Measuring and modeling the proximity effect in direct-write electron-beam lithography kinoforms

The proximity effect in successively developed direct-write electron-beam lithography gratings is measured. The grating relief shapes are obtained from the measured power in several of the gratings' diffraction orders. Describing the proximity effect by a convolution with a double Gaussian point-spread function, we determine the parameters of the point-spread function. The writing part of the point-spread function is found to increase significantly with increasing development time, the background part much less.     

Theory of the superconducting proximity effect

The Gor'kov theory of the proximity effect has been solved to a high degree of accuracy by a numerical variational method for a system of two metallic films of finite thickness. The results are analyzed in the context of exact sum rules for the Gor'kov theory and compared with those obtained from various approximate theories, including diffusion theories. It is found that for thin films the order parameter varies considerably more rapidly than is predicted by local theories (e.g., Ginzburg-Landau) and that the different diffusion theories are quite unreliable.Supported by the National Research Council of Canada.     

On the proximity effect in unconventional superconductors

Boundary conditions are derived that link the quasiclassical Green's functions of a singlet and triplet superconductor across a proximity contact. The boundary conditions are used to calculate the order parameter and the tunneling spectra of proximity junctions specifically arranged to study unconventional superconductors for possible triplet pairing.     

Interlacing technique approach for the synthesis of kinoforms

An interlacing technique algorithm is proposed for the synthesis of kinoforms. The conventional iterative methods are quite powerful for optimizing kinoforms, but there is still a large reconstruction error for a quantized kinoform. We suggest the use of a number of subkinoforms interlaced together to synthesize a multikinoform for reconstructing the desired image. The idea of our interlacing technique is to increase the size of a kinoform to reduce the reconstruction error. The first subkinoform is generated from the desired image. Other subkinoforms are generated from the error images between the desired image and the image reconstructed from the previous subkinoforms. A theoretical analysis shows that the reconstruction error will be reduced as the number of subkinoforms is increased. Simulation results show that our interlacing method can reduce the reconstruction error more than do the conventional iterative methods and that the reconstructed image can be improved.     

Superconducting proximity effect of NbN

Proximity effect sandwiches with NbN as type II superconductor and copper, silver, or gold as normal conductors were prepared. The transition temperature of the double layers investigated by resistivity measurements decreases with increasing thickness of the normal conductor. A maximal decline of 0.5 K was observed. The results are in qualitative agreement with theoretical expectations.Dedicated to Professor Dr. phil. Eduard Justi on his seventieth birthday.This work was supported by the Deutsche Forschungsgemeinschaft.     

The proximity effect of the normal-superconducting superlattice

We propose an iterative procedure to solve the linearized Gorkov equation and calculate the superconducting critical temperature of superconductor-normal metal superlattice in the whole region of layer thickness. We also give a discussion of the validity range of the local approximation made on the proximity effect of SN superlattice.     

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 (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 _F ^* /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 (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 (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.     

Generalized McMillan tunneling model of the superconducting proximity effect

We have extended the McMillan tunneling model (MTM) of the superconducting proximity effect to the case of strongly coupled normal-superconducting composites. The generalization has been carried out by treating the tunneling Hamiltonian to all orders of self-consistent perturbation theory. The electronic density of states, the real and imaginary parts of the order parameter, the energy gap, and the transition temperatureT _c calculated in the generalized model are significantly different from the MTM results. Generally, the peak inN _N (E) is broader, shifted, and damped. The energy gap is lower and the depression inT _c is softened.Work supported in part by a grant from the National Research Council of Canada.     

Phonon structure in the tunneling characteristics of thin proximity effect sandwiches

The tunneling spectra of Al-oxide-N-S proximity effect junctions show phonon structure not only from the superconducting layerS, but also from the normal metal layerN. We report experimental results for junctios with Ag, Cu, and Al normal metal layers (100–400 Å thick) and Pb or Sn superconducting layers (3000 Å thick). We compare these experimental results with the McMillan geometrical resonance model of the proximity effect and with an extended McMillan tunneling model. The extension includes the effect of the frequency-dependent electron-phonon interaction in the normal metal. Finally, we estimate the electron-phonon coupling constant for the normal metals from the size of their phonon structure in the tunneling spectra.Research supported by National Science Foundation under Grant #NSF DMR 73-7518 A01.Research supported by National Science Foundation under Grant #NSF DMR 75-19544.Research supported by National Science Foundation under Grant #GH 37239.     

Generalized de Gennes-Takahashi-Tachiki proximity effect theory

We generalize the de Gennes-Takahashi-Tachiki theory of coupled dirty superconductors to include the effects of orbital diamagnetism, spin (Pauli) paramagnetism, spin-orbit scattering, and magnetic impurity scattering. Some new expressions for thin film sandwiches and superlattices are obtained.     

Transition temperature of superconducting-magnetic proximity effect sandwiches

The diffusion approximation of Werthamer and de Gennes for the calculation ofT _c of proximity effect sandwiches is generalized to include the effect of a barrier with a finite penetration probability.T _c is higher than the one calculated for a transparent barrier. The effect of the barrier enters through generalized boundary conditions for the pair potentials. It is found that within the diffusion approximation, the presence of magnetic impurities on the normal side does not affect the boundary conditions.     

