Abrikosov-Gorkov Theory Research Articles

Superconductivity in Heusler compound ScAu2Al

We report superconducting state properties and electronic structure of a full Heusler material ScAu2Al. The resistivity measurement indicates a zero-field (at nominal Earth’s field) superconducting transition temperature, T c = 5.12 K (in contrary to the previously reported value of 4.4 K), which falls in the highest T c -regime among the Heusler superconductors. The magnetization data shows that ScAu2Al is a moderate type-II superconductor, where the critical field values can be estimated from the Ginzburg–Landau–Abrikosov–Gorkov theory. The field-dependent magnetization response further shows signatures of flux jump in ScAu2Al. A sharp jump in the temperature dependent specific heat (C p ) data confirms bulk superconductivity. We report that the electron–phonon coupling constant, λ e–ph = 0.77, suggesting a moderate electron–phonon coupling in ScAu2Al. Further, we show that the observed λ e–ph value in ScAu2Al is the highest amongst the reported Heusler superconductors, indicating strong correlation between T c and λ e–ph values and significant role of electron–phonon coupling in mediating superconductivity in Heusler superconductors. Finally, we discuss the electronic properties and reveal the existence of van Hove singularity near the Fermi level in ScAu2Al.

Signatures of Surface Magnetic Disorder in Niobium Films

We present our studies on the evolution of the normal and superconducting properties with thickness of thin Nb films with a low level of non-magnetic disorder (k <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</sub> l ≈ 150 for the thickest film in the set). The analysis of the superconducting behavior points to the presence of magnetic moments, hidden in the native oxide on the surface of Nb films. Using the Abrikosov-Gorkov theory, we obtain the density of surface magnetic moments of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">13</sup> cm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2</sup> , which is in agreement with the previously reported data for Nb films.

The Superconductivity Mechanism in Nd-1111 Iron-Based Superconductor Doped by Calcium

We describe the effect of nonmagnetic impurity on the superconductivity behavior of the NdFeAsO0.8F0.2 iron-based superconductor. The resistivity measurements show that the superconductivity is suppressed upon increasing the low amounts of calcium impurity (x ≤ 0.05). Also, the TC decreases with the increase in the residual resistivity. Such behavior is qualitatively described by the Abrikosov–Gorkov theory and confirms that these impurities act as scattering centers. Moreover, we present the phase diagram of our synthesized samples for the various calcium dopings. We find that according to the increase in the calcium impurities and the decrease in the spin-density wave transition tempeθrature (TSDW), Fe ions are arranged stripe-antiferromagnetic at lower temperatures and also the superconducting transition temperature (TC) declines. Based on our results and in agreement with the available theories as is explained in the text, since the S++ state has no effect on the impurity-doped samples, and for low amounts of calcium, the S± state that is attributed to the spin-fluctuations causes the superconductivity suppression. So, it confirms the role of the spin-fluctuations as a dominant pairing mechanism in our synthesized samples.

Analysis of Electronic Properties from Magnetotransport Measurements on Ba(Fe1xNix)2As2 Thin Films.

We performed a detailed structural, magnetotransport, and superconducting analysis of thin epitaxial Ba(Fe1−xNix)2As2 films with Ni doping of x = 0.05 and 0.08, as prepared by pulsed laser deposition. X-ray diffraction studies demonstrate the high crystalline perfection of the films, which have a similar quality to single crystals. Furthermore, magnetotransport measurements of the films were performed in magnetic fields up to 9 T. The results we used to estimate the density of electronic states at the Fermi level, the coefficient of electronic heat capacity, and other electronic parameters for this compound, in their dependence on the dopant concentration within the framework of the Ginzburg–Landau–Abrikosov–Gorkov theory. The comparison of the determined parameters with measurement data on comparable Ba(Fe1−xNix)2As2 single crystals shows good agreement, which confirms the high quality of the obtained films.

