Tomasch Effect Research Articles

Ultralong-distance quantum correlations in three-terminal Josephson junctions

The production of entangled pairs of electrons in ferromagnet-superconductor-ferromagnet or normal metal-superconductor-normal metal three-terminal structures has aroused considerable interest in the last twenty years. In these studies, the distance between the contacts is limited by the zero-energy superconducting coherence length. Here, we demonstrate nonlocality and quantum correlations in voltage-biased three-terminal Josephson junctions over the ultralong distance that exceeds the superconducting coherence length by orders of magnitude. The effect relies on} the interplay between the time-periodic Floquet-Josephson dynamics, Cooper pair splitting and long-range coupling similar to the two-terminal Tomasch effect. We find cross-over between the ``Floquet-Andreev quartets'' (if the spatial separation is smaller than the superconducting coherence length), and the ``ultralong-distance Floquet-Tomasch clusters of Cooper pairs'' if the separation exceeds the superconducting coherence length, possibly reaching the same $\simeq 30\,\mu$m as in the Tomasch experiments. The effect can be detected with DC-transport and zero-frequency quantum current-noise cross-correlation experiments, and it can be used for fundamental studies of superconducting quasiparticle quantum coherence in the circuits of quantum engineering.

Tomasch effect in nanoscale superconductors

The Tomasch effect (TE) is due to quasiparticle interference (QPI) as induced by a nonuniform superconducting order parameter, which results in oscillations in the density of states (DOS) at energies above the superconducting gap. Quantum confinement in nanoscale superconductors leads to an inhomogenerous distribution of the Cooper-pair condensate, which we found triggers the manifestation of a new TE. We investigate the electronic structure of the nanoscale superconductors by solving the Bogoliubov-de Gennes (BdG) equations self-consistently and obtain the TE determined by two types of processes, involving two or three particle QPIs. Both types of QPIs result in additional BCS-like Bogoliubov-quasiparticles and BCS-like energy gaps leading to oscillations in the DOS and modulated wave patterns in the local density of states. These effects are strongly related to the symmetries of the system. A reduced $4 \times 4$ inter-subbands BdG Hamiltonian is established in order to describe analytically the TE of two-QPIs. Our study is relevant to nanoscale superconductors, either nanowires or thin films, Bose-Einsten condensates and confined systems such as two-dimensional electron gas interface superconductivity.

Study of transport properties in superconducting junctions of double insulating barrier

Abstract In this work we analyze the Tomasch effect in double barrier insulating superconducting N1ISIN2 (N: normal metal, I: insulator and S: superconductor) junctions. From the solution of the Bogoliubov–de Gennes equations we find that the differential conductance presents resonances when the applied voltage changes. These resonances are originated by the formation of quasibound states in the superconducting region and depend on the symmetry of the pair potential. We develop an analytical model in order to find the quasibound states energies and its lifetimes. This model allows us to calculate the voltage at which each resonance appears and the resonance widths. We calculate and analyze the dependence of the transmission coefficients with the thickness of the superconducting layer.

Local Density of States in Anisotropic N-S-N and N-I-S-N Junctions

The local density of states of an anisotropic normal metal-superconductor-normal metal and normal metal-insulator-superconductor-normal metal junction is investigated using the Green's function method. The Green's functions are determined by means of the solutions of the Bogoliubov-de Gennes equations for anisotropic superconductors. We find oscillations in the local density of states which in the isotropic case coincide with those found in the Tomasch effect. The local density of states depends on the potential symmetry, the strength of the insulating barrier, and the width of the superconducting region. The influence of different pair potential symmetries on the local density of states is analyzed in detail. In particular, s and d symmetries are considered.

Current-voltage relation of a normal-metal-superconductor junction.

