Dr. Mihajlo Vanević
Condensed Matter Theory Group
Research interests
- • Mesoscopic transport, noise, statistics of charge transfer (full counting statistics)
- • Circuit theory of mesoscopic transport
- • Superconductivity; field theoretical methods in condensed matter physics
- • Quantum entanglement in many-body systems
- • Electronic properties of graphene and related nanostructures
- • Polarons in solid-state systems
Teaching
Publications
[check arXiv;
show/hide abstracts]
Continuous-variable entanglement test in driven quantum contacts
Hongxin Zhan, Mihajlo Vanević, and Wolfgang Belzig
[abstract; arXiv:1711.11461]
The standard entanglement test using the Clauser-Horne-Shimony-Holt inequality is known to fail in mesoscopic junctions at finite temperatures. Since this is due to the bidirectional particle flow, a similar failure is expected to occur in an ac-driven contact. We develop a continuous-variable entanglement test suitable for electrons and holes that are created by the ac drive. The generalized Bell inequality is violated in junctions with low conductance or small number of transport channels and with ac voltages which create few electron-hole pairs per cycle. Our ac-entanglement test depends on the total number of electron-hole pairs and on the distribution of probabilities of pair creations similar to the Fano factor.
Electron and electron-hole quasiparticle states in a driven quantum contact
Mihajlo Vanević, Julien Gabelli, Wolfgang Belzig, and Bertrand Reulet
[abstract; arXiv:1506.03878; Physical Review B 93, 041416(R) (2016)]
We study the many-body electronic state created by a time-dependent drive of a mesoscopic contact. The many-body state is expressed manifestly in terms of single-electron and electron-hole quasiparticle excitations with the amplitudes and probabilities of creation which depend on the details of the applied voltage. We experimentally probe the time dependence of the constituent electronic states by using an analog of the optical Hong-Ou-Mandel correlation experiment where electrons emitted from the terminals with a relative time delay collide at the contact. The electron wave packet overlap is directly related to the current noise power in the contact. We have confirmed the time dependence of the electronic states predicted theoretically by measurements of the current noise power in a tunnel junction under harmonic excitation.
Electron and electron-hole excitations in a driven Fermi sea
Mihajlo Vanević, Yuli V. Nazarov, and Wolfgang Belzig
[abstract; Physica Status Solidi B (2016), special issue "Single-electron control in solid-state devices"]
We study the response of a degenerate Fermi sea to a time-dependent perturbation. We find that elementary excitations in the Fermi sea are electrons and electron-hole quasiparticle pairs whose number and probability of creation depend on amplitude and shape of the drive. The excitations can be probed in a driven quantum contact at low temperature. While electronic excitations are associated to the dc voltage offset and determine the average current, the electron-hole pairs are created by the ac voltage component and give rise to the excess current noise power in a junction. We also obtain the many-body state created by the drive. The constituent quasiparticle states can be probed by a Hong-Ou-Mandel type of correlation experiment where electrons emitted from the terminals with a relative time delay collide at the contact. The current noise power in the junction is directly related to the wave packet overlap at the contact. This is confirmed in recent experiments, demonstrating an unprecedented degree of control of elementary excitations in a coherent mesoscopic junction.
Elementary Andreev processes in a driven superconductor - normal metal contact
Wolfgang Belzig and Mihajlo Vanević
[abstract; arXiv:1508.07039; Physica E 75, 22 (2016)]
We investigate the full counting statistics of a voltage-driven normal metal (N) - superconductor (S) contact. In the low-bias regime below the superconducting gap, the NS contact can be mapped onto a purely normal contact, albeit with doubled voltage and counting fields. Hence, in this regime the transport characteristics can be obtained by the corresponding substitution of the normal metal results. The elementary processes are single Andreev transfers and electron- and hole-like Andreev transfers. Considering Lorentzian voltage pulses we find an optimal quantization for half-integer levitons.
