Effect of a tunnel barrier on the scattering from a Majorana bound state in an Andreev billiard

•Scattering and time-delay matrices j.p.d.f. of certain random quantum dots is found.•Heat current do not probe unpaired Majorana states if one mode is ballistic coupled.•Ballistic coupling may not lift unpaired Majorana states dependence in dI /dV curve.•The Majorana phase transition can be probed in nanowire-random quantum dot geometry.

We calculate the joint distribution P(S,Q)P(S,Q) of the scattering matrix S   and time-delay matrix Q=−iℏS†dS/dEQ=−iℏS†dS/dE of a chaotic quantum dot coupled by point contacts to metal electrodes. While S and Q are statistically independent for ballistic coupling, they become correlated for tunnel coupling. We relate the ensemble averages of Q and S and thereby obtain the average density of states at the Fermi level. We apply this to a calculation of the effect of a tunnel barrier on the Majorana resonance in a topological superconductor. We find that the presence of a Majorana bound state is hidden in the density of states and in the thermal conductance if even a single scattering channel has unit tunnel probability. The electrical conductance remains sensitive to the appearance of a Majorana bound state, and we calculate the variation of the average conductance through a topological phase transition.