We calculate the electrical and heat currents flowing through a narrow, mesoscopic, normal-metal--superconductor (NS) junction containing a single point impurity. The NS junction exhibits large subgap conduction steps in its current-voltage (I-V) relation when the impurity is located in the normal metal. Our new subgap step appears in the I-V characteristic whenever an additional quasiparticle is trapped between the impurity and NS interface. Locating the impurity inside the superconductor produces both an ``excess current'' and oscillations in the I-V characteristic analogous to the Tomasch effect. A maximum excess current of 2e\ensuremath{\Delta}/h, slightly less than the 8e\ensuremath{\Delta}/3h found for a ballistic NS junction obtains when the impurity is located a few coherence lengths inside the superconductor.

Compound geometrical resonances in superconducting proximity-effect sandwiches

In superconducting-normal-metal sandwich structures in which the pair potential has abrupt spatial changes, Andreev scattering occurs in which electronlike and holelike excitations are intermixed. The Tomasch effect, Rowell-McMillan oscillations, and de Gennes-Saint James bound states result from interference of these excitations within a single film. In this paper we discuss compound resonances which can occur in both films of a sandwich structure in which each film is clean and has a thickness on the order of its coherence length. Two of the compound-resonance effects have the following interpretations: (1) Below the energy gap, the de Gennes-Saint James bound states split into energy bands. (2) Above the energy gap, resonances occurring with different frequencies result in a beating of the oscillations of the density of states as a function of energy. A number of previously puzzling or poorly understood features of published experimental data are shown to be consistent with compound-resonance effects. Suggestions are given for systematic experiments.

Study of a superconductor—normal-metal interface in a finite geometry

A finite geometry of a superconductor---normal-metal ($S\ensuremath{-}N$) system is studied, assuming a constant BCS interaction in the superconductor. The Green's functions of the finite system are constructed from the wave-function solutions, satisfying boundary conditions at the free surfaces. The tunneling electronic density of states in $S$ and $N$ is calculated and it is shown that there are geometrical resonance effects, associated with the variation of the pair potential at the interface, due to the finiteness of both slabs. The effect of a finite mean free path is demonstrated. The result reproduces, as special cases, the Tomasch effect, the Rowell-McMillan effect, and the de Gennes-Saint James bound states below the energy gap. The transition temperature, in the case where both metals are much thinner than the coherence length (the Cooper limit) is calculated, and the effect of a finite mean free path on the Cooper-limit result is found.

Tunneling effect and lifshitz transitions

The possibility of using the Tomasch effect to observe the topological transitions of the Fermi surface of a metal (Lifshitz transitions) is investigated. It is shown that the appearance of a new constant-energy cavity results in a qualitative change of the Tomasch oscillations in the tunnelling conductance.

Effect of a barrier at the superconducting-normal metal interface

A superconducting-normal metal sandwich, in which there is a potential barrier at the interface, is considered. The Tomasch effect amplitude is shown to be reduced by a factor related to the barrier transmission coefficient. We then derive from first principles the boundary conditions obeyed by the kernel in the Gor'kov gap equation at the interface, within the diffusion approximation. This represents the first ab initio calculation of the “second” de Gennes boundary condition.

Tomasch effect and superconducting proximity

Tunneling conductance measurements (dI/dV)(V) on thick and clean super-conducting films (S 1 ) backed by normal metals (M 2 ) show geometrical resonance effects, which have been associated with the variation δΔ(x) of the pair potential near the interface. We analyze both theoretically and experimentally the three factors involved in the resonance effect on S 1 /M 1 sandwiches where one of the films is superconducting. The first factor, period, has already received considerable attention and is connected with the ratioV /d 1 of the renormalized Fermi velocity and the thicknessd 1 of the film S 1 . We show that generally subharmonic effects, related to multiple interference effects, are expected. This result is demonstrated experimentally on Pb/M 2 sandwiches. The phase depends on the sign of δΔ(x), the detailed shape of its variation, and on the nature of the tunneling process. Unfortunately the interplay among these parameters is difficult to analyze, although directive tunneling is found to fit better with experiments than the generally assumed diffuse tunneling. The amplitude of the resonance should be proportional to the change of δΔ(x). The ambiguity in the experimental results is pointed out and an experiment that shows clearly the role of the superconducting proximity mechanisms is described where the proximity on several simultaneously prepared Pb/Al films is controlled by deposition of various third upper layers. The theoretical discussion is an extension of the Ishii matrix formulation, itself based on an application of the Andreev scattering process.

