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Stationary Scattering in Planar Confining Geometries

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Control of Magnetotransport in Quantum Billiards

Part of the book series: Lecture Notes in Physics ((LNP,volume 927))

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Abstract

In the Landauer-Büttiker formalism developed previously, the multiterminal transmission function of a mesoscopic device constitutes the core of the description of coherent electron transport. In this chapter it will be seen how the asymptotic scattering matrix of the system as well as spatially resolved quantities of interest such as the full scattering wave function can be formally determined and practically calculated from the system Hamiltonian. This is achieved within the Green function formalism in terms of an effective, energy-dependent and non-Hermitian Hamiltonian describing the scattering region connected to the peripheral leads. The theoretical framework is reviewed from the particular viewpoint of (planar) confinement with generic, geometrically defined asymptotic scattering channels, highlighting the involved concepts and the main observable interference effects in transmission, Fano resonances and Aharonov-Bohm oscillations.

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Notes

  1. 1.

    The subband energy of the vertical z-confinement is here included in the potential, and the energy E is thus the total energy of the in-plane motion, that is, offset to the ground level ε 0 of heterojunction well.

  2. 2.

    In order to avoid a multitude of namings for \(\mathcal{G}\), since it is encountered as the Green operator acting on states, as the Green function in spatial representation, or even as the Green (function) matrix in the discretized (spatial) representation to follow, we will use the relatively unconventional though unifying term ‘Greenian’, just as is commonly done for the Hamiltonian.

  3. 3.

    The symbol \(\sum \!\!\!\!\!\!\!\int _{n,k}\) is here used explicitly to denote the summation over discrete eigenstates n together with the integration over continuous (momentum) eigenvalues k. For notational simplicity, we also use 1 throughout to denote the identity operator.

  4. 4.

    Although the adaptation region for the magnetic field attenuation is usually implemented in practice to describe a homogeneous applied field, it is by far no extraordinary challenge for current technology to produce local magnetic field gradients, even at the nanoscale [17]. This can be achieved, e.g., by fabrication of hybrid structures with superconducting nanopatterned components [18] which expel the magnetic field locally, or by using ferromagnetic microstructures which produce field variation patterns [19, 20] with large field gradients.

  5. 5.

    This approximation was employed by Born in the same paper [29] where, in a footnote, the interpretation of the squared absolute wave function amplitude as a probability (density) was originally proposed.

  6. 6.

    In a derivation of the Fisher-Lee formula based on the scattering formalism of the present chapter, which will not be repeated here, flux normalized asymptotic lead states can be used throughout: Since the scattering state is initially expressed in terms of the spatially represented Greenian [13, 14] and then projected on the lead eigenstates to obtain the S-matrix (and not vice versa), the lead indices present in the corresponding fluxes enter symmetrically among the leads. In the (perhaps more intuitive) derivation presented in Sect. 4.2.2, the selective evaluation of the Green function from a selected lead n carries this index via the flux normalization in this lead, which would replace \(\sqrt{v_{n }^{(q) }v_{m }^{(\,p)}}\) with v n (q) in (4.47). To avoid this technical issue, the rescaled (in general non-unitary) S-matrix equation (4.43) has to be used in (4.42).

  7. 7.

    On the other hand, if the total confining potential is included in \(\mathcal{V}\), as was done in Chap. 3, δ β α would have no meaning since no leads would exist for \(\mathcal{V} = 0\).

  8. 8.

    Treating all leads on an equal footing, we have dropped here the index q of an individual lead, and describe all leads collectively by \(\mathcal{H}_{L}\).

  9. 9.

    Moreover, one of the reasons for maintaining a representation-independent description here is to separate the notion of the effective scatterer propagator from its discrete spatial (matrix) representation to be used in the next chapter. Using the more common [3, 6] matrix formulation from the beginning might suggest that its validity is subject to the approximative tight-binding approach, which is not the case.

  10. 10.