Unusual evolution from a superconducting to an antiferromagnetic ground state in Y1−xGdxPb3 (0 ≤ x ≤ 1)

We have studied the evolution of ${\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{Gd}}_{x}{\mathrm{Pb}}_{3}$ ($0\ensuremath{\le}x\ensuremath{\le}1$) from its superconducting ($x=0$) to antiferromagnetic ($x=1$) states with critical temperatures ${T}_{c}=4.6(2)\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ and ${T}_{N}=15.7(2)\phantom{\rule{0.16em}{0ex}}\mathrm{K}$, respectively. At relatively low Gd concentrations ($x\ensuremath{\le}0.03$) ${T}_{c}$ presents a weak but linear suppression with increasing $x$, as expected from the Abrikosov-Gorkov (AG) theory describing the Cooper pair breaking due to the exchange interaction between the impurity localized magnetic moment and the conduction electrons ($ce$). This linear ${T}_{c}$ suppression rate leads to an effective exchange parameter of ${\ensuremath{\langle}{J}^{2}(\mathbf{q})\ensuremath{\rangle}}_{AG}^{1/2}=0.20(7)\phantom{\rule{0.16em}{0ex}}\mathrm{meV}$ between the $4f$ localized electrons and the $ce$. For intermediate Gd concentrations ($0.03\ensuremath{\le}x\ensuremath{\le}0.15$), although no Gd clustering is observed, there is an unusual inflection and leveling off trend in ${T}_{c}$, indicating that the superconducting phase persists until the highest Gd concentration in this region where magnetic interactions are evidenced by increasingly negative values of the Curie-Weiss temperature ${\mathrm{\ensuremath{\Theta}}}_{C}$. Analysis of low-$T\phantom{\rule{4pt}{0ex}}{\mathrm{Gd}}^{3+}$ electron spin resonance (ESR) experiments and their ESR line-shape simulations, in conjunction with the trend in ${T}_{c}$, leads us to suggest that an exchange bottleneck mechanism between the ${\mathrm{Gd}}^{+3}$ localized magnetic moment and the $ce$ may be the reason behind the inhibition of the Cooper pair-breaking mechanism, in favor of a magnetic interaction via $ce$ polarization. Therefore, our superconducting and ESR results suggest a scenario where superconducting and magnetic phases coexist extensively in the ${\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{Gd}}_{x}{\mathrm{Pb}}_{3}$ system, mediated by different types of $ce$.

Crossover from impurity-controlled to granular superconductivity in (TMTSF)2ClO4

Using a proper cooling procedure, a controllable amount of non-magnetic structural disorder can be introduced at low temperature in (TMTSF)2ClO4. Here we performed simultaneous measurements of transport and magnetic properties of (TMTSF)2ClO4 in its normal and superconducting states, while finely covering three orders of magnitude of the cooling rate around the anion ordering temperature. Our result reveals, with increasing density of disorder, the existence of a crossover between homogeneous defect-controlled d-wave superconductivity and granular superconductivity. At slow cooling rates, with small amount of disorder, the evolution of superconducting properties is well described with the Abrikosov-Gorkov theory, providing further confirmation of non-s-wave pairing in this compound. In contrast, at fast cooling rates, zero resistance and diamagnetic shielding are achieved through a randomly distributed network of superconducting puddles embedded in an normal conducting background and interconnected by proximity effect coupling. The temperature dependence of the AC complex susceptibility reveals features typical for a network of granular superconductors. This makes (TMTSF)2ClO4 a model system for granular superconductivity where the grain size and their concentration are tunable within the same sample.

Modulation of pairing symmetry with bond disorder in unconventional superconductors

We study a two-orbital $t$-$J_1$-$J_2$ model, originally developed to describe iron-based superconductors at low energies, in the presence of bond disorder (via next-nearest-neighbor $J_2$-bond dilution). By using the Bogoliubov--de Gennes approach, we self-consistently calculate the local pairing amplitudes and the corresponding density of states, which demonstrate a change of dominant pairing symmetry from $s_\pm$ wave to $d$ wave when increasing disorder strength as long as $J_1\lesssim J_2$. Moreover, the combined pairing interaction and strong bond disorder lead to the formation of $s_\pm$ wave "islands" with length scale of the superconducting coherence length embedded in a $d$ wave "sea." This picture is further complemented by the disorder-averaged pair-pair correlation functions, distinct from the case with potential disorder, where the "sea" is insulating. Due to this inevitable formation of spatial inhomogeneity, the superconducting $T_c$ determined by the superfluid density $\rho_s(T)$ obviously deviates from the value predicted by the conventional Abrikosov-Gorkov theory, where the pairing amplitudes are viewed as uniformly suppressed as the disorder increases.

Weak itinerant ferromagnetism and clean superconductivity in Y9Co7

We present low-temperature magnetic properties for a high-quality polycrystalline sample of the first ferromagnetic superconductor Y9Co7. The results of susceptibility and magnetization measurements show the coexistence of a weak itinerant ferromagnetic order with a Curie temperature of TC ≅ 4.5 K and superconductivity below TS = 3 K. Several electronic quantities and parameters characterizing the superconducting and normal state are calculated within the Ginzburg–Landau–Abrikosov–Gorkov theory. The data reveal clean limit superconductivity in Y9Co7 due to the good chemical purity and structural order of the specimen.