Tomasch-effect-induced negative-resistance phenomena in single-crystal superconducting Pb films

A negative-resistance region has been observed just above the gap in tunnel junctions fabricated on 110>-oriented single-crystal Pb films. This phenomenon was found to be present only when tunneling into single-crystal Pb films about 1.4 \ensuremath{\mu}m thick. When these junctions were raised to temperatures above 3.7\ifmmode^\circ\else\textdegree\fi{}K or placed in a parallel applied magnetic field above 200 G the negative resistance was not present although a deep minimum in conductance persisted. The structure was always accompanied by large Tomasch oscillations and appears to be related to the presence of the Tomasch effect and the details of the effect as they occur in 100>-oriented single-crystal Pb films. The negative resistance and accompanying structure were reproduced in some respects by theoretical calculations using a superconductor density of states modified by the Tomasch term.

Obtaining densities of states from tunneling between two superconductors

We consider tunneling between two superconductors, SA and SB, in the approximation that SA is a BCS superconductor and that thermal excitations are unimportant. Inversion formulas are developed which permit numerical evaluation of the tunneling density of states for SB when the ac conductance, or its voltage derivative, is known as a function of bias voltage. The method is applied to finely spaced (∼30 μV) conductance structure produced by quasiparticle interference phenomena in a thick (∼30 μm) In film at 0.3 K (so-called Tomasch effect).

Tomasch Oscillations in the Density of States of Superconducting Films

The mechanism which gives rise to the oscillatory structure in the quasiparticle density of states of superconducting films with spatially varying electron-electron interaction is discussed. It is shown that the Tomasch effect results from processes in which a quasiparticle is condensed into the sea of Cooper pairs, leaving behind a different but degenerate quasiparticle. The tunneling density of states is obtained for a composite two-region superconductor. The nature of the density-of-states structure depends strongly upon the ratio of the energy gaps of the two regions. In the low-energy range, bound eigenstates characterized by the quantization of the difference of the momenta of the degenerate quasiparticles are found for infinite mean free path. Explicit results are presented for the composite film systems In-Al and In-Pb for several values of the elctron mean free path.

Surface States and Additional Structure in the Mcmillan-Anderson Model for the Tomasch Effect

Surface States and Additional Structure in the Mcmillan-Anderson Model for the Tomasch Effect

Tomasch Effect as a Probe of the Dispersion Relation of Electrons in "Gapless" Superconductors

The recent theory of McMillan and Anderson is used to discuss the effect of nonmagnetic impurities and depairing mechanisms on the period and damping of the Tomasch oscillations. These oscillations in the density of states seem to provide a unique way of directly measuring the dispersion relation of electronic excitations in superconductors [$\ensuremath{\omega}$ versus $\mathrm{Re}{({\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}^{2}\ensuremath{-}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Delta}}}^{2})}^{\frac{1}{2}}\ensuremath{\propto}$ pseudomomentum ${k}^{*}$, where $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}$ and $\ensuremath{\Delta}$ are the usual renormalized diagonal and off-diagonal self-energies]. We shall discuss the theoretical dispersion curves for films with a small concentration of magnetic impurities and for dirty films ($l\ensuremath{\ll}{\ensuremath{\xi}}_{0}$) in the presence of a parallel magnetic field. The two situations lead to quite different dispersion curves at small values of ${k}^{*}$, for the same value of the depairing parameter. Using the Tomasch effect to plot out the dispersion curves becomes more difficult as $\ensuremath{\omega}$ decreases, since the damping of the oscillations [$\ensuremath{\propto}\mathrm{Im}{({\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}^{2}\ensuremath{-}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Delta}}}^{2})}^{\frac{1}{2}}$] increases.