarily strong in-plane electric field and a perpendicular magnetic field in the frame of the time-dependent Ginzburg-Landau theory. The fluctuation Hall conductivity is suppressed in high electric fields much stronger than the longitudinal one. For high-temperature superconductors we predict a pronounced non-Ohmic behavior of the excess Hall effect near the critical temperature in moderate magnetic fields and electric fields of the ord…
Phase fluctuations of the superconducting order parameter play a larger role in the cuprates than in conventional BCS superconductors because of the low superfluid density of a doped insulator. In this paper, we analyze an XY model of classical phase fluctuations in the high temperature superconductors using a low-temperature expansion and Monte Carlo simulations. In agreement with experiment, the value of the superfluid density at temperature T=0 is a quite robust predictor of Tc, and the evolution of the superfluid density with T, including its T-linear behavior at low temperature, is insens…
The short wavelength cutoff has been introduced in the calculation of ac fluctuation conductivity of superconductors. It is shown that a finite cutoff leads to a breakdown of the scaling property in frequency and temperature. Also, it increases the phase phi of the complex conductivity (tan(phi)=sigma_2/sigma_1) beyond pi/4 at T_c. Detailed expressions containing all essential parameters are derived for 3D isotropic and anisotropic fluctuation conductivity. In the 2D case we obtain individual expressions for the fluctuation conductivity for each term in the sum over discrete wavevectors perpen…
It is conjectured that the anomalous spin dynamics observed in the normal state of cuprate superconductors might find its origin in a nearly ordered spin system which is kept in motion by thermally meandering charged domain walls. `Temperature sets the scale' finds a natural explanation, while a crossover to a low temperature quantum domain wall fluid is implied.
We study the superconducting transition temperature, T_c, of an array of fluctuating, maximally gapped interacting ladders, embedded in a disordered plane. Renormalization group analysis indicates that geometrical fluctuations mitigate the suppression of the intra-ladder pairing correlations by the disorder. In addition, the fluctuations enhance the Josephson tunneling between the ladders. Both effects lead to an increase of T_c in the Josephson coupled array. These findings may be relevant to the understanding of the 1/8 anomaly in the lanthanum based high-temperature superconductors.
Neutron scattering has been used to study the spin fluctuations in the YBa$_2$Cu$_3$O$_{7-x}$ and Bi$_2$Sr$_2$CaCu$_2$O$_8$ materials. Evidence is found for both incommensurate fluctuations and a commensurate resonance excitation. Measurements on the lattice dynamics for YBa$_2$Cu$_3$O$_{6.6}$ show incommensurate structure that appears to stem from charge fluctuations that are associated with the spin fluctuations.
We present macroscopic experimental evidence for field-induced microscopic quantum fluctuations in different hole- and electron-type cuprate superconductors with varying doping levels and numbers of CuO$_2$ layers per unit cell. The significant suppression of the zero-temperature in-plane magnetic irreversibility field relative to the paramagnetic field in all cuprate superconductors suggests strong quantum fluctuations due to the proximity of the cuprates to quantum criticality.
This is a Comment on a recent Letter (Phys.Rev.Lett.93, 247002 (2004)) written by Dalidovich and Yang. It is pointed out that any Pauli-limited approach neglecting the vortices is inapplicable to describing the FFLO state in CeCoIn5.
We study order parameter fluctuation effects in the superconducting state as a possible precursor to the pseudogap phenomena. Using a low-energy effective theory in the d-wave BCS model, we self-consistently calculate the single-particle properties. We find that the fluctuations reduce the spectral gap, and the shape of the gap is deformed, showing reduced slope of the gap at the nodes. We also show the angle-dependence of the quasiparticle lifetime due to the fluctuations.