[Reprinted in the special issue
"Frontiers in quantum electronic transport", in memory of Markus Büttiker]
Interface superconductivity in cuprates defies Fermi-liquid description
Z. Radović, M. Vanević, J. Wu, A. Bollinger, and I. Božović
[abstract; Journal of Superconductivity and Novel Magnetism (2016)]
La_{2-x}Sr_{x}CuO_{4}/La_{2}CuO_{4} bilayers show interface superconductivity that originates from accumulation and depletion of mobile charge carriers across the interface. Surprisingly, the doping level can be varied broadly (within the interval 0.15<x<0.47) without affecting the transition temperature, which stays essentially constant and equal to that in optimally doped material, T_{c} ≈ 40 K. We argue that this finding implies that doping up to the optimum level does not shift the chemical potential, unlike in ordinary Fermi liquids. We discuss possible physical scenarios that can give doping-independent chemical potential in the pseudogap regime: electronic phase separation, formation of charge density waves, strong Coulomb interactions, or self-trapping of mobile charge carriers.
Coherent dynamics in long fluxonium qubits
Gianluca Rastelli, Mihajlo Vanević, and Wolfgang Belzig
[abstract; arXiv:1403.4565; New Journal of Physics 17, 053026 (2015)]
We analyze the coherent dynamics of a fluxonium device (Manucharyan et al 2009 Science 326 113) formed by a superconducting ring of Josephson junctions in which strong quantum phase fluctuations are localized exclusively on a single weak element. In such a system, quantum phase tunnelling by 2pi occurring at the weak element couples the states of the ring with supercurrents circulating in opposite directions, while the rest of the ring provides an intrinsic electromagnetic environment of the qubit. Taking into account the capacitive coupling between nearest neighbors and the capacitance to the ground, we show that the homogeneous part of the ring can sustain electrodynamic modes which couple to the two levels of the flux qubit. In particular, when the number of Josephson junctions is increased, several low-energy modes can have frequencies lower than the qubit frequency. This gives rise to a quasiperiodic dynamics, which manifests itself as a decay of oscillations between the two counterpropagating current states at short times, followed by oscillation-like revivals at later times. We analyze how the system approaches such a dynamics as the ring's length is increased and discuss possible experimental implications of this non-adiabatic regime.
Transmon-based simulator of nonlocal electron-phonon coupling: a platform for observing sharp small-polaron transitions
Vladimir M. Stojanović, Mihajlo Vanević, Eugene Demler, and Lin Tian
[abstract; arXiv:1401.4783; Physical Review B 89, 144508 (2014)]
We propose an analog superconducting quantum simulator for a one-dimensional model featuring momentum-dependent (nonlocal) electron-phonon couplings of Su-Schrieffer-Heeger and "breathing-mode" types. Because its corresponding vertex function depends on both the electron- and phonon quasimomenta, this model does not belong to the realm of validity of the Gerlach-Loewen theorem that rules out any nonanalyticities in single-particle properties. The superconducting circuit behind the proposed simulator entails an array of transmon qubits and microwave resonators. By applying microwave driving fields to the qubits, a small-polaron Bloch state with an arbitrary quasimomentum can be prepared in this system within times several orders of magnitude shorter than the typical qubit decoherence times. We demonstrate that in this system -- by varying the circuit parameters -- one can readily reach the critical coupling strength required for observing the sharp transition from a nondegenerate (single-particle) ground state corresponding to zero quasimomentum (Kgs=0) to a twofold-degenerate small-polaron ground state at nonzero quasimomenta Kgs and -Kgs. Through exact numerical diagonalization of our effective Hamiltonian, we show how this nonanalyticity is reflected in the relevant single-particle properties (ground-state energy, quasiparticle residue, average number of phonons). The proposed setup provides an ideal testbed for studying quantum dynamics of polaron formation in systems with strongly momentum-dependent electron-phonon interactions.
Graphene's morphology and electronic properties from discrete differential geometry
A. P. Sanjuan, Z. Wang, H. P. Imani, M. Vanević, and S. Barraza-Lopez
[abstract; arXiv:1402.3751; Physical Review B 89, 121403(R) (2014)]
The geometry of two-dimensional crystalline membranes dictates their mechanical, electronic, and chemical properties. The local geometry of a surface is determined from the two invariants of the metric and the curvature tensors. Here we discuss those invariants directly from atomic positions in terms of angles, areas, and vertex and normal vectors from carbon atoms on the graphene lattice, for arbitrary elastic regimes and atomic conformations, and without recourse to an effective continuum model. The geometrical analysis of graphene membranes under mechanical load is complemented with a study of the local density of states (LDOS), discrete induced gauge potentials, velocity renormalization, and nontrivial electronic effects originating from the scalar deformation potential. The asymmetric LDOS is related to sublattice-specific deformation potential differences, giving rise to the pseudomagnetic field. The results here enable the study of geometrical, mechanical, and electronic properties for arbitrarily shaped graphene membranes in experimentally relevant regimes without recourse to differential geometry and continuum elasticity.
