Ballistic quantum transport at high energies and high magnetic fields

Rotter, Stefan; Weingartner, B.; Rohringer, Nina; Burgdörfer, Joachim

doi:10.1103/physrevb.68.165302

Cited by 49 publications

(48 citation statements)

References 66 publications

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“…[18] considered the Hall effect in a 2D cross (a specific code was developed to handle this geometry). Modular algorithms [19,20] allow to compute the properties of 2D quantum ballistic billiards. Most references with multiterminals involve direct inversions with small system sizes; others adaptation of the algorithm to the specific problem at hand [21,22].…”

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confidence: 99%

Knitting algorithm for calculating Green functions in quantum systems

2008

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We propose a fast and versatile algorithm to calculate local and transport properties such as conductance, shot noise, local density of state or local currents in mesoscopic quantum systems. Within the non equilibrium Green function formalism, we generalize the recursive Green function technique to tackle multiterminal devices with arbitrary geometries. We apply our method to analyze two recent experiments: an electronic Mach-Zehnder interferometer in a 2D gas and a Hall bar made of graphene nanoribbons in quantum Hall regime. In the latter case, we find that the Landau edge state pinned to the Dirac point gets diluted upon increasing carrier density.The field of quantum transport at the nanometer scale now includes a large number of systems involving very different physics. Examples include for instance mesoscopic devices in two dimensional heterostructures [1], graphene nanoribbons [2], superconducting weak links [3], molecular electronic devices [4] or ferromagnetic multilayers nanopillars [5,6]. Although those systems have different structures and geometries, they are all quantum systems connected to the macroscopic world through electrodes and, consequently, formalisms developed to describe one of them can often be adapted to the others. This is in particular the case of the widely used Landauer-Büttiker formalism [7] which focuses on the scattering properties of the system. The formalism is very intuitive and general. However, it is not well suited when one is interested in what happens inside the sample or for performing a microscopic calculation for a given device. An alternative mathematically equivalent approach is referred as the NEGF (non equilibrium Green function) formalism [8,9]. NEGF, which is derived from the Keldysh formalism [10], provides a simple route to compute the physical observables from a microscopic model. It is now an extremely popular numerical approach to a very wide class of physical problems (see references in [11]). For instance, all the references mentioned in examples above correspond to calculation done with this technique either from ab-initio or from phenomenological models [1,2,3,4,5,6,12,13]. At the core of NEGF is the calculation of the retarded Green function G of the mesoscopic region in presence of the (semiinfinite) electrodes. A straightforward method consists of direct inversion of the Hamiltonian H. However, when doing so, one is restricted to rather small systems of a few thousand sites: for a system of size L in dimension d, the computing time scales as L 3d while the needed memory scales as L2d . An alternative algorithm, known as the recursive Green function technique [1,5,14], takes advantage of the structure of H to reduce drastically the computing time down to L 3d−2 , putting systems of a few million sites within reach. In its original version [14,15,16], only the transport properties of the device could be computed, but recent progresses made it possible to get access to observables inside the sample (like local electronic density or local currents [1,17]...

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confidence: 99%

Knitting algorithm for calculating Green functions in quantum systems

2008

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“…While for particular cases general transport algorithms, such as the RGF algorithm, cannot compete with more specialized algorithms, such as the modular recursive Green's function technique [35,36] that is optimized for special geometries, they are very versatile and easily adapted to two-terminal geometries-provided that the leads are arranged collinearly. Amongst other things, this restriction will be lifted by the approach presented in this work.…”

Section: Introductionmentioning

confidence: 99%

Optimal block-tridiagonalization of matrices for coherent charge transport

Wimmer

Richter

2009

Journal of Computational Physics

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Numerical quantum transport calculations are commonly based on a tight-binding formulation. A wide class of quantum transport algorithms requires the tight-binding Hamiltonian to be in the form of a block-tridiagonal matrix. Here, we develop a matrix reordering algorithm based on graph partitioning techniques that yields the optimal block-tridiagonal form for quantum transport. The reordered Hamiltonian can lead to significant performance gains in transport calculations, and allows to apply conventional two-terminal algorithms to arbitrary complex geometries, including multi-terminal structures. The block-tridiagonalization algorithm can thus be the foundation for a generic quantum transport code, applicable to arbitrary tight-binding systems. We demonstrate the power of this approach by applying the block-tridiagonalization algorithm together with the recursive Green's function algorithm to various examples of mesoscopic transport in two-dimensional electron gases in semiconductors and graphene.

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“…Nonretracing electron-hole orbits leave their mark on the Andreev wave functions which we determine by solving the Bogoliubov-de Gennes equation with the help of the modular recursive Green's function method (MRGM). 28,29 We find that Andreev states which correspond to nonretracing orbits break the close correspondence between the electron and hole wave function patterns. The analysis of the eigenstates allows us to check the merits and limitations of the BS approximation against the exact quantum mechanical approach.…”

Section: Introductionmentioning

confidence: 99%

Non-retracing orbits in Andreev billiards

2006

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The validity of the retracing approximation in the semiclassical quantization of Andreev billiards is investigated. The exact energy spectrum and the eigenstates of normal-conducting, ballistic quantum dots in contact with a superconductor are calculated by solving the Bogoliubov-de Gennes equation and compared with the semiclassical Bohr-Sommerfeld quantization for periodic orbits which result from Andreev reflections. We find deviations that are due to the assumption of exact retracing electron-hole orbits rather than the semiclassical approximation, as a concurrently performed Einstein-Brillouin-Keller quantization demonstrates. We identify three different mechanisms producing non-retracing orbits which are directly identified through differences between electron and hole wave functions.

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Ballistic quantum transport at high energies and high magnetic fields

Cited by 49 publications

References 66 publications

Knitting algorithm for calculating Green functions in quantum systems

Knitting algorithm for calculating Green functions in quantum systems

Optimal block-tridiagonalization of matrices for coherent charge transport

Non-retracing orbits in Andreev billiards

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