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Discrete versus continuous wires on quantum networks.

Amnon Aharony1, Ora Entin-Wohlman

  • 1Department of Physics, Ben Gurion University, Beer Sheva, Israel. aaharony@bgu.ac.il

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|August 13, 2009
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Summary
This summary is machine-generated.

Comparing two methods for modeling quantum networks, this study finds that the continuum and tight-binding approaches yield different results. They only agree under specific conditions, even when the tight-binding lattice constant approaches zero.

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Area of Science:

  • Quantum physics
  • Mesoscopic systems
  • Materials science

Background:

  • Mesoscopic systems and large molecules are frequently modeled as graphs of interconnected 1D wires.
  • Solutions to the Schrödinger equation on these graphs, termed "quantum networks," are crucial for analyzing electron energy spectra and transmission in finite systems.
  • Understanding electron behavior in quantum networks is vital for developing novel electronic devices.

Purpose of the Study:

  • To compare the accuracy and applicability of two common methods for solving the Schrödinger equation on quantum networks: the continuum approach and the tight-binding model.
  • To identify the conditions under which these two approaches yield consistent results.
  • To demonstrate the practical implications of discrepancies between the two models using a T-shaped scatterer example.

Main Methods:

  • The study employs the continuum approach, solving the 1D Schrödinger equation on wires with Neumann-Kirchoff-de Gennes matching conditions at vertices.
  • It also utilizes the tight-binding model, discretizing wires into a finite number of atoms.
  • Both methods are applied to analyze electron behavior in quantum networks, with a focus on comparing their outcomes.

Main Results:

  • The continuum and tight-binding approaches do not generally produce identical results for quantum networks.
  • Discrepancies persist even in the limit where the tight-binding model's lattice constant approaches zero.
  • The two methods only coincide when specific, highly constrained relationships exist between the tight-binding parameters.

Conclusions:

  • The choice between the continuum and tight-binding models for quantum networks is critical and depends on the specific system and desired accuracy.
  • The study highlights the limitations of the tight-binding model in accurately representing quantum networks without careful parameter selection.
  • Further research is needed to reconcile the differences and establish precise conditions for the equivalence of these modeling techniques.