Anomalous independence of interface superconductivity from carrier density
J. Wu, O. Pelleg, G. Logvenov, A. T. Bollinger, Y. Sun, G. S. Boebinger, M. Vanević, Z. Radović, and I. Božović
[abstract; Nature Materials 12, 877 (2013)]
The recent discovery of superconductivity at the interface of two non-superconducting materials has received much attention. In cuprate bilayers, the critical temperature (Tc) can be significantly enhanced compared with single-phase samples. Several explanations have been proposed, invoking Sr interdiffusion, accumulation and depletion of mobile charge carriers, elongation of the copper-to-apical-oxygen bond length, or a beneficial crosstalk between a material with a high pairing energy and another with a large phase stiffness. From each of these models, one would predict Tc to depend strongly on the carrier density in the constituent materials. Here, we study combinatorial libraries of La2-xSrxCuO4-La2CuO4 bilayer samples -- an unprecedentedly large set of more than 800 different compositions. The doping level x spans a wide range, 0.15 < x < 0.47, and the measured Hall coefficient varies by one order of magnitude. Nevertheless, across the entire sample set, Tc stays essentially constant at about 40 K. We infer that doping up to the optimum level does not shift the chemical potential, unlike in ordinary Fermi liquids. This result poses a new challenge to theory -- cuprate superconductors have not run out of surprises.
[Interface Superconductivity Withstands Variations in Atomic Configuration]
[Nature Materials cover; Nature Materials News & Views article]
Strain-engineering of graphene's electronic structure beyond continuum elasticity
S. Barraza-Lopez, A. P. Sanjuan, Z. Wang, and M. Vanević
[abstract; arXiv:1310.3622; Solid State Communications 166, 70 (2013), fast track article]
We present a new first-order approach to strain-engineering of graphene's electronic structure where no continuous displacement field u(x,y) is required. The approach is valid for negligible curvature. The theory is directly expressed in terms of atomic displacements under mechanical load, such that one can determine if mechanical strain is varying smoothly at each unit cell, and the extent to which sublattice symmetry holds. Since strain deforms lattice vectors at each unit cell, orthogonality between lattice and reciprocal lattice vectors leads to renormalization of the reciprocal lattice vectors as well, making the K and K' points shift in opposite directions. From this observation we conclude that no K-dependent gauges enter in a first-order theory. In this formulation of the theory the deformation potential and pseudo-magnetic field take discrete values at each graphene unit cell. We illustrate the formalism by providing strain-generated fields and local density of electronic states in graphene membranes with large numbers of atoms. The present method complements and goes beyond the prevalent approach, where strain engineering in graphene is based upon first-order continuum elasticity.
Early stages of magnetization relaxation in superconductors
Mihajlo Vanević, Zoran Radović, and Vladimir G. Kogan
[abstract; arXiv:1302.4312; Physical Review B 87, 144501 (2013)]
Magnetic flux dynamics in type-II superconductors is studied within the model of a viscous nonlinear diffusion of vortices for various sample geometries. We find that time dependence of magnetic moment relaxation after the field is switched off can be accurately approximated by \( m(t)\propto 1-\sqrt{t/\tilde\tau} \) in the narrow initial time interval and by \( m(t)\propto (1+t/\tau)^{-1} \) at later times before the flux creep sets in. The characteristic times \( \tilde\tau \) and \( \tau \) are proportional to the viscous drag coefficient \( \eta \). Quantitative agreement with available experimental data is obtained for both conventional and high-temperature superconductors with \( \eta \) exceeding by many orders of magnitude the Bardeen-Stephen coefficient for free vortices. Huge enhancement of the drag, as well as its exponential temperature dependence, indicate a strong influence of pinning centers on the flux diffusion. Notwithstanding complexity of the vortex motion in the presence of pinning and thermal agitation, we argue that the initial relaxation of magnetization can still be considered as a viscous flux flow with an effective drag coefficient.
