Time-delay matrix, midgap spectral peak, and thermopower of an Andreev billiard

We derive the statistics of the time-delay matrix (energy derivative of the scattering matrix) in an ensemble of superconducting quantum dots with chaotic scattering (Andreev billiards), coupled ballistically to $M$ conducting modes (electron-hole modes in a normal metal or Majorana edge modes in a superconductor). As a first application we calculate the density of states ${\rho }_0$ at the Fermi level. The ensemble average $\langle {\rho }_0\rangle ={\delta }_0^{-1}M{[max(0,M+2\alpha /\beta )]}^{-1}$ deviates from the bulk value $1/{\delta }_0$ by an amount depending on the Altland-Zirnbauer symmetry indices $\alpha ,\beta$. The divergent average for $M=1,2$ in symmetry class D ($\alpha =-1$, $\beta =1$) originates from the midgap spectral peak of a closed quantum dot, but now no longer depends on the presence or absence of a Majorana zero mode. As a second application we calculate the probability distribution of the thermopower, contrasting the difference for paired and unpaired Majorana edge modes.

Figure 1

Ensemble-averaged density of states (1) of an Andreev billiard in symmetry class C (${\rho }_{-}$, dashed curve), or class D without a Majorana zero mode (${\rho }_{+}$, solid curve). The class-D billiard with a Majorana zero mode has the smooth density of states ${\rho }_{-}$ together with the delta-function contribution from the zero mode. In this paper we investigate how the midgap spectral peak or dip evolves when the billiard is opened via a ballistic point contact to a metallic reservoir. We find that the distinction between classes C and D remains, but the signature of the Majorana zero mode is lost.

Figure 2

Andreev-billiard geometry to measure the thermopower $\mathcal{S}$ of a semiconductor quantum dot coupled to chiral Majorana modes at the edge of a topological superconductor. A temperature difference $\delta T$ induces a voltage difference $V=-\mathcal{S}\delta T$ under the condition that no electrical current flows between the contacts. For a random-matrix theory we assume that the Majorana modes are uniformly mixed with the modes in the point contact, by chaotic scattering events in the quantum dot.

Figure 3

Probability distributions of the Fermi-level density of states, for $M=1$ and $M=2$ modes coupling the quantum dot to the continuum, in symmetry classes C and D. The ensemble average diverges for class D; see Eq. (22).

Figure 4

Histograms: Probability distributions of the Fermi-level density of states in symmetry class D for $M=1$, $M_0=140,141$ and $M=2$, $M_0=200,201$, calculated numerically from Eq. (27) by averaging the Hamiltonian over the Gaussian ensemble. For each dimensionality $M$ of the scattering matrix we compare an even-dimensional Hamiltonian, without a Majorana zero mode, to an odd-dimensional Hamiltonian with a zero mode. The black curve is the analytical result (23) for the circular scattering matrix ensemble, predicting no effect from the Majorana zero mode for this case of ballistic coupling. Notice that there is no fit parameter in this comparison between numerics and analytics.

Figure 5

Probability distribution of the dimensionless thermopower $p=\mathcal{S}\times \hslash /t_0{\mathcal{S}}_0$ in symmetry class C (black solid curve, bottom and left axes), and in class D (blue dashed curve, top and right axes). These are results for the quantum dot of Fig. 2 connecting a single-channel point contact to the unpaired Majorana edge mode of a chiral $p$-wave superconductor (class D), or to the paired Majorana mode of a chiral $d$-wave superconductor (class C).

Figure 6

Geometry to detect a Majorana zero mode by a measurement of the Andreev conductance of a ballistic point contact to a superconducting quantum dot. The probability distribution of the conductance depends on the presence or absence of the Majorana zero mode, while the distribution of the density of states does not.