Disorder effects in multiorbitals±-wave superconductors: Implications for Zn-doped BaFe2As2compounds

Recent experiments on Zn-doped 122-type iron pnictides, Ba(Fe$_{1-x-y}$Co$_y$Zn$_x$)$_2$As$_2$, are challenging our understanding of electron doping the 122s and the interplay between doping and impurity scattering. To resolve this enigma, we investigate the disorder effects of nonmagnetic Zn impurities in the strong (unitary) scattering limit on various properties of the system in the $s_{\pm}$-wave superconducting pairing state. The lattice Bogoliubov-de Gennes equation (BdG) is solved self-consistently based on a minimal two-orbital model with an extended range of impurity concentrations. We find that Zn impurity is best modeled as a defect, where charge is mainly localized, but scattering is extended over a few lattice sites. With increasing Zn concentration the density of states shows a gradual filling of the gap, revealing the impurity-induced pair breaking effect. Moreover, both the disorder configuration-averaged superconducting order parameter and the superfluid density are dramatically suppressed towards the dirty limit, indicating the violation of the Anderson theorem for conventional s-wave superconductors and the breakdown of the Abrikosov-Gorkov theory for impurity-averaged Green's functions. Furthermore, we find that the superconducting phase is fully suppressed close to the critical impurity concentration of roughly $n_\text{imp}\approx 10%$, in agreement with recent experiments.

Magnetic penetration-depth measurements of a suppressed superfluid density of superconducting Ca0.5Na0.5Fe2As2single crystals by proton irradiation

We report on the dramatic effect of random point defects, produced by proton irradiation, on the superfluid density $\rho_{s}$ in superconducting Ca$_{0.5}$Na$_{0.5}$Fe$_2$As$_2$ single crystals. The magnitude of the suppression is inferred from measurements of the temperature-dependent magnetic penetration depth $\lambda(T)$ using magnetic force microscopy. Our findings indicate that a radiation dose of 2$\times10^{16}$cm$^{-2}$ produced by 3 MeV protons results in a reduction of the superconducting critical temperature $T_{c}$ by approximately 10%. % with no appreciable change in the slope of the upper critical fields. In contrast, $\rho_{s}(0)$ is suppressed by approximately 60%. This break-down of the Abrikosov-Gorkov theory may be explained by the so-called "Swiss cheese model", which accounts for the spatial suppression of the order parameter near point defects similar to holes in Swiss cheese. Both the slope of the upper critical field and the penetration depth $\lambda(T/T_{c})/\lambda(0)$ exhibit similar temperature dependences before and after irradiation. This may be due to a combination of the highly disordered nature of Ca$_{0.5}$Na$_{0.5}$Fe$_2$As$_2$ with large intraband and simultaneous interband scattering as well as the $s^\pm$-wave nature of short coherence length superconductivity.

Analytical density of states in the Abrikosov–Gorkov theory

Since the early 1960s, Abrikosov–Gorkov theory has been used to describe superconductors with paramagnetic impurities. Interestingly, the density of states resulting from the theoretical framework has to date only been known approximately, as a numeric solution of a complex polynomial. Here we introduce an exact analytical solution for the density of states of a superconductor with paramagnetic impurities. The solution is valid in the whole regime of Abrikosov–Gorkov theory independent of the presence of an energy gap. While of significant fundamental interest, we argue that this solution also has computational benefits in the evaluation of integrals for tunnelling conductances and allows for an analytic description of materials with densities of states that are modeled from the basic Abrikosov–Gorkov density of states.

Using AG theory to model a S/I/N tunnel junction

We have measured the spin polarized tunneling conductance of a thin Al/Al 2O 3/amorphous Gd x Si 1− x planar tunnel junction at T=25 mK in parallel magnetic field H⩽3.0 T. Abrikosov–Gorkov theory is a better model for the thin superconducting Al electrode than previously used Bardeen–Cooper–Schrieffer theory. As the magnetic field is increased, we find the best fit to the data by modelling a transition from a superconductor with gap ( α<1) to a gapless superconductor ( α>1).

Superconducting Fluctuation and Pseudogap in Disordered Short Coherence Length Superconductor

We investigate the role of disorder on the superconducting (SC) fluctuation in short coherence length d-wave superconductors. The particular intetest is focused on the disorder-induced microscopic inhomogeneity of SC fluctuation and its effect on the pseudogap phenomena. We formulate the self-consistent 1-loop order theory for the SC fluctuation in inhomogeneous systems and analyze the disordered $t$-$t'$-$V$ model. The SC correlation function, electronic DOS and the critical temperature are estimated. The SC fluctuation is localized like a nanoscale granular structure when the coherence length is short, namely the transition temperature is high. This is contrasted to the long coherence length superconductors where the order parameter is almost uniform in the microscopic scale. In the former case, the SC fluctuation is enhanced by the disorder in contrast to the Abrikosov-Gorkov theory. These results are consistent with the STM, NMR and transport measurements in high-$T_{\rm c}$ cuprates and illuminate the essential role of the microscopic inhomogeneity. We calculate the spacial dependence of DOS around the single impurity and discuss the consistency with the NMR measurements.

Superconducting gap and pair breaking in CeRu2 studied by point contacts

The superconducting gap in a CeRu2 single crystal is investigated by point contacts. BCS-like behavior of the gap Δ in the temperature range below Tc* (Tc*&amp;lt;Tc, where Tc is the critical temperature) is established, indicating the presence of a gapless superconductivity region (between Tc* and Tc). The pair-breaking effect of paramagnetic impurities, supposedly Ce ions, is taken into consideration using the Scalski–Betbeder–Matibet–Weiss approach based on Abrikosov–Gorkov theory. It allows us to recalculate the superconducting order parameter Δα (in the presence of paramagnetic impurities) and the gap Δp (in the pure case) for the single crystal and for the previously studied polycrystalline CeRu2. The value 2Δp(0)≈2 meV, with 2Δp(0)/kBTc≈3.75, is found in both cases, indicating that CeRu2 is a “moderate” strong-coupling superconductor.

Microwave properties of RE–Ni 2B 2C (RE=Y, Er) superconducting thin films

A comparative study of the microwave properties of YNi 2B 2C and ErNi 2B 2C superconducting films is reported. The measurements were performed using a microstrip resonator ( f=2 GHz) and a microwave cavity ( f=87 GHz). The analysis of the temperature dependence of the surface impedance Z s confirms that YNi 2B 2C compound is a conventional BCS (s-wave) superconductor. On the contrary, ErNi 2B 2C samples show an anomalous behaviour in both the surface resistance and penetration depth. This is tentatively ascribed to pair breaking effects on the superconducting density of states. A simplified generalisation of the Abrikosov–Gorkov theory, proposed by Machida, is successfully applied to describe the penetration depth data in the antiferromagnetic compound, while for the surface resistance granular effects cannot be ruled out.

Superconducting state parameters of C-16 structured Zr 2Rh

Low temperature heat capacity, resistivity and dc magnetisation measurements were performed to characterize the superconducting state of C-16 structured Zr 2Rh intermetallic compound. Using parametrised expressions based on the Ginzburg–Landau–Abrikosov–Gorkov theory, we conclude that Zr 2Rh is a strongly coupled superconductor with 2 Δ(0)/ k B T c∼4.7. It has a κ∼44 and the measured H c2/ H c1 ratio is 275.5. We also discuss the effect of hydrogenation on the superconducting transition temperature and the irreversibility line of this intermetallic compound.

Absence of effect of paramagnetic impurities on flux quantization in superconductors

In this work we investigate flux quantization effects in uniform superconducting rings doped with paramagnetic impurities. Several theories predicted that the impurity spins in weak links affect superconductivity by inducing phase shifts, which might result in half-integral fluxoid quantization. We tested whether these ideas could be extended to the case of uniform Mo rings doped with Fe, which is paramagnetic, but did not observe any such effects. We explain the absence of such effects within the de Gennes version of the Abrikosov-Gorkov theory of pair breaking.

Cooper pair breaking and isotope effect coefficient variation in high-T c superconductors

The effect of pair breaking on the isotope effect coefficientα=−d lnT c/d lnM in La2−x Sr x CuO4 and Pr-, Ca-, and Zn-doped YBa2Cu3O7−x and EuBa2Cu3O7−x is studied using the generalized Abrikosov-Gorkov theory recently employed by Singh and Kishore for superconductivity. It is argued that the isotope effect coefficientα can be further enhanced, in agreement with experimental observations, by considering the dependence of the characteristic scattering timeτ s for Cooper pairs on the concentrationn of impurities (both magnetic and nonmagnetic) and the disorder ignored by them (J. Supercond. 8, 9 (1995).

Transition from superconductivity to heavy-fermion behavior in U-doped Lu2Fe3Si5.

In the superconductor ${\mathrm{Lu}}_{2}$${\mathrm{Fe}}_{3}$${\mathrm{Si}}_{5}$ (${\mathit{T}}_{\mathit{c}}$=6.25 K) lutetium can be replaced by uranium up to a maximum of 40% [(${\mathrm{Lu}}_{0.6}$${\mathrm{U}}_{0.4}$${)}_{2}$${\mathrm{Fe}}_{3}$${\mathrm{Si}}_{5}$], maintaining the original P4/mnc structure. When the uranium content in (${\mathrm{Lu}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{U}}_{\mathit{x}}$${)}_{2}$${\mathrm{Fe}}_{2}$${\mathrm{Si}}_{5}$ is increased the superconductivity vanishes and a heavy-fermion state is built up. Already for small amounts of uranium there is a strong suppression of superconductivity (${\mathit{T}}_{\mathit{c}}$=1.8 K for x=0.01). According to the Abrikosov-Gorkov theory this suppression can be described by a strong exchange coupling between the uranium 5f electrons and the conduction electrons. In agreement with this high exchange coupling we detect drastically reduced magnetic (uranium) moments. For low dopings (x0.1) there is no significant change of the Sommerfeld parameter \ensuremath{\gamma}. When the U content exceeds x&gt;0.1 there is a strong \ensuremath{\gamma} increase. The enhanced specific heat remains unchanged when the sample is exposed to a magnetic field of 13 T. This field-independent specific heat, the screened magnetic moments, and the Wilson ratio \ensuremath{\chi}/\ensuremath{\gamma} corroborate that the increase in \ensuremath{\gamma} is indeed indicative of an enhanced effective mass and not a result of magnetic entropy.Just like in other U-based systems (e.g., ${\mathrm{UPt}}_{3}$ and ${\mathrm{UBe}}_{13}$) the evolution of the heavy-fermion state in (${\mathrm{U}}_{\mathit{x}}$${\mathrm{Lu}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${)}_{2}$${\mathrm{Fe}}_{3}$${\mathrm{Si}}_{5}$ cannot merely be explained by a pure single-ion Kondo effect. The fact that the \ensuremath{\gamma} enhancement begins only after the U content exceeds x&gt;0.1 points out that collective phenomena (i.e., interaction between the U ions) play a decisive role for the evolution of the heavy-fermion state in this system. Cerium-doped samples (${\mathrm{Lu}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Ce}}_{\mathit{x}}$${)}_{2}$${\mathrm{Fe}}_{3}$${\mathrm{Si}}_{5}$ revealed a distinctly weaker ${\mathit{T}}_{\mathit{c}}$ suppression (${\mathit{T}}_{\mathit{c}}$=1.9 K for x=0.1), less reduced magnetic moments, and no significant \ensuremath{\gamma} enhancement.

Superconductivity of the Pr-doped (Y,Eu)Ba2Cu4O8 system.

A systematic study of the Tc dependence of the disordered R1-xPrxBa2Cu4O8 (RPr-124) system (R=Y,Eu) on the Pr concentration x is reported here. Like (Y,Eu)1-xPrxBa2Cu3O7 (RPr-123), Tc of these systems decreases monotonically with Pr concentration. dTc/dx, however, is found to be smaller in RPr-124 than in RPr-123. Moreover, unlike RPr-123, dTc/dx does not vary monotonically with x at least in the YPr-124 system, although the relative variation of Tc with the rare-earth (Y,Eu) ionic radius is similar in both the cases. The observed results are discussed in terms of the hole-filling model and the Abrikosov-Gorkov (AG) theory. It is found that the AG theory is not adequate to explain the observed Tc variation with x in the RPr-124 systems. We have attributed the failure of the AG theory in the RPr-124 systems to the linear x dependence of the carrier-impurity scattering rate. A modified AG theory considering the observed carrier-impurity scattering rate is found to provide a reasonable explanation of Tc variation in the RPr-124 systems. The different Tc variations in the YPr-124 and EuPr-124 systems are qualitatively understood on the basis of an indirect effect of the Pr impurity on the coupling of the Y,Eu state with the Cu3d–O 2p state. The relatively smaller value of dTc/dx in Y,Eu-124 than in Y,Eu-123 is understood on the basis of the larger separation of the CuO2-(Y,Eu/Pr)-CuO2 regions in the former system.