The iron-based superconductor FeSe offers a unique possibility to study the interplay of superconductivity with purely nematic as well magnetic-nematic order by pressure (p) tuning. By measuring specific heat under p up to 2.36GPa, we study the multiple phases in FeSe using a thermodynamic probe. We conclude that superconductivity is bulk across the entire p range and competes with magnetism. Our analysis suggests that superconducting and magnetic fluctuations exist over a wide temperature range above the respective bulk transition temperatures, whenever magnetism is present. These observation…
We measured the static uniform spin susceptibility of Ba(Fe$_{1-x}$Rh$_x$)$_2$As$_2$ iron-based superconductors, over a broad range of doping ($0.041\leq x\leq 0.094$) and magnetic fields. At small fields ($H \le$ 1 kOe) we observed, above the transition temperature $T_c$, the occurrence of precursor diamagnetism, which is not ascribable to the Ginzburg-Landau theory. On the contrary, our data fit a phase fluctuation model, which has been used to interpret a similar phenomenology occurring in the high-$T_c$ cuprate superconductors. On the other hand, in presence of strong fields the unconventi…
We present the results of a microscopic calculation of the longitudinal thermal conductivity of quasiparticles, $κ_{xx}$, in a 2D d-wave superconductor in the vortex state. Our approach takes into account both impurity scattering and a contribution to the thermal transport lifetime due to the scattering of quasiparticles off of vortices. We compare the results with the experimental measurements on high-T$_c$ cuprates and organic superconductors.
We study fluctuation conductivity in a single crystal of BaFe$_{1.9}$Ni$_{0.1}$As$_2$ superconductor ($T_c$ = 20 K) as a function of temperature and applied magnetic field. Magneto-conductivity curves, $Δσ$ vs. $T$, were analyzed in terms of -1/(dln($Δσ$)/d$T$) vs. $T$ plots, which allow to study different fluctuations regimes and to estimate exponent values and temperature widths of each regime. The analysis of magneto-conductivity curves evidence the existence of only two fluctuations regimes, a possible critical one (of glass-like type) going from the irreversible temperature to above $T_c(…
We report on fluctuations in the electron system, Cooper pairs and quasiparticles, of a superconducting aluminium film. The superconductor is exposed to pair-breaking photons (1.54 THz), which are coupled through an antenna. The change in the complex conductivity of the superconductor upon a change in the quasiparticle number is read out by a microwave resonator. A large range in radiation power can be chosen by carefully filtering the radiation from a blackbody source. We identify two regimes. At high radiation power, fluctuations in the electron system caused by the random arrival rate of th…
Motivated by recent experiments on isovalent-doped 1111 iron-based superconductors LaFeAs$_{1-{x}}$P$_{x}$O$_{1-{y}}$F$_{y}$ and the theoretical study that followed, we investigate, within the five orbital model, the correlation between the spin fluctuation and the superconducting transition temperature, which exhibits a double dome feature upon varying the Fe-As-Fe bond angle. Around the first dome with higher $T_c$, the low energy spin fluctuation and $T_c$ are not tightly correlated because the finite energy spin fluctuation also contributes to superconductivity. On the other hand, the stre…
We study the phase transition between the normal and non-uniform (Fulde-Ferrell-Larkin-Ovchinnikov) superconducting state in quasi two-dimensional d-wave superconductors at finite temperature. We obtain an appropriate Ginzburg-Landau theory for this transition, in which the fluctuation spectrum of the order parameter has a set of minima at non-zero momenta. The momentum shell renormalization group procedure combined with dimensional expansion is then applied to analyze the phase structure of the theory. We find that all fixed points have more than one relevant directions, indicating the transi…
We show that magnetic fluctuations can destroy the Hebel-Slichter peak in conventional superconductors. The Hebel-Slichter peak has previously been expected to survive even in the presence of strong electronic interactions. However, we show that antiferromagnetic fluctuations suppress the peak at $\bf{q}=0$ in the imaginary part of the magnetic susceptibility, $χ_{+-}''\left(\bf{q},ω\right)$, which causes the Hebel-Slichter peak. This is of general interest as in many materials superconductivity is found near a magnetically ordered phase, and the absence of a Hebel-Slichter peak is taken as ev…
We have theoretically investigated the spin and charge fluctuations in the quasi-one dimensional organic superconductor (TMTSF)2ClO4. Using the extended multi-site Hubbard model, which contains four sites in a unit cell and the transfer energies obtained by the extended Hückel method, we calculate the linearized gap equation with the random phase approximation, to find novel order parameters of superconductivity due to several kinds of charge fluctuations induced by the anisotropic intersite repulsive interactions. For the singlet state, the order parameter with line nodes appears in the case …