On the electron tunneling study of the superconducting proximity effect in Pb-Cd

We show that the recent experimental data of Toplicar and Finnemore on the superconducting proximity effect in Pb-Cd thin films can be understood within the McMillan tunneling model by using our new transition temperature (T _c) formula and determining the parameters self-consistently. Agreement between the theoretical and the experimental values of T _c, the gap, and the general shape and the height of the electronic density of state N _s(E) curves is very good. In agreement with experiment, we find no low-energy peak in the density of states.Work supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.     

Influence of the proximity effect on the energy resolution of STJs

The generating function describing statistical processes of X-ray registration in a superconducting tunnel junction (STJ) with proximized electrodes, and the influence of the proximity effect on the mean value of the signal and the energy resolution of the STJ is derived.     

Error reduction of quantized kinoforms by means of increasing the kinoform size

There are two kinds of method that utilize the redundancy in kinoform domains for reducing the reconstruction errors of quantized kinoforms. One is the iterative-dummy area (IDA) method, which increases the kinoform size indirectly by the addition of a dummy area to the desired image. The other is the interlacing technique (IT), which increases the kinoform size directly by the interlacing of a number of subkinoforms whose sizes are the same as the desired image. We compare the error reduction of quantized kinoforms between these two methods. Simulation results show that reconstruction errors from the IT method can be reduced further and faster than those from the IDA method when the kinoform size is increased to larger than 4 x 4 times the size of the desired image.     

On the proximity effect in a superconductive slab bordered by metal

The first Ginzburg-Landau equation for the order parameter in the absence of magnetic fields is solved analytically for a superconducting slab of thickness 2d bordered by semi-infinite regions of normal metal at each face. The real-valued normalized wave function f=/ depends only on the transversal spatial coordinate x, normalized with respect to the coherence length of the superconductor, provided the de Gennes boundary condition df/dx=f/b is used. The closed-form solution expresses x as an elliptic integral of f, depending on the normalized parameters d and b. It is predicted theoretically that, for b< and dd_c=arctan(1/b), the proximity effect is so strong that the superconductivity is completely suppressed. In fact, in this case, the first Ginzburg-Landau equation possesses only the trivial solution f0.     

The long-range proximity effect: Pair echoes

Recent experiments on N-N-S junctions with high-temperature cuprate superconductors have shown that superconductivity can be induced in the adjacent, normal metal at distances many times greater than the few hundred Ångstroms range of the conventional proximity effect. We show that this arises in the particular geometry used in these experiments through the constructive interference of the tails of the de Gennes-Saint-James bound and quasi-bound states of the normal metal N that penetrate into the third metal, N. This constructive interference results in the re-appearance of a pair amplitude in this metal which is analogous to the spin echo of NMR and which we name a pair echo.     

Use of proximity effect in iridium-gold superconducting phase transition thermometers

We have studied the proximity effect in bilayers of thin films of iridium covered by gold. These structures were evaporated onto sapphire single crystals for use as phase transition thermometers in cryogenic particle detectors. By varying the thicknesses of the iridium and gold layers, we achieved critical temperatures as low as 33 mK. The critical temperature of the bilayers is lower than predicted by the theory of de Gennes-Werthamer, but adding a free parameter to the theory allows good agreement. The transitions of the bilayers typically had widths of a few mK, with the narrowest reaching 0.2 mK, and were always sharper than those of pure iridium films evaporated simultaneously.     

A microscopic model for the nonequilibrium proximity effect in superconductor-normal metal junctions

The current-voltage characteristics of a superconductor-normal metal junction are affected by the proximity superconductivity induced on the normal side. We study this effect using the quasiclassical theory. We first present a unified derivation of the previous results obtained by time-dependent Ginzburg-Landau (TDGL) theories. In order to study the effect beyond the scope of the TDGL theory, we approximate the induced order parameter by a step function. The amplitude and the phase of the induced order parameter as well as the width of the step are determined self-consistently. The calculation is made in the pure limit, but no restrictions are set on the temperature or on the transmission coefficient of the junction besides the ones implicit in the step approximation. A time-dependent solution for the order parameter is found at all currents or voltages. It is found that the results in the TDGL region and outside are relatively similar, although different mechanisms are responsible for the conversion from the normal to supercurrent in each case.     

Transport properties in a superconducting proximity sandwich. Kondo effect

Temperature-dependent properties are evaluated in a superconducting Kondo alloy induced by the proximity effect. These include (i) the order parameter, (ii) the dc Josephson current, and (iii) the thermal conductivity and ultrasonic attenuation, in the case when there is reentrance of superconductivity. The calculations are based on the theory given by Kaiser, in which the McMillan tunneling model and the Müller-Hartmann-Zittartz theory are combined. The temperature dependence of pair-breaking is reflected directly in the above properties in much the same way as in an intrinsic Kondo superconductor, indicating that proximity-effect studies may provide detailed information on the Kondo effect in superconductivity.     

Design of fan-out kinoforms in the entire scalar diffraction regime with an optimal-rotation-angle method

An algorithm for the design of diffractive optical phase elements (kinoforms) that give rise to fan-out (i.e., spot) patterns was developed and tested. The algorithm is based on the Helmholtz-Kirchhoff rigorous scalar diffraction integral for the evaluation of the electric field behind the kinoform. The optimization of the kinoform phase modulation is performed with an efficient optimal-rotation-angle method. The algorithm permits any spatial configuration of the locations of the desired spots. For example, the spots (all or some) can be located at large angles to the optical axis (nonparaxial case) or they can be located in the near near field of the kinoform, i.e., where the Fresnel approximation is no longer valid. Two examples of fabricated kinoforms designed with this algorithm are presented.