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.

## Friday, August 24, 2007

### Maxim Vavilov: Quantum Disorder in Andreev Billiards

Posted by Dirk Morr at 11:25 AM

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## 1 comment:

I wonder why I did not knew this before.... it was a good read. You are working on a very good blog.. I look forward to visit your blog again....

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