    In the considered 2D system the charge density naturally has units [area]−1 (integrated over some area it gives the average number of enclosed electrons), though it is occasionally explicitly multiplied also with electronic charge e (in order to give the enclosed charge when integrated).

  11. 11.

    \(\mathcal{G}_{ij}^{<}(t,t')\) and \(\mathcal{G}_{ij}^{>}(t,t')\) are the so-called lesser and greater Green functions, respectively, defined as the corresponding expectation values of products of creation and annihilation operators (that is, as correlation functions) in the second quantization picture for many-body systems. They are related, owing to the fluctuation-dissipation theorem, to the retarded and advanced Green functions through the relation

    $$\displaystyle{ \mathcal{G}^{>} -\mathcal{G}^{<} = \mathcal{G}^{+} -\mathcal{G}^{-}, }$$
    (4.122)

    which connects the occupancy of states (represented by \(\mathcal{G}^{\gtrless }\)) to their spectral features (represented by \(\mathcal{G}^{\pm }\)).

  12. 12.

    Note that the symbol \(d\boldsymbol{r}\) is used simply as a shorthand for the 2D volume element of integration \(d\boldsymbol{r} \equiv dxdy\) (and not to denote an infinitesimal vector); ds is the 1D surface element on \(\partial \mathbb{D}_{S}\) with outward normal unit vector \(\hat{\boldsymbol{n}}_{\partial \mathbb{D}_{S}}\) coinciding with the direction \(\hat{\boldsymbol{x}}_{p}\) in each (straight) lead p.

  13. 13.

    We here use \(\frac{d} {dt}\mathinner{\vert \psi \rangle }\mathinner{\langle \psi \vert } = ( \frac{d} {dt}\mathinner{\vert \psi \rangle })\mathinner{\langle \psi \vert } +\mathinner{ \vert \psi \rangle }( \frac{d} {dt}\mathinner{\langle \psi \vert })\) and the fact that the Hamiltonian of the isolated scatterer is hermitian.

  14. 14.

    Equation (4.142) is often derived directly from the Fisher-Lee formula  (4.49) by considering explicit self-energy matrices in a specific representation, usually within a certain approximation scheme such as the tight-binding approach which we will employ in practice in the following chapters.

  15. 15.

    The type of coupling between states in different channels is specific to the system considered; Fano used the general notion of (many-body) configuration interaction between the quasi-bound state and the continuum in the context of atomic auto-ionization [61]. In the simplest case, the coupling amounts to the (real-space) projection between the state wave functions via the (single-particle effective) Hamiltonian in the reaction region of the scattering potential [64–66].

  16. 16.

    Fano originally [67, 68] considered a single channel and neglected the energy shift of the quasi-bound state, and treated the subject later more rigorously [61]. A generalization to multiple channels and overlapping resonances was provided by Feshbach [69] in the context of nuclear reaction theory, using a projection scheme as here in Sect. 4.3.1 but in state space (into closed and open channels) instead of configuration space. The subject of discrete states coupled to the continuum by configuration interaction was firstly treated, however, by Majorana in 1931 [70–72].

  17. 17.

    The normalization is chosen such that \(\vert \psi _{1}(\boldsymbol{r}_{\text{in}}) +\psi _{2}(\boldsymbol{r}_{\text{in}})\vert ^{2} = 1\) at the entrance point \(\boldsymbol{r}_{\text{in}}\) of the loop.

  18. 18.

    Note that this is the ‘first order’ AB effect for the loop system considered: There will in principle be contributions from any number of windings of paths around the loop, including half windings if reflection at the nodes is included [2]. As a result, higher frequencies appear in the AB oscillations; these are, however, suppressed in experimental spectra due to the finite electronic coherence length.

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    Christian V. Morfonios & Peter Schmelcher

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Morfonios, C.V., Schmelcher, P. (2017). Stationary Scattering in Planar Confining Geometries. In: Control of Magnetotransport in Quantum Billiards. Lecture Notes in Physics, vol 927. Springer, Cham. https://doi.org/10.1007/978-3-319-39833-4_4

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