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We calculate the probability distribution of the Andreev reflection eigenvalues R_n at the Fermi level in the circular ensemble of random-matrix theory. Without spin-rotation symmetry, the statistics of the electrical conductance G depends on the topological quantum number Q of the superconductor. We show that this dependence is nonperturbative in the number N of scattering channels, by proving that the p-th cumulant of G is independent of Q for p<N/d (with d=2 or d=1 in the presence or in the absence of time-reversal symmetry). A large-N effect such as weak localization cannot, therefore, probe the topological quantum number. For small N we calculate the full distribution P(G) of the conductance and find qualitative differences in the topologically trivial and nontrivial phases.
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We investigate transport and shot noise in lateral N-TI-S contacts, where N is a normal metal, TI is a Bi-based three dimensional topological insulator (3D TI), and S is an s-type superconductor. In normal state, the devices are in the elastic diffusive transport regime, as demonstrated by a nearly universal value of the shot noise Fano factor $F_{rm N}approx1/3$ in magnetic field and in reference normal contact. In the absence of magnetic field, we identify the Andreev reflection (AR) regime, which gives rise to the effective charge doubling in shot noise measurements. Surprisingly, the Fano factor $F_{rm AR}approx0.22pm0.02$ is considerably reduced in the AR regime compared to $F_{rm N}$, in contrast to previous AR experiments in normal metals and semiconductors. We suggest that this effect is related to a finite thermal conduction of the proximized, superconducting TI owing to a residual density of states at low energies.
An effective random matrix theory description is developed for the universal gap fluctuations and the ensemble averaged density of states of chaotic Andreev billiards for finite Ehrenfest time. It yields a very good agreement with the numerical calculation for Sinai-Andreev billiards. A systematic linear decrease of the mean field gap with increasing Ehrenfest time $tau_E$ is observed but its derivative with respect to $tau_E$ is in between two competing theoretical predictions and close to that of the recent numerical calculations for Andreev map. The exponential tail of the density of states is interpreted semi-classically.
Using the non-equilibrium Green function method, we study the Andreev reflection in a Y-shaped graphene-superconductor device by tight-binding model. Considering both the zigzag and armchair terminals, we confirm that the zigzag terminals are the better choice for detecting the Andreev reflection without no external field. Due to scattering from the boundaries of the finite-size centre region, the difference between Andreev retroreflection and specular reflection is hard to be distinguished. Although adjusting the size of the device makes the difference visible, to distinguish them quantitatively is still impossible through the transport conductance. The problem is circumvented when applying a perpendicular magnetic field on the centre region, which makes the incident electrons and the reflected holes propagate along the edge or the interface. In this case, the retroreflected and specular reflected holes from the different bands have opposite effective masses, therefore the moving direction of one is opposite to the other. Which external terminal the reflected holes flow into depends entirely on the kind of the Andreev reflection. Therefore, the specular Andreev reflection can be clearly distinguished from the retroreflected one in the presence of strong magnetic field, even for the device with finite size.
I. Introduction (What is new in RMT, Superconducting quasiparticles, Experimental platforms) II. Topological superconductivity (Kitaev chain, Majorana operators, Majorana zero-modes, Phase transition beyond mean-field) III. Fundamental symmetries (Particle-hole symmetry, Majorana representation, Time-reversal and chiral symmetry) IV. Hamiltonian ensembles (The ten-fold way, Midgap spectral peak, Energy level repulsion) V. Scattering matrix ensembles (Fundamental symmetries, Chaotic scattering, Circular ensembles, Topological quantum numbers) VI. Electrical conduction (Majorana nanowire, Counting Majorana zero-modes, Conductance distribution, Weak antilocalization, Andreev resonances, Shot noise of Majorana edge modes) VII. Thermal conduction (Topological phase transitions, Super-universality, Heat transport by Majorana edge modes, Thermopower and time-delay matrix, Andreev billiard with chiral symmetry) VIII. Josephson junctions (Fermion parity switches, 4{pi}-periodic Josephson effect, Discrete vortices) IX. Conclusion
We report the study of ballistic transport in normal metal/graphene/superconductor junctions in edge-contact geometry. While in the normal state, we have observed Fabry-P{e}rot resonances suggesting that charge carriers travel ballistically, the superconducting state shows that the Andreev reflection at the graphene/superconductor interface is affected by these interferences. Our experimental results in the superconducting state have been analyzed and explained with a modified Octavio-Tinkham-Blonder-Klapwijk model taking into account the magnetic pair-breaking effects and the two different interface transparencies, textit{i.e.},between the normal metal and graphene, and between graphene and the superconductor. We show that the transparency of the normal metal/graphene interface strongly varies with doping at large scale, while it undergoes weaker changes at the graphene/superconductor interface. When a cavity is formed by the charge transfer occurring in the vicinity of the contacts, we see that the transmission probabilities follow the normal state conductance highlighting the interplay between the Andreev processes and the electronic interferometer.
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