Control of electron-hole pair generation by biharmonic voltage drive of a quantum point contact
Mihajlo Vanević and Wolfgang Belzig
[abstract; arXiv:1210.7541; Physical Review B 86, 241306(R) (2012)]
A time-dependent electromagnetic field creates electron-hole excitations in a Fermi sea at low temperature. We show that the electron-hole pairs can be generated in a controlled way using harmonic and biharmonic time-dependent voltages applied to a quantum contact and obtain the probabilities of the pair creations. For a biharmonic voltage drive, we find that the probability of a pair creation decreases in the presence of an in-phase second harmonic. This accounts for the suppression of the excess noise observed experimentally [Gabelli and Reulet, arXiv:1205.3638] proving that dynamic control and detection of elementary excitations in quantum conductors are within the reach of the present technology.
Quantum phase slips in superconducting wires with weak inhomogeneities
Mihajlo Vanević and Yuli V. Nazarov
[abstract; arXiv:1108.3553; Physical Review Letters 108, 187002 (2012)]
Quantum phase slips are traditionally considered in diffusive superconducting wires which are assumed homogeneous. We present a definite estimate for the amplitude of phase slips that occur at a weak inhomogeneity in the wire where local resistivity is slightly increased. We model such a weak link as a general coherent conductor and show that the amplitude is dominated by topological part of the action. We argue that such weak links occur naturally in apparently homogeneous wires and adjust the estimate to that case. The fabrication of an artificial weak link would localize phase slips and facilitate a better control of the phase-slip amplitude.
[selected for Virtual Journal of Nanoscale Science & Technology]
Electron-phonon coupling in graphene antidot lattices: an indication
of polaronic behavior
N. Vukmirović, V. M. Stojanović, and M. Vanević
[abstract; arXiv:0909.2179; Physical Review B
81, 041408(R) (2010)]
We study graphene antidot lattices --
superlattices of perforations (antidots) in a graphene sheet -- using
a model that accounts for the phonon-modulation of the pi-electron
hopping integrals. We calculate the phonon spectra of selected
antidot lattices using two different semi-empirical interatomic
potentials. Based on the adopted model and the obtained phonon modes,
we quantify the nature of charge-carriers in the system by computing
the quasiparticle spectral weight due to the electron-phonon
interaction for an excess electron in the conduction band. We show
that the phonon-induced renormalization is much stronger than in
graphene, with the effective electron masses exhibiting an
interesting nonmonotonic dependence on the superlattice period for a
given antidot diameter. Our study provides an indication of polaronic
behavior and points to the necessity of taking into account the
inelastic degrees of freedom in future studies of electronic
transport in graphene antidot lattices.
[selected as PRB Editors' suggestion]
[selected for Virtual Journal of Nanoscale Science & Technology]
Effects of metallic contacts on electron transport through graphene
S. Barraza-Lopez, M. Vanević, M. Kindermann, and M. Y. Chou
[abstract; arXiv:1001.5257; Physical Review Letters
104, 076807 (2010)]
We report on a first-principles study of the
conductance through graphene suspended between Al contacts as a
function of junction length, width, and orientation. The charge
transfer at the leads and into the freestanding section gives rise to
an electron-hole asymmetry in the conductance and in sufficiently
long junctions induces two conductance minima at the energies of the
Dirac points for suspended and clamped regions, respectively. We
obtain the potential profile along a junction caused by doping and
provide parameters for effective model calculations of the junction
conductance with weakly interacting metallic leads.
[Georgia Tech research news: Making
contact...]
Character of electronic states in graphene antidot lattices: Flat
bands and spatial localization
M. Vanević, V. M. Stojanović, and M. Kindermann
[abstract; arXiv:0903.0918; Physical Review B
80, 045410 (2009)]
Graphene antidot lattices have recently been
proposed as a new breed of graphene-based superlattice structures. We
study electronic properties of triangular antidot lattices, with
emphasis on the occurrence of dispersionless (flat) bands and the
ensuing electron localization. Apart from strictly flat bands at
zero-energy (Fermi level), whose existence is closely related to the
bipartite lattice structure, we also find quasi-flat bands at low
energies. We predict the real-space electron density profiles due to
these localized states for a number of representative antidot
lattices. We point out that the studied low-energy, localized states
compete with states induced by defects on the superlattice scale in
this system which have been proposed as hosts for electron spin
qubits. We furthermore suggest that local moments formed in these
midgap zero-energy states may be at the origin of a surprising
saturation of the electron dephasing length observed in recent weak
localization measurements in graphene antidot lattices.
