Abstract
In this chapter we will address the actual determination of the propagator of a system, in terms of which all quantities of interest for coherent transport are derived. To maintain a high flexibility in setup variations, the numerical computation is performed on a tight-binding grid upon which arbitrary device confining potentials can be defined. After a brief review of relevant computational schemes, we introduce the matrix form of the discretized theory, and then develop a block-partitioning technique for computing transport as well as local density properties of multiterminal systems with arbitrary geometry and topology. The approach constitutes an extended version of the recursive Green function method based on the assembly of multiply connected structures from given inter- and intra-connected subsystems with multiple leads. It is combined with a block-reordered recursive computation of subsystem propagators, thus enabling the efficient investigation of a large diversity of system setups in a highly resolved parameter space.
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Notes
- 1.
Note that, for a system infinitely extended in both x- and y-directions, like the generic multiterminal scatterer attached to semi-infinite leads, the coordinates (s, r) can in general not be counted in a slice- or row-major scheme (e.g., bottom to top and then left to right) by a single site index α, since some slice or row may contain infinite sites. The only alternative would be a rather inconvenient outward spiral-like counting scheme for α. With the decomposition scheme used here, together with the tight-binding approximation to follow, only sites of the finite scatterer domain will be used in the description, such that a single-index counting is well defined.
- 2.
Bold upright letters (\(\boldsymbol{\mathsf{H}}\),\(\mathsf{\boldsymbol{\varPsi }}\)), possibly with sub- or superscripts, will be used to denote matrices represented on the spatial grid, with their thin variant (H α β ,\(\mathsf{\Psi }_{\alpha }\)) denoting individual matrix elements.
- 3.
On the grid, one can think of the boundary \(\partial \mathbb{D}\) as drawn in the middle between gridpoints (see inset of Fig. 5.1), so that hard-wall (Dirichlet) boundaries for lead p are implemented by setting ψ(x p , y p = −a 0∕2) = ψ(x p , y p = N w a 0 + a 0∕2) = 0 at the gridpoints just outside the lead; the effective width of the lead is thus w = (N w p + 1)a 0.
- 4.
Note that, in the tight-binding grid representation, and for a uniform grid, each matrix product is accompanied by a constant factor a 0 2, corresponding to the element of 2D spatial integration of matrix elements in the continuum limit. To simplify notation, we choose to absorb these constants in the corresponding multiplied matrices; for example, the symbol \(\boldsymbol{\mathsf{G}}\) will denote the grid-represented Green function multiplied by the surface element a 0 2.
- 5.
Such structures are called ‘antidots’ in the context of nanoelectronic systems, since they expel the electrons instead of trapping them like quantum dots do. If their (negative) potential is low, then their appearance may depend on the quasi-Fermi level in the 2DEG , leading to fluctuating Aharonov-Bohm-like loops, as discussed in Sect. 4.4.2 For strong and steep enough potential, the antidot can be modeled by a correspondingly shaped closed hard wall, with the enclosed sites discarded from the Hamiltonian matrix, as done in Figs. 5.1 and 5.2.
Fig. 5.2 
(a ) Toy device with 26 internal sites (blue) and 16 surface sites (red) of which 14 are connected to four terminals (yellow) and two represent decoherence probes. The scatterer also has a hole of four discarded sites. The legend on the right indicates the effective Hamiltonian on-site and hopping matrix elements describing the truncated scatterer, where lead sites (yellow) participate via the their Green function in the self-energy \(\mathsf{\boldsymbol{\varSigma }}\). (b ) Effective Hamiltonian matrix \(\tilde{\boldsymbol{\mathsf{H}}}\) (or equivalently \(\tilde{\mathsf{\boldsymbol{\varDelta }}} =\boldsymbol{ \mathsf{E}} -\tilde{\boldsymbol{\mathsf{H}}}\)) of the lead-connected scatterer with all sites (including red surface sites) indexed from left to right by slice numbers s = 1: 9 and from bottom to top by row number r = 1: N r (s) in each slice. Self-energies (on-site = red, nearest-neighbor coupling = orange, remote coupling = brown) contribute full diagonal blocks for horizontal leads (attached on the left or right) and scattered off-diagonal blocks for vertical leads (attached on the bottom or top). (c ) Block-reordered \(\tilde{\mathsf{\boldsymbol{\varDelta }}}\)-matrix by indexing first internal sites and then surface sites as numbered in (a ), with leads numbered as encountered from left to right and bottom to top. The reordered matrix has block-tridiagonal internal superblock \(\tilde{\mathsf{\boldsymbol{\varDelta }}}^{\iota \iota }\), block-diagonal surface superblock \(\tilde{\mathsf{\boldsymbol{\varDelta }}}^{\sigma \sigma }\) and surface-internal coupling \(\tilde{\mathsf{\boldsymbol{\varDelta }}}^{\iota \sigma } = [\tilde{\mathsf{\boldsymbol{\varDelta }}}^{\sigma \iota }]^{\dag }\). Indicated subblocks \(\tilde{\mathsf{\boldsymbol{\varDelta }}}_{s,s}^{\iota \iota }\), \(\tilde{\mathsf{\boldsymbol{\varDelta }}}_{s,s+1}^{\iota \iota }\) and \(\tilde{\mathsf{\boldsymbol{\varDelta }}}_{s}^{\iota \sigma } = [\tilde{\mathsf{\boldsymbol{\varDelta }}}_{s}^{\sigma \iota }]^{\dag }\) are used in sth iteration of the BGE process
- 6.
