Wigner transport equation with finite coherence length

Abstract

The use of the Wigner function for the study of quantum transport in open systems is subject to severe criticisms. Some of the problems arise from the assumption of infinite coherence length of the electron dynamics outside the system of interest. In the present work the theory of the Wigner function is revised assuming a finite coherence length. A new dynamical equation is found, corresponding to move the Wigner momentum off the real axis, and a numerical analysis is performed for the case of study of the one-dimensional potential barrier. In quantum device simulations, for a sufficiently long coherence length, the new formulation does not modify the physics in any finite region of interest but it prevents mathematical divergence problems.

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Fig. 1

Notes

  1. 1.

    The Journal of Optics B has published a Wigner Centennial issue (J. Opt. B 5(3) (2003)), where many references can be found.

  2. 2.

    An alternative derivation seems to avoid this problem (private communication by F. Rossi and by L. Demeio), but in this case we need to assume the existence of the Fourier transform and of its anti-transform in a generalised sense, and this is not always rigorously justified.

  3. 3.

    This result must be compared with a previous paper [18], where different conclusions are reached.

  4. 4.

    For numerical purposes a finite coherence length has been sometimes introduced in the potential term of the WF, see e.g. [30, 31].

  5. 5.

    In the derivation of Eq. (11) the damping factor e −|s|/λ has been attributed to the WF to obtain an expression in terms of f wc . With an equivalent procedure the damping factor could be attributed to the potential difference [V(+)−V(−)], thus leading to a damped potential \(\mathcal{V}_{wc}\). In other words, it can be proved that: \(\int\mathcal{V}_{w} f_{wc} dp'=\int\mathcal{V}_{wc} f_{w} dp'\). This means that the potential term in (11) does not contribute to ∂f wc /∂t in points x far from the region where the potential V(x) is not constant (e.g. for a potential step). A close analysis of the integral in the form given in Eq. (11) shows that the integrand in such points has negligible contributions because of the very fast oscillations in \(\mathcal{V}_{w}\).

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Acknowledgements

The authors are extremely grateful to Fausto Rossi, Lucio Demeio and Alberto Guandalini for stimulating and fruitful discussions.

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Correspondence to Paolo Bordone.

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Jacoboni, C., Bordone, P. Wigner transport equation with finite coherence length. J Comput Electron 13, 257–263 (2014). https://doi.org/10.1007/s10825-013-0510-7

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Keywords

  • Wigner equation
  • Quantum transport
  • Open systems