### Abstract

We demonstrate that the exact quantum mechanical calculations are in good agreement with the semiclassical predictions for rectangular Andreev billiards, and therefore for a large number of open channels it is sufficient to investigate the Bohr-Sommerfeld approximation of the density of states. We present exact calculations of the classical path length distribution P(s), which is a nondifferentiable function of s, but whose integral is a smooth function with logarithmically dependent asymptotic behavior. Consequently, the density of states of rectangular Andreev billiards has two contributions on the scale of the Thouless energy: one which is well-known and is proportional to the energy, and the other which shows a logarithmic energy dependence. It is shown that the prefactors of both contributions depend on the geometry of the billiards but have universal limiting values when the width of the superconductor tends to zero.

Original language | English |
---|---|

Article number | 172506 |

Pages (from-to) | 1725061-1725064 |

Number of pages | 4 |

Journal | Physical Review B - Condensed Matter and Materials Physics |

Volume | 67 |

Issue number | 17 |

Publication status | Published - May 2003 |

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### ASJC Scopus subject areas

- Condensed Matter Physics

### Cite this

*Physical Review B - Condensed Matter and Materials Physics*,

*67*(17), 1725061-1725064. [172506].

**Logarithmic contribution to the density of states of rectangular Andreev billiards.** / Kormányos, A.; Kaufmann, Z.; Cserti, J.; Lambert, C. J.

Research output: Contribution to journal › Article

*Physical Review B - Condensed Matter and Materials Physics*, vol. 67, no. 17, 172506, pp. 1725061-1725064.

}

TY - JOUR

T1 - Logarithmic contribution to the density of states of rectangular Andreev billiards

AU - Kormányos, A.

AU - Kaufmann, Z.

AU - Cserti, J.

AU - Lambert, C. J.

PY - 2003/5

Y1 - 2003/5

N2 - We demonstrate that the exact quantum mechanical calculations are in good agreement with the semiclassical predictions for rectangular Andreev billiards, and therefore for a large number of open channels it is sufficient to investigate the Bohr-Sommerfeld approximation of the density of states. We present exact calculations of the classical path length distribution P(s), which is a nondifferentiable function of s, but whose integral is a smooth function with logarithmically dependent asymptotic behavior. Consequently, the density of states of rectangular Andreev billiards has two contributions on the scale of the Thouless energy: one which is well-known and is proportional to the energy, and the other which shows a logarithmic energy dependence. It is shown that the prefactors of both contributions depend on the geometry of the billiards but have universal limiting values when the width of the superconductor tends to zero.

AB - We demonstrate that the exact quantum mechanical calculations are in good agreement with the semiclassical predictions for rectangular Andreev billiards, and therefore for a large number of open channels it is sufficient to investigate the Bohr-Sommerfeld approximation of the density of states. We present exact calculations of the classical path length distribution P(s), which is a nondifferentiable function of s, but whose integral is a smooth function with logarithmically dependent asymptotic behavior. Consequently, the density of states of rectangular Andreev billiards has two contributions on the scale of the Thouless energy: one which is well-known and is proportional to the energy, and the other which shows a logarithmic energy dependence. It is shown that the prefactors of both contributions depend on the geometry of the billiards but have universal limiting values when the width of the superconductor tends to zero.

UR - http://www.scopus.com/inward/record.url?scp=0037561509&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037561509&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0037561509

VL - 67

SP - 1725061

EP - 1725064

JO - Physical Review B-Condensed Matter

JF - Physical Review B-Condensed Matter

SN - 0163-1829

IS - 17

M1 - 172506

ER -