D-Wave Superconductors near Surfaces and Interfaces: A Scattering Matrix Approach within the Quasiclassical Technique

A recently developed method [A. Shelankov and M. Ozana, Phys. Rev. B 61, 7077 (2000)] is applied to investigate d-wave superconductors in the vicinity of (rough) surfaces. While this method allows the incorporation of arbitrary interfaces into the quasiclassical technique, we discuss, as examples, diffusive surfaces and boundaries with small tilted mirrors (facets). The properties of the surface enter via the scattering matrix in the boundary condition for the quasiclassical Green's function. The diffusive surface is described by an ensemble of random scattering matrices. We find that the fluctuations of the density of states around the average are small; the zero bias conductance peak broadens with increasing disorder. The faceted surface is described in the model where the scattering matrix couples m in- and m out-trajectories (m>=2). No zero bias conductance peak is found for [100] surfaces; the relation to the model of Fogelstrom et al. [Phys. Rev. Lett. 79, 281 (1997)] is discussed.Comment: RevTeX, 19 pages, 18 figure

Non-Equilibrium Quasiclassical Theory for Josephson Structures

We present a non-equilibrium quasiclassical formalism suitable for studying linear response ac properties of Josephson junctions. The non-equilibrium self-consistency equations are satisfied, to very good accuracy, already in zeroth iteration. We use the formalism to study ac Josephson effect in a ballistic superconducting point contact. The real and imaginary parts of the ac linear conductance are calculated both analytically (at low frequencies) and numerically (at arbitrary frequency). They show strong temperature, frequency, and phase dependence. Many anomalous properties appear near phi = pi. We ascribe them to the presence of zero energy bound states.Comment: 11 pages, 9 figures, Final version to appear in PR

Thermodynamics of a d-wave Superconductor Near a Surface

We study the properties of an anisotropically paired superconductor in the presence of a specularly reflecting surface. The bulk stable phase of the superconducting order parameter is taken to have $d_{x^2-y^2}$ symmetry. Contributions by order parameter components of different symmetries vanish in the bulk, but may enter in the vicinity of a wall. We calculate the self-consistent order parameter and surface free energy within the quasiclassical formulation of superconductivity. We discuss, in particular, the dependence of these quantities on the degree of order parameter mixing and the surface to lattice orientation. Knowledge of the thermodynamically stable order parameter near a surface is a necessary precondition for calculating measurable surface properties which we present in a companion paper.Comment: 12 pages of revtex text with 12 compressed and encoded figures. To appear in J. Low Temp. Phys., December, 199

ac Josephson effect in superconducting d-wave junctions

We study theoretically the ac Josephson effect in superconducting planar d-wave junctions. The insulating barrier assumed to be present between the two superconductors may have arbitrary strength. Many properties of this system depend on the orientation of the d-wave superconductor: we calculate the ac components of the Josephson current. In some arrangements there is substantial negative differential conductance due to the presence of mid-gap states. We study how robust these features are to finite temperature and also comment on how the calculated current-voltage curves compare with experiments. For some other configurations (for small barrier strength) we find zero-bias conductance peaks due to multiple Andreev reflections through midgap states. Moreover, the odd ac components are strongly suppressed and even absent in some arrangements. This absence will lead to a doubling of the Josephson frequency. All these features are due to the d-wave order parameter changing sign when rotated $90^{\circ}$. Recently, there have been several theoretical reports on parallel current in the d-wave case for both the stationary Josephson junction and for the normal metal-superconductor junction. Also in our case there may appear current density parallel to the junction, and we present a few examples when this takes place. Finally, we give a fairly complete account of the method used and also discuss how numerical calculations should be performed in order to produce current-voltage curves

Effects of gap anisotropy upon the electronic structure around a superconducting vortex

An isolated single vortex is considered within the framework of the quasiclassical theory. The local density of states around a vortex is calculated in a clean type II superconductor with an anisotropy. The anisotropy of a superconducting energy gap is crucial for bound states around a vortex. A characteristic structure of the local density of states, observed in the layered hexagonal superconductor 2H-NbSe2 by scanning tunneling microscopy (STM), is well reproduced if one assumes an anisotropic s-wave gap in the hexagonal plane. The local density of states (or the bound states) around the vortex is interpreted in terms of quasiparticle trajectories to facilitate an understanding of the rich electronic structure observed in STM experiments. It is pointed out that further fine structures and extra peaks in the local density of states should be observed by STM.Comment: 11 pages, REVTeX; 20 PostScript figures; An Animated GIFS file for the star-shaped vortex bound states is available at http://mp.okayama-u.ac.jp/~hayashi/vortex.htm

