Knitting algorithm for calculating Green functions in quantum systems

K. Kazymyrenko and X. Waintal

Phys. Rev. B 77, 115119 – Published 17 March 2008

Abstract

We propose a fast and versatile algorithm to calculate local and transport properties such as conductance, shot noise, local density of state, and local currents in mesoscopic quantum systems. Within the nonequilibrium 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 two-dimensional gas and a Hall bar made of graphene nanoribbons in a 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.

Received 19 February 2008

DOI:https://doi.org/10.1103/PhysRevB.77.115119

Authors & Affiliations

K. Kazymyrenko and X. Waintal

Service de Physique de l’Etat Condensé, DSM/DRECAM/SPEC, CEA Saclay, 91191 Gif Sur Yvette Cedex, France

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Vol. 77, Iss. 11 — 15 March 2008

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Images

Figure 1
(Color online) The system scheme and notations. The sites are labeled according to the order in which they are added to the system. The boxes define the three electrodes coupled to the mesoscopic system. The various letters stand for the added site $(A)$ , electrode sites ( $α$ , $β$ , and $γ$ ), connected sites ( $σ_{1}$ and $σ_{2}$ ), and interface sites ( $i$ and $j$ ), as discussed in the text. Bold circles indicate sites whose GF elements are updated at the current knitting step. The thick dashed line separates the part already included (left) from the part that is still to be knitted to the system (right).Reuse & Permissions
Figure 2
(Color online) Mach–Zehnder interferometer in a 2D gas modeled by a scalar tight-binding model. The mobility is $μ = 5 \times 10^{5} {cm}^{2} ∕ V s$ , the electron density is $n_{s} = 10^{10} {cm}^{- 2}$ , and the magnetic field is $B = 0.2 T$ . (a) Local current intensity when a bias voltage is applied to lead 0 and the other contacts are grounded ( $1.2 \times 10^{6}$ sites; blue corresponds to no current and red to maximum current). (b) Differential conductance $d I_{3} ∕ d V_{0}$ as a function of the number of flux quanta $ϕ$ through the hole. (c) Visibility coefficients $V_{1}$ and $V_{2}$ (see text) as a function of the electron mobility $μ$ .Reuse & Permissions
Figure 3
(Color online) Same as Fig. 2a for two dirtier samples with mobility $μ = 3 \times 10^{5} {cm}^{2} ∕ V s$ (a) and $μ = 2.5 \times 10^{5} {cm}^{2} ∕ V s$ (b), electron density $n_{s} = 10^{10} {cm}^{- 2}$ , and magnetic field $B = 0.2 T$ . Zoom of (a) shows the current that escapes the first contact and makes the second loop. It is responsible for the presence of a second visibility harmonic $V_{1} = 1$ $V_{2} = 0.1$ . Similar zoom of (b) shows backscattered current, which is recovered by contact 2. $V_{1} = 0.99$ and $V_{2} = 0.01$ .Reuse & Permissions
Figure 4
(Color online) Quantum Hall effect in graphene. Hall conductance $σ_{x y}$ and longitudinal resistance $ρ_{x x}$ are plotted as a function of inverse magnetic field $1 ∕ Φ$ (a) and carrier density $n$ (b) in the presence of a small disordered potential (10% of the hopping matrix elements). In (a) the carrier density is $0.18 e$ /hexagon, while in (b) a magnetic flux $Φ = 0.014 h ∕ e$ per hexagon is applied. The inset of (b) shows a zoom of the transition between plateaus. (c) shows the local current intensity when current is injected from both contacts 1 and 4 ( $3 \times 10^{5}$ hexagons, $N = 2$ ).Reuse & Permissions
Figure 5
(Color online) Comparison of the edge channels for a graphene (a) and a semiconductor heterostructure (b) two-dimensional electron gases. The plots show the local current density in the $y$ direction $d I_{y} (x) ∕ d V$ as a function of the distance $x$ from the border of the sample for four consecutive Hall plateaus corresponding to $N = 0, 1, 2, 3$ for graphene (a) and $n = 1, 2, 3, 4$ for the heterostructure (b). All graphs are translated for clarity from bottom to top. The magnetic field is $B = 14 T$ .Reuse & Permissions

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