For example, if next-to-nearest-neighbor coupling were included (that is, via a higher order, nine-point stencil approximation to the 2D Laplacian ), the coupling matrix \(\boldsymbol{\uptau }\) would ‘reach’ further (by one more site in each direction) across the interfaces to the leads, but since \(\boldsymbol{\uptau }^{\dag }\) projects back onto the scatterer domain [see (4.84)], the matrix \(\mathsf{\boldsymbol{\varSigma }}\) remains of the size of \(\boldsymbol{\mathsf{H}}\).
- 7.
The coupling of the leads themselves to reservoirs is here implicit, with a corresponding imaginary term iη absorbed in \(\boldsymbol{\mathsf{H}}_{L}\) which makes \(\boldsymbol{\mathsf{g}}\) convergent, as shown in Sect. A.2 in Appendix A.
- 8.
Note that the \(\mathsf{\boldsymbol{\varPsi }}\) is determined from the effective Hamiltonian \(\tilde{\boldsymbol{\mathsf{H}}}\), and only the evaluation of the current at surface points is skipped here for simplicity, since they do not affect the current streamline pattern in the interior which is of interest. If leadpoints were added as Büttiker decoherence probes in the bulk of the scatterer, then the current should be evaluated at those sites as well, including the corresponding self-energy couplings on the links.
- 9.
The symbol diag( ) denotes (with a single matrix argument) the column vector of the diagonal elements or (with multiple arguments) the (block-) diagonal matrix with elements (matrices) on the diagonal.
- 10.
Note that, in contrast to the units used in the theory of previous chapters, here we do not set c = 1 for the speed of light. In fact, since [length] = a 0 and time = ℏ∕[E] = m eff a 0 2∕ℏ, velocity scales as a 0 −1.
- 11.
Note that this is in accordance with the Landauer-Büttiker framework for transport, on which the formulation of the scattering problem is based: Recall that semi-infinite leads merely represent an (ideal form of) electron reservoirs , which in turn correspond to electrodes attached to the transport device. Coupling (that is, hopping elements) between surface (lead-connected) sites of two different leads would, in the continuum limit a 0, correspond to a connection between the respective electrodes, in which case they would equilibrate (short-circuit) to the same chemical potential and effectively constitute a single attached electrode, to be modeled by a single semi-infinite lead.
- 12.
For simplicity, we consider only horizontal or vertical semi-infinite leads attached to the computational box containing the scatterer, for which the lead Greenians are easily evaluated. That surface sites connected to one lead are then y- or x-collinear, respectively, leading the tridiagonal \(\boldsymbol{\mathsf{H}}_{p}\). Leads at arbitrary angles can be implemented by ‘adiabatic bending’ into a horizontal or vertical lead by enlarging the computational box accordingly, as shown schematically (for very low grid resolution) in Fig. 5.1. With sufficiently smooth bending, the Fano resonance width of quasi-bound states in the bent wire becomes negligible (that is, affects the transmission profile of the system only at distinct points in energy). If the lattice Greenian for a tilted lead is known, then attaching the lead can be trivially implemented in the present scheme and would simply introduce zeros on the side-diagonals of \(\boldsymbol{\mathsf{H}}_{p}\).
- 13.
Note here that the proportionality factor is affected by the matrix additions, scaling as [N r (s)]2, but mostly by the fact that the offdiagonal blocks are usually not square.
- 14.
Note that conservation of flux implies T 21 11 + T 31 11 + R 1 = 1 in the first channel.
- 15.
The resolution in this transmission spectrum is not fine enough to resolve all Fano resonances, and the ones that are visible are also not resolved in full detail.
- 16.
We have here in fact also separated the lead stubs from the previous three-terminal module, which now serve as separate bridge modules, in order to be able to vary the bridge length without re-computing the ellipse module. This increases the number of inter-connections in the assembly, but does not practically affect the computation time.
- 17.
This is about twice the time ≈ 0. 75 s needed for each ellipse module (the original one and its mirror image) plus some additional time ≈ 0. 31 s for a 1024 gridpoints long magnetic field adaptation module intervening between the billiard and each attached lead.
- 18.
Note here that, due to the geometrical x and y mirror symmetry (or symmetry under an in-plane rotation through π) of the setup, transmission between leads 1 and 4 is symmetric in B, T 41(B) = T 41(−B). This is a consequence of the reciprocity relation T 41(−B) = T 14(B) and the fact that the symmetry operation exchanging leads 1 and 4 brings \(\boldsymbol{B} \parallel \hat{\boldsymbol{ z}}\) to itself.
- 19.
To have a qualitative picture of the effect of moderate to strong fields, recall that classical trajectories are deflected anticlockwise for B > 0 and clockwise for B < 0.
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Morfonios, C.V., Schmelcher, P. (2017). Computational Quantum Transport in Multiterminal and Multiply Connected Structures. 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_5
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