Zero-modes in the random hopping model

If the number of lattice sites is odd, a quantum particle hopping on a bipartite lattice with random hopping between the two sublattices only is guaranteed to have an eigenstate at zero energy. We show that the localization length of this eigenstate depends strongly on the boundaries of the lattice, and can take values anywhere between the mean free path and infinity. The same dependence on boundary conditions is seen in the conductance of such a lattice if it is connected to electron reservoirs via narrow leads. For any nonzero energy, the dependence on boundary conditions is removed for sufficiently large system sizes.Comment: 12 pages, 11 figure

Theory of charge transport in diffusive normal metal / unconventional singlet superconductor contacts

We analyze the transport properties of contacts between unconventional superconductor and normal diffusive metal in the framework of the extended circuit theory. We obtain a general boundary condition for the Keldysh-Nambu Green's functions at the interface that is valid for arbitrary transparencies of the interface. This allows us to investigate the voltage-dependent conductance (conductance spectrum) of a diffusive normal metal (DN)/ unconventional singlet superconductor junction in both ballistic and diffusive cases. For d-wave superconductor, we calculate conductance spectra numerically for different orientations of the junctions, resistances, Thouless energies in DN, and transparencies of the interface. We demonstrate that conductance spectra exhibit a variety of features including a $V$-shaped gap-like structure, zero bias conductance peak (ZBCP) and zero bias conductance dip (ZBCD). We show that two distinct mechanisms: (i) coherent Andreev reflection (CAR) in DN and (ii) formation of midgap Andreev bound state (MABS) at the interface of d-wave superconductors, are responsible for ZBCP, their relative importance being dependent on the angle $\alpha$ between the interface normal and the crystal axis of d-wave superconductors. For $\alpha=0$, the ZBCP is due to CAR in the junctions of low transparency with small Thouless energies, this is similar to the case of diffusive normal metal / insulator /s-wave superconductor junctions. With increase of $\alpha$ from zero to $\pi/4$, the MABS contribution to ZBCP becomes more prominent and the effect of CAR is gradually suppressed. Such complex spectral features shall be observable in conductance spectra of realistic high-$T_c$ junctions at very low temperature

Theory of charge transport in diffusive normal metal / conventional superconductor point contacts

Tunneling conductance in diffusive normal metal / insulator / s-wave superconductor (DN/I/S) junctions is calculated for various situations by changing the magnitudes of the resistance and Thouless energy in DN and the transparency of the insulating barrier. The generalized boundary condition introduced by Yu. Nazarov [Superlattices and Microstructures 25 1221 (1999)] is applied, where the ballistic theory by Blonder Tinkham and Klapwijk (BTK) and the diffusive theory by Volkov Zaitsev and Klapwijk based on the boundary condition of Kupriyanov and Lukichev (KL) are naturally reproduced. It is shown that the proximity effect can enhance (reduce) the tunneling conductance for junctions with a low (high) transparency. A wide variety of dependencies of tunneling conductance on voltage bias is demonstrated including a $U$-shaped gap like structure, a zero bias conductance peak (ZBCP) and a zero bias conductance dip (ZBCD)

Traffic and Related Self-Driven Many-Particle Systems

Since the subject of traffic dynamics has captured the interest of physicists, many astonishing effects have been revealed and explained. Some of the questions now understood are the following: Why are vehicles sometimes stopped by so-called ``phantom traffic jams'', although they all like to drive fast? What are the mechanisms behind stop-and-go traffic? Why are there several different kinds of congestion, and how are they related? Why do most traffic jams occur considerably before the road capacity is reached? Can a temporary reduction of the traffic volume cause a lasting traffic jam? Under which conditions can speed limits speed up traffic? Why do pedestrians moving in opposite directions normally organize in lanes, while similar systems are ``freezing by heating''? Why do self-organizing systems tend to reach an optimal state? Why do panicking pedestrians produce dangerous deadlocks? All these questions have been answered by applying and extending methods from statistical physics and non-linear dynamics to self-driven many-particle systems. This review article on traffic introduces (i) empirically data, facts, and observations, (ii) the main approaches to pedestrian, highway, and city traffic, (iii) microscopic (particle-based), mesoscopic (gas-kinetic), and macroscopic (fluid-dynamic) models. Attention is also paid to the formulation of a micro-macro link, to aspects of universality, and to other unifying concepts like a general modelling framework for self-driven many-particle systems, including spin systems. Subjects such as the optimization of traffic flows and relations to biological or socio-economic systems such as bacterial colonies, flocks of birds, panics, and stock market dynamics are discussed as well.Comment: A shortened version of this article will appear in Reviews of Modern Physics, an extended one as a book. The 63 figures were omitted because of storage capacity. For related work see http://www.helbing.org