Maxim Vavilov: Quantum Disorder in Andreev Billiards

Maxim Vavilov discussed the effects of quantum disorder in Andreev
Billiards. These billiards consist of a small grain of normal state
material that is brought into contact with a superconducting
reservoir. These systems are realized, for example, by connecting a
quantum dot to a superconducting leads

Maxim first discussed the various energy scales that are relevant
for this problem. The largest energy scale is set by the (isotropic)
superconducting gap, Delta_sc, which implies perfect Andreev
reflection at the interface between the normal and superconducting
systems. The next smaller energy scale is set by the Thouless energy
E_T=hbar/tau_f where tau_f=L/v_F is the flight time of the
electrons, and L is the size of the normal state grain. Another
energy scale is set by E_g=hbar/tau_d where tau_d=tau_f*L/b is the
dwell time of the electrons, and b is the length of the interface
between the normal and superconducting systems. The last energy
scale is set by the mean level spacing, delta_I, of the normal state
system. The relative order of energy scales for the system that
Maxim studied is given by

Delta_sc >> E_T >> E_g >> delta_I

The objective of Maxim's work was to study the properties of the
electrons in the normal state grain, which are reflected in the
averaged density of states (DOS). Of particular interest is the
question of whether Andreev scattering off the interface leads to a
suppression of the normal state DOS at low energies. Maxim then
proceeded to outline a calculation using Random matrix theory (RMT) (see "Induced superconductivity distinguishes chaotic from integrable billiards", J. A. Melsen, P. W. Brouwer, K. M. Frahm, C. W. J. Beenakker Europhys. Lett. 35 (1996) 7) and a Gaussian Orthogonal Ensemble, which can be exactly solved in the
limit hbar/(tau_f * delta_I) -> 00. In this case, the DOS opens up a
hard gap at low energies up to an energy scale set by E_g, and
increases as DOS ~ sqrt(w - E_g) for energies w>E_g. At this point
Daniel Sheehy asked whether this result is achieved by averaging
over ensembles. Maxim answered that in the case he considered,
averaging over ensembles is equal to averaging over many energy
levels. Hence the RMT result should be valid for the average DOS of a
single normal grain.

Maxim then proceeded to outline a different calculation based on the
Eilenberger equations developed with Anatoly Larkin ("Quantum Disorder and Quantum Chaos in Andreev Billiards", M.G. Vavilov, A.I. Larkin, Phys. Rev. B 67, 115335 (2003)). This approach corresponds to the semiclassical approximation only if impurity scattering is not taken
into account. Without disorder, this approach yields an averaged DOS
in the normal grain that is suppressed at low energies (below E_g),
but does not show a hard gap, in contrast to the results of the
random matrix theory. Finally, Maxim considered the effects of
disorder, as realized by a distribution of short range impurities.
In the limit of strong disorder, when the scattering time is
comparable with the dwell time, the Eilenberger approach recovers
the RMT result, and a hard gap opens in the DOS up to a frequency of
E_g. However, even in the case of weak disorder, a gap opens in the
DOS.

Daniel Sheehy asked whether the Andreev reflection at the interface
is perfect. Maxim answered that this is the case as long as the
superconducting gap is the largest energy scale in the problem, and
in particular, as long as Delta_sc >> E_T.

Andrey Chubukov asked whether this averaged DOS can be measured
experimentally. Maxim pointed out that in general, it can be
measured by studying quantum dots connected to superconducting
leads. However, the main experimental problem seems to be the
interface between the superconducting and normal state materials.
Finally, Maxim remarked that while his theory was developed for
two-dimensional grains, the effect might be more easily observable
in three dimensional systems.