The complete set of diagrams for conventional fluctuation corrections to the normal state of an ultraclean superconductor is derived using the functional-integral approach. The standard diagrams which characterize the fluctuation phase of a superconductor, namely, Aslamazov-Larkin, Maki-Thompson, and Density of States, are obtained and proved to be insufficient to produce a gauge-invariant electromagnetic response. An additional diagram is derived, and it is found to be essential for resolving a problematic and underappreciated aspect of the conventional fluctuation formalism. It is only with …
Recently, compressed H$_2$S has been shown to become superconducting at 203 K under a pressure of 155 GPa. One might expect fluctuations to dominate at such temperatures. Using the magnetisation critical current, we determine the ground-state London penetration depth, $λ_0$=189 nm, and the superconducting energy gap, $Δ_0$=27.8 meV, and find these parameters are similar to those of cuprate superconductors. We also determine the fluctuation temperature scale, $T_{\textrm{fluc}}=1470$ K, which shows that, unlike the cuprates, $T_c$ of the hydride is not limited by fluctuations. This is due to it…
In a two-band superconductor, two qualitatively different fluctuation modes related to the gap modules contribute to free energy and heat capacity, in addition to the phase fluctuations. The first mode has divergent temperature behaviour since it accounts for the critical fluctuations around the phase transition point, Tc, along with pseudo-critical ones associated with former instability of the weaker-superconductivity component. The involvement of these two factors, competing under interband interaction, results in the Ginzburg number which increases with Tc non-monotonically, allowing the r…
There exists a wide temperature region $({\rm Gi}T<T-T_c<T\sqrt{\rm Gi})$, where the influence of fluctuations on the thermodynamic properties of superconductors can be taken into account in the linear (Gaussian) approximation, while their influence on the kinetic properties is strongly nonlinear. Maki-Thompson contribution to conductivity saturates in this region. However, Aslamazov-Larkin contribution becomes even more singular. Such an enhancement is connected with the fact that nonlinear effects yield an increase in the lifetime of fluctuating pairs. Pairbreaking and energy relaxation proc…
In underdoped cuprates, T_c is thought to be determined by Cooper pair phase fluctuations because of the small superfluid density n_s. Experimentally, T_c is found to scale with n_s. The fluctuation-exchange approximation (FLEX) in its standard form fails to predict this behavior of T_c since it does not include phase fluctuations. We therefore extend the FLEX to include them selfconsistently. We present results for T_c[n_s,x], where x is the doping.
The coupling of Josephson oscillations in layered superconductors is studied with help of a tunneling Hamiltonian formalism. The general form of the current density across the barriers between the superconducting layers is derived. The induced charge fluctuations on the superconducting layers lead to a coupling of the Josephson oscillations in different junctions. A simplified set of equations is then used to study the non-linear dynamics of the system. In particular the influence of the coupling on the current-voltage characteristics is investigated and upper limits for the coupling strength …
The fluctuation conduction of NSN and SNS contacts above T_c are analyzed. For NSN contacts, both Aslamazov-Larkin and Maki- Thompson corrections to the conduction are found to be of the same order and diverge for T_c^* < T_c according to the law (T - T_c^*)^-1. For SNS contacts, the Aslamazov - Larkin correction vanishes,while the Maki - Thompson correction is essential for contacts shorter than the phase-breaking length.
We study a two-dimensional superconductor in close proximity to a two-dimensional metallic sheet. The electrons in the superconducting sheet are coupled to those in the metallic sheet by the Coulomb interaction only. We obtain an effective phase-only action for the superconductor by integrating out all the electronic degrees of freedom in the problem. The Coulomb drag of the normal electrons in the metallic sheet is found to make the spectrum of phase-fluctuations in the superconductor, dissipative at long wavelengths. The dissipative co-efficient $η$ is shown to be simply related to the norma…
We study the effect of thermal fluctuations of the s-wave order parameter of a quasi two dimensional superconductor on the nuclear spin relaxation rate near the transition temperature Tc. We consider both the effects of the amplitude fluctuations and the Berezinskii-Kosterlitz-Thouless (BKT) phase fluctuations in weakly coupled layered superconductors. In the treatment of the amplitude fluctuations we employ the Gaussian approximation and evaluate the longitudinal relaxation rate 1/T1 for a clean s-wave superconductor, with and without pair breaking effects, using the static pair fluctuation p…
