Theory of ballistic quantum transport in the presence of localized defects

We present an efficient numerical approach for treating ballistic quantum transport across devices described by tight-binding (TB) Hamiltonians designated to systems with localized potential defects. The method is based on the wave function matching approach, Lippmann-Schwinger equation (LEQ), and the scattering matrix formalism. We show that the number of matrix elements of the Green's function to be evaluated for the unperturbed system can be essentially reduced by projection of the time reversed scattering wave functions on LEQ which radically improves the speed and lowers the memory consumption of the calculations. Our approach can be applied to quantum devices of an arbitrary geometry and any number of degrees of freedom or leads attached. We provide a couple of examples of possible applications of the theory, including current equilibration at the $p\text{−}n$ junction in graphene and scanning gate microscopy mapping of electron trajectories in the magnetic focusing experiment on a graphene ribbon. Additionally, we provide a simple toy example of electron transport through 1D wire with added onsite perturbation and obtain a simple formula for conductance showing that Green's function of the device can be obtained from the conductance versus impurity strength characteristics.

Figure 1

(a) A sketch of a quantum scatterer described by ${\mathbf{H}}_0$ coupled to the three semi-infinite leads by self-energy matrices ${\mathrm{\Sigma }}_l$. The electron comes from the source represented by ${\mathrm{\Gamma }}_l$ source vector. The arrows point the possible direction of the scattering electron for this specific example. (b) Block tridiagonal partitioning of the Hamiltonian near the leads from which ${\mathrm{\Sigma }}_l$ and ${\mathrm{\Gamma }}_l$ can be computed. The green area denotes the first slice which belongs to the isolated system—the semi-infinite lead and quantum device interface atoms.

Figure 2

(a) Schematics of two uncoupled infinite channels horizontal $A$ and vertical $B$. (b) System $B$ is glued to system $A$ with coupling matrix ${\mathbf{V}}_{AB}$ which removes the self-energy term from the Hamiltonian $B$ and connects proper sites of both systems.

Figure 3

(a) Schematic picture of two uncoupled infinite channels describing different materials. (b) After gluing with proper coupling matrix the systems form a quantum junction.

Figure 4

(a) Sketch of two infinite systems which define $p$ and $n$ parts of the graphene $p\text{−}n$ junction. Two systems are then glued with coupling matrix ${\mathbf{V}}_{\mathrm{pn}}$. (b) The $p\text{−}n$ junction after gluing process. The green areas denote the lead unit cells.

Figure 5

(a) Averaged conductance as a function of Fermi level energy and potential energy ${\mathbf{V}}_{\mathrm{LG}}$ in the $p$ region obtained for $N_{\mathrm{samp}}=10000$ random configuration of on-site disorder at $p\text{−}n$ junction interface. (b) Same as (a) but for clean junction (no averaging). (c) Analytical prediction adapted from Ref. [27].

Figure 6

The cross sections along vertical (a) and horizontal (b) lines in Fig. 5. The black dashed lines on each plot show the analytical prediction for fully equilibrated currents [Fig. 5]. Thick color lines correspond to averaged conductance from Fig. 5. Dashed color lines correspond to the conductance of clean $p\text{−}n$ junction from Fig. 5.

Figure 7

The sketch of the magnetic focusing device. The large reservoir $A$ is built from 191 000 atoms and is connected with two smaller vertical leads $L_{1/2}$ with coupling matrices ${\mathbf{V}}_{m{L}_{1/2},\mathrm{A}}$ separated by distance $d=320$ Å. The blue disk shows the SGM tip influence radius. Only the atoms below the blue area are affected by the SGM potential.

Figure 8

Conductance between vertical leads as a function of magnetic field amplitude. The insets indicate the trajectories behind the conductance peaks.

Figure 9

(a) The scattering electron density for electron incoming from lead ${\mathbf{L}}_1$ at $B=27\mathrm{T}$. The black dashed line shows the SGM scan area. (b) Simulated SGM image for the case from (a). Dashed arc corresponds to the classical cyclotron orbit. (c),(d) same as (a),(b) but obtained for $B=40\mathrm{T}$. The electron densities of (a),(c) are calculated from direct solution of the scattering problem and the SGM map with the method introduced in this paper.