Green’s function technique for studying electron flow in two-dimensional mesoscopic samples

Abstract

In a recent series of scanning probe experiments, it became possible to visualize local electron flow in a two-dimensional electron gas. In this paper, a Green's function technique is presented that enables efficient calculation of the quantity measured in such experiments. Efficient means that the computational effort scales like M 3 N (M is the width of the tight-binding lattice used, and N is its length), which is a factor M N better than the standard recursive technique for the same problem. Moreover, within our numerical framework it is also possible to calculate (with the same computational effort M 3 N ) the local density of states, the electron density, and the current distribution in the sample, which are not accessible with the standard recursive method. Furthermore, an imaging method is discussed where the scanning tip can be used to measure the local chemical potential. The numerical technique is used to study electron flow through a quantum point contact. All features seen in experiments on this system are reproduced and a new interference effect is observed resulting from the crossing of coherent beams of electron flow.

Exactly solvable topological chiral spin liquid with random exchange

“…We computed σ th xy numerically for specific disorder realizations and then averaged over 100 realizations. Since (2) with (3) is exactly diagonalizable, the conductivity can be determined within the Keldysh Green's function formalism which has been successfully applied to mesoscopic transport in quantum wires 11,17,18 . We implemented this technique by coupling our disordered system to two ideal "leads" which are modeled as clean versions of the system with j ′ = j such that chiral edge modes are supported in the leads.…”

“…To linear order in T , σ th xy is given by a Chester-Thellung-Kubo-Greenwood formula 11,[16][17][18] ,…”

“…3 for large W disorder realizations in the non-trivial topological phase ν = ±1 by adapting the methods of Ref. 18 . The I th ij are determined from the non-equilibrium Green's function G < in the linear response regime of small T .…”

Local density of states in mesoscopic samples from scanning gate microscopy

“…15 From a theoretical point of view, numerical simulations based on the Keldysh-Green's functions formalism have been exploited to demonstrate wavefunction imaging in open quantum dots 16,17 and to associate conductance images to electron flow in quantum point contacts. 18,19 Recently, isophase lines for electrons in an electrostatic Aharonov-Bohm (AB) experiment 20 and local-densityof-states (LDOS) mapping inside a coherent AB ring 21 have been reported. In the latter work, both experimental curves and numerical simulations have found a linear dependence of the conductance variation on the tip voltage.…”

Scanning-gate-induced effects in nonlinear transport through nanostructures

“…Even in the linear regime, the interpretation of the resulting scans is delicate [21,22]. On one hand, from various experimental and theoretical works focused on a QPC probed by a strongly charged tip, the conductance change appears to be closely related to the local current density [15,16,23,24]. On the other, it has been shown [22] that only under quite restrictive conditions (a spatially symmetric QPC tuned to a conductance plateau) the tip-induced conductance change is directly related to the current density at the tip position.…”