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Journal of Physics Condensed Matter

Analysis of electron transport in a two-dimensional structure using quantal trajectories

Tomas Lundberg, Erik Sjöqvist, Karl‐Fredrik Berggren

Published Jun 29, 1998• 8 citations
Publisher

Abstract

Ballistic and dissipative electron transport through a two-dimensional geometry is studied in the de Broglie-Bohm quantal trajectory model. The dissipative effect, incorporated to simulate inelastic scattering, is introduced via an imaginary potential term in the Hamiltonