Energy gap,transition temperature and electron-electron interaction of the weak coupling superconductors tin and indium as a function of the mean free path of the electrons

The energy gap and the transition temperature of the weak coupling superconductors tin and indium are measured as a function of the mean free path of the electrons. Both of them increase with decreasing mean free path. The ratioα=2Δ ₀/kT_c increases proportionally to the reciprocal mean free path from 3.6 to 4.0 for tin and from 3.6 to 3.95 for indium. A theoretical consideration is presented showing that the electron-electron interaction is increased in dirty superconductors. Without any fit we find from the theoretical calculation an increase inNV for indium ofδNV=5 · 10^?10 m/l _tr and for tinδNV=2 · 10^?10 m/l _tr (l_tr is the transport mean free path of the electrons). The experiments give an increase inNV for indium ofδNV=4.2 · 10^?10 m/l_tr and for tinδNV=1.7 · 10^?10 m/l _tr. The agreement is surprisingly good.     

The mass enhancement and the specific heat of disordered and amorphous metals

The reduction of the mean free path in metals causes a change of the electronphonon interaction and the Eliashberg function α²(Ω)F(Ω). The values of α²(Ω)F(Ω) for pure, disordered and amorphous lead are used to calculate the enhancement of the density of states and the electronic specific heat in these materials. In pure lead the cyclotron mass (1+λ) has a maximum at 11 °K and the Sommerfeld constant (γ) has a maximum at 5 °K. In disordered lead with a mean free path of about 70 Å, both, 1+λ and γ are almost constant in this temperature range. For amorphous lead there is no temperature range where 1+λ and γ are sufficiently constant. A measurement of the specific heat in disordered non-superconducting metals could answer the question whether these metals are closer to the superconducting state when their mean free path is reduced, a fact which is true for all weak-coupling superconductors.     

Energy gap and transition temperature of highly disordered and amorphous thallium- and indium-films

Thin and pure Tl-films and In-films with different impurities (Ag, Te, Ge, Sb) are condensed onto a cooled substrate at 4 K. Measurements of the energy gap by means of the tunnel effect and the transition temperature of the weak-coupling superconductor thallium are carried out as a function of the degree of disorder of the films. The ratioα=2Δ ₀/kT _c increases proportional to the reciprocal mean free path from 3.5 for the annealed film up to 3.8 for the highly disordered film. For In-films condensed by quenching with impurity additions, one finds a linear relation between energy gap and transition temperature. In-films with Sb-additive are obtained in an amorphous phase with a ratioα=2Δ ₀/kT _c of 4.4. The amorphous state of the In/Sb-films is confirmed by measurements of the electric conductivity and the Hall-effect.     

d-Wave superconductivity via buckling-like phonon mode

We follow the classic strong-coupling theory of superconductivity through electron-phonon interaction with a buckling-like phonon mode. We find a nonzero d-wave order parameter in the sense of the Eliashberg theory. We derive a zero temperature gap Δ(0,π) at the gap edge versus the electron-phonon coupling strength g² relation. We find that large enough value for Δ(0,π) as compared to those of high-T_c superconductors cannot be realized in the electron-buckling-like-phonon coupling on the CuO₂ planes.     

Comparison between measured and calculatedRF losses in the superconducting state

Results of experiments and of the BCS-theory on the surface resistance of superconductors are presented. It will be shown that agreement between theory and experiment can be achieved by using revised material parameters. In the case of lead and niobium we will show that therf absorption decreases markedly with decreasing mean free pathl. This dependence on mean free path can become a method to measurel in a surface sheath.     

The upper critical field of quenched condensed Sn,Tl, In and Hg with extremely short mean free path

Quenched condensed films of Sn, Tl, In and Hg are fine crystalline and have very short mean free path. The upper critical field of these disordered superconductors is measured. The initial slope,dB _c 2/dT is compared with a strong coupling calculation. We use α² F(ω) which is measured for these superconductors in the pure state and for Sn in the quenched state. The strong coupling behavior enhances the upper critical field by 15–20%. We find a fair (about 10%) agreement between the experimental and theoretical values ofdB _c 2/dT except for Hg. In Hg one has either to consider band structure effects or the α² F(ω) of quenched condensed Hg is quite different from that of pure Hg.     

Critical temperature and energy gap in superconductors with singular density of states

The influence of-functional (symmetric as well as asymmetric) singularities in density of states (DOS) on the critical temperature and zero temperature energy gap is calculated. Surprisingly, we have obtained the same function for the off-symmetry of the peak position in DOS on the corresponding critical temperature as for the temperature dependence of the energy gap in the strong-coupling limit. The enhancement of the critical temperature due to the singularity (compared with the constant DOS near the Fermi surface) is much lower for strong-coupling superconductors than in the weak-coupling limit. Hence, the singularity in DOS cannot be the exclusive reason for large values of critical temperatures in highTc superconductors.This work was supported by the grant GA SAV 188/1991.     

Strong-coupling planar diagrams,random surfaces,and the large-N phase transition in U(N) lattice gauge theories

The strong-coupling expansion of U(N) gauge theory on a D-dimensional lattice is reformulated in the limit N → ∞ through a set of diagrammatic rules directly for the free energy and Wilson loops. The strong-coupling planar diagrams are interpreted as surfaces embedded in the lattice. The large-N phase transition is related to the entropy of these surfaces. It is shown that the strong-coupling phase of the U(∞) gauge theory terminates with a phase transition of Gross-Witten type only in 2 and 3 dimensions. When D⩾4 the large-N singularity takes place in a metastable phase because of an earlier first-order transition to the weak-coupling phase of the theory.     

Effect of paramagnetic fluctuations on superconductivity in ferromagnetic superconductors

The superconducting transition temperature T_c is calculated for ferromagnetic superconductors taking into account explicitly elastic scattering between electrons and paramagnetic fluctuations. Depending on the strength and range of the Heisenberg interaction between localized magnetic moments, the superconducting gap may or may not remain finite at the onset of long range ferromagnetic ordering. In a dirty system, the mean free path reduces the depairing effect of fluctuations. The Heisenberg interaction is assumed to be of other than RKKY origin.     

Coulomb repulsion-induced hyperbolic pairing as a mechanism of high-T c superconductivity

The hyperbolic metric of the dispersion law (the effective mass tensor components of carriers are opposite in sign) in the vicinity of the Fermi contour in high-T _c superconducting cuprates in the case of repulsive interaction gives rise to a superconducting state characterized by the condensate of pairs with a large total momentum (hyperbolic pairing). The gain in the energy of the superconducting state over the normal state is due to the fact that a change in the kinetic energy of pairs (because of the negative light component of the effective mass) dominates over the change in the potential energy (corresponding to energy loss). The shift of the chemical potential upon the transition to the superconducting phase is substantial in this case. With increasing repulsive interaction, the superconducting gap δ_K increases and the resulting gain in energy changes to an energy loss at a certain critical value of the repulsive potential. The low temperature T _c of the superconducting transition and the large value of δ _K in this region of potential values are the reasons for the high value of the 2δ_K/T _c ratio and for the developed quantum fluctuations that are observed in underdoped cuprate superconductors.     

The upper critical magnetic field of lead as a function of the mean free path of the electrons

The transition curves of quenched lead films are measured in a perpendicular magnetic field as a function of the field strength. We vary the mean free path of the electrons by different condensation temperatures and by the annealing of the films. An influence of the edge of the film on the resistance measurement is suppressed by the superposition of iron onto the lead film. We will show, that the measured critical fieldB _{c ⊥} of the film is the upper critical fieldB _c2 of a bulk superconductor with the sameeffective mean free path. As a result of the measurements we obtain ?B _c2/?T as a function of the degree of disorder (the mean free path of the electrons) of the films. The results are compared with the theory of Gor'kov for the weak coupling superconductor and the theory of Eilenberger and Ambegaokar for the strong coupling superconductor. They disagree with both theories. From the magnetic measurement we can determine the coherence length of lead. An independent determination at the same film is given by the proximity effect of the system Pb/Fe. The results of both methods are compared.     

Übergangstemperatur zur Supraleitung,Energielücke und Widerstandsverlauf amorpher und kristallisierter Thallium-Tellur-Schichten

Thin Tl/Te-films are condensed onto a cooled substrate at 4 K. An amorphous phase is obtained. The maximum superconducting transition temperature of the amorphous Tl/Te-films is 4.2 K. Measurements of the energy gap by superconducting tunneling yield a ratio 2Δ/k T _c =4.6 for amorphous Tl/Te-films and 2Δ/k T _c =3.8 for disordered, pure Tl-films.     

On the relation between the energy gap and the critical temperature

It is shown that the ratio β = ϵ/T_c (ϵ is the energy gap at T = OK) saturates with increasing strength of the electron-phonon interaction; the value β_max is calculated. The special case of a multigap superconductor is discussed. New high-T_c superconductors present a unique oppurtunity to observe a multigap structure.     

Stripes, electron fractionalization, and ARPES

Although structurally the high temperature superconductors are quasi-two-dimensional, there is both theoretical and experimental evidence of a substantial range of temperatures in which ‘stripe’ correlations make the electronic structure locally quasi-one-dimensional. We consider an array of Josephson coupled, spin gapped one dimensional electron gases as a model of the high temperature superconductors. For temperatures above T_c, this system exhibits electron fractionalization, yielding a single particle spectral response which is sharp as a function of momentum, but broad as a function of energy. For temperatures below the spin gap but above T_c, there are enhanced one-dimensional superconducting fluctuations and pseudogap phenomena. Pair tunneling induces a crossover to three-dimensional physics as T_c is approached. Below T_c, solitons are confined in multiplets with quantum numbers which are simply related to the electron, and a coherent piece of the single particle spectral function appears. The weight of this coherent piece vanishes in the neighborhood of T_c in proportion to a positive power of the interchain superfluid density. This behavior is highly reminiscent of recent ARPES measurements on the high temperature superconductors.     

Kohärenzeffekte in Doppelschichten aus verschiedenen Supraleitern

Films of lead are deposited on a superconducting substrate. The transition temperatureT _c of this sandwich is measured as a function of the lead thicknessD _Pb . In different experiments the properties (thickness, transition temperature, mean free path of electrons) of the substrate are varied. It is especially interesting that the initial slope (small lead thickness) of theT _c (D _Pb )-curve is independent of the electronic mean free path of the superconducting substrate and inversely proportional to its thickness. The results for different thicknesses of the components in the system indium — lead can be well described by a theory ofdeGennes andWerthamer.     

Systematic trends in the transport mean free path with electron energy and atomic number

The inelastic mean free path, λ_i, and the transport mean free path, λ_tr, hold the key to understanding the effects of elastic scattering of electrons in electron spectroscopy techniques such as Auger electron spectroscopy (AES) and X-ray photoelectron spectroscopy (XPS). Systematic trends in the variation of λ_tr and χ=λ_i/λ_tr with atomic number, Z, and electron energy, E, are explored using the differential elastic scattering cross-section data of Czyzewski et al. (J. Appl. Phys., 68 (1990) 3066). At low Z, λ_tr increases with energy in accord with the predictions of the Born approximation, but at higher Z, a more complex behaviour is revealed. In the first and second transition metal series, χ varies little with energy over much of the kinetic energy range probed in XPS in accord with the energy scaling ideas from the semi-classical scattering theory of Tilinin (Soviet Physics JETP, 67 (1992) 1570). In the third transition series, a pronounced minimum is found at around 200 eV, with χ for Au lower than for the corresponding transition metals. The results identify regions of the periodic table where elastic scattering effects are particularly pronounced.     

Effective interactions and the nature of Cooper instability of spin polarons in the 2D Kondo lattice

It has been shown that two-and three-center interactions arise in the strong-coupling regime for the 2D Kondo lattice; these interactions both induce and suppress the Cooper instability. It is important that, in contrast to the t-J* model, the three-center interactions promote the Cooper pairing and ensure the appearance of the superconducting phase with a high critical temperature T _c. The calculated concentration dependences of T _c agree well with the experimental data for the cuprate superconductors.     

Static properties of type-II superconductors at arbitrary temperature and induction

A concise expression for the free energy of clean type-II superconductors is given from which magnetization curves, gap and field are easily calculated for 0 < T < T_c and 0 < B < H_c2.     

Effect of disorder on the transport properties of the high-T c superconductor Nd2− x CexCuO4+δ

The temperature dependences ρ_ab(T) of Nd_2?x Ce_xCuO_4+δ single crystals with 0≤x≤0.20 are studied and analyzed on the basis of the concepts in the theory of disordered 2D systems. The results are compared with the data obtained for other copper-oxide HTSC. It is found that a transition to the superconducting state in the optimal doping region 0.14≤x≤0.18 occurs only in crystals with a fairly small degree of disorder (k _Fl≥2, where l is the mean free path). This transition is compatible with the weak 2D-localization mode as long as the localization radius is longer than the characteristic size of a Cooper pair. The superconducting transition temperature in the optimal doping region increases monotonically with the parameter k _Fl characterizing the degree of disorder in the crystal. The degradation of superconducting properties upon a further increase in the doping level (x>0.18) is apparently associated with a transition from 2D to 3D conductivity in the single crystal.