[selected for Virtual Journal of Nanoscale Science & Technology]
Elementary charge-transfer processes in mesoscopic conductors
M. Vanević, Yu. V. Nazarov, and W. Belzig
[abstract; arXiv:0808.3370; Physical Review B
78, 245308 (2008)]
We determine charge-transfer statistics in a
quantum conductor driven by a time-dependent voltage and identify the
elementary transport processes. At zero temperature unidirectional
and bidirectional single-charge transfers occur. The unidirectional
processes involve electrons injected from the source terminal due to
excess dc bias voltage. The bidirectional processes involve
electron-hole pairs created by time-dependent voltage bias. This
interpretation is further supported by the charge-transfer statistics
in a multiterminal beam-splitter geometry in which injected electrons
and holes can be partitioned into different outgoing terminals. The
probabilities of elementary processes can be probed by noise
measurements: the unidirectional processes set the dc noise level,
while bidirectional ones give rise to the excess noise. For ac
voltage drive, the noise oscillates with increasing the driving
amplitude. The decomposition of the noise into the contributions of
elementary processes reveals the origin of these oscillations: the
number of electron-hole pairs generated per cycle increases with
increasing the amplitude. The decomposition of the noise into
elementary processes is studied for different time-dependent
voltages. The method we use is also suitable for systematic
calculation of higher-order current correlators at finite
temperature. We obtain current noise power and the third cumulant in
the presence of time-dependent voltage drive. The charge-transfer
statistics at finite temperature can be interpreted in terms of
multiple-charge transfers with probabilities which depend on energy
and temperature.
[selected for Virtual Journal of Nanoscale Science & Technology]
Quantum-entanglement aspects of polaron systems
V. M. Stojanović and M. Vanević
[abstract; arXiv:0808.2632; Physical Review B
78, 214301 (2008)]
We describe quantum entanglement inherent to
the polaron ground states of coupled electron-phonon (or, more
generally, particle-phonon) systems based on a model comprising both
local (Holstein-type) and nonlocal (Peierls-type) couplings. We study
this model using a variational method supplemented by the exact
numerical diagonalization on a system of finite size. By way of
subsequent numerical diagonalization of the reduced density matrix,
we determine the particle-phonon entanglement as given by the von
Neumann and linear entropies. Our results are strongly indicative of
the intimate relationship between the particle
localization/delocalization and the particle-phonon entanglement. In
particular, we find a compelling evidence for the existence of a
nonanalyticity in the entanglement entropies with respect to the
Peierls-coupling strength. The occurrence of such nonanalyticity - not
accompanied by an actual quantum phase transition - reinforces
analogous conclusion drawn in several recent studies of entanglement
in the realm of quantum-dissipative systems. In addition, we
demonstrate that the entanglement entropies saturate inside the
self-trapped region where the small-polaron states are nearly
maximally mixed.
[selected as PRB Editors' suggestion]
[selected for Virtual Journal of Nanoscale Science & Technology]
Circuit theory of charge transport in mesoscopic conductors
M. Vanević (PhD Thesis, February 2008, Basel)
[summary; full text; e-Diss@UNI
BASEL]
This Thesis is devoted to the circuit theory of
mesoscopic transport. The emphasis is put on its extension which
provides a method to obtain the complete statistics of the
transferred charge. To accomplish this task, several topics have to
be combined: a mathematical description of the charge transfer
statistics, the scattering approach to mesoscopic transport, and the
nonequilibrium Keldysh-Green's function technique. Although the
underlying theory is rather complex, the circuit-theory rules which
are obtained at the end are in fact very simple. They resemble
Kirchhoff's laws for conventional macroscopic conductors, with
currents and voltages replaced by their mesoscopic counterparts. An
important difference is that the mesoscopic "currents'' and
"voltages'' acquire matrix structure, and that the
"current''-"voltage'' relation is in general nonlinear. The matrix
structure originates from the Keldysh-Green's function formalism
which is needed to account for the many-body quantum state of the
electrons in the system. The circuit theory is applicable to
multiterminal mesoscopic structures with terminals of different
types, e.g., normal metals, superconductors, and ferromagnets. The
junctions within the structure can be different also, e.g.,
transparent quantum point contacts, tunnel barriers, disordered
interfaces, diffusive wires, etc.
The Thesis is organized as follows. In Chapter I, we provide
introductory information on noise. We discuss early experiments on
noise in vacuum tubes and the Schottky result which relates the
spectral density of current fluctuations and the average current. We
summarize some important results on noise in mesoscopic conductors
which can be obtained within circuit theory. Chapters II and III are
devoted to theoretical prerequisites needed for the circuit theory.
In Chapter II, we define the notion of the cumulant generating
function and its relation to statistically independent processes. In
Chapter III, we introduce the scattering approach to mesoscopic
transport and the method of Keldysh-Green's functions. The circuit
theory is presented in Chapter IV focusing on the extension which
provides complete information on the charge transfer statistics. The
method is illustrated by calculation of the transmission distribution
in 2-terminal junctions, and by studying current cross correlations
in a superconductor-beam splitter geometry. In Chapters V - VII we
apply the general template of the circuit theory and obtain the
charge transfer statistics in several physical systems of interest: a
cavity coupled to a superconductor and a normal terminal, several
junctions in series, and a voltage driven mesoscopic junction. The
knowledge of the charge transfer statistics enables us to identify
the elementary charge transfer processes in these systems. The
conclusion is given in Chapter VIII.
Elementary events of electron transfer in a voltage-driven quantum
point contact
M. Vanević, Yu. V. Nazarov, and W. Belzig
[abstract; cond-mat/0701282;
Physical Review Letters
99, 076601 (2007)]
We show that the statistics of electron
transfer in a coherent quantum point contact driven by an arbitrary
time-dependent voltage is composed of elementary events of two kinds:
unidirectional one-electron transfers determining the average current
and bidirectional two-electron processes contributing to the noise
only. This result pertains at vanishing temperature while the
extended Keldysh-Green's function formalism in use also enables the
systematic calculation of the higher-order current correlators at
finite temperatures.
[selected for Virtual Journal of Nanoscale Science & Technology]
Quasiparticle transport in arrays of chaotic cavities
M. Vanević and W. Belzig
[abstract; cond-mat/0605329;
Europhysics Letters
75, 604 (2006)]
We find the distribution of transmission
eigenvalues in a series of identical junctions between chaotic
cavities using the circuit theory of mesoscopic transport. This
distribution rapidly approaches the diffusive wire limit as the
number of junctions increases, independent of the specific scattering
properties of a single junction. The cumulant generating function and
the first three cumulants of the charge transfer through the system
are obtained both in the normal and in the superconducting
state.
Full counting statistics of Andreev scattering in an asymmetric
chaotic cavity
M. Vanević and W. Belzig
[abstract; cond-mat/0412320;
Physical Review B
72, 134522 (2005)]
We study the charge transport statistics in
coherent two-terminal double junctions within the framework of the
circuit theory of mesoscopic transport. We obtain the general
solution of the circuit-theory matrix equations for the Green's
function of a chaotic cavity between arbitrary contacts. As an
example we discuss the full counting statistics and the first three
cumulants for an open asymmetric cavity between a superconductor and
a normal-metal lead at temperatures and voltages below the
superconducting gap. The third cumulant shows a characteristic sign
change as a function of the asymmetry of the two quantum point
contacts, which is related to the properties of the Andreev
reflection eigenvalue distribution.
[selected for Virtual Journal of Applications of
Superconductivity]
[selected for Virtual Journal of Nanoscale Science & Technology]
Quasiparticle states in superconducting superlattices
M. Vanević and Z. Radović
[abstract; cond-mat/0502180;
European Physical Journal B
46, 419 (2005)]
The energy bands and the global density of
states are computed for superconductor / normal-metal superlattices
in the clean limit. Dispersion relations are derived for the general
case of insulating interfaces, including the mismatch of Fermi
velocities and effective band masses. We focus on the influence of
finite interface transparency and compare our results with those for
transparent superlattices and trilayers. Analogously to the rapid
variation on the atomic scale of the energy dispersion with layer
thicknesses in transparent superlattices, we find strong oscillations
of the almost flat energy bands (transmission resonances) in the case
of finite transparency. In small-period transparent superlattices the
BCS coherence peak disappears and a similar subgap peak is formed due
to the Andreev process. With decreasing interface transparency the
characteristic double peak structure in the global density of states
develops towards a gapless BCS-like result in the tunnel limit. This
effect can be used as a reliable STM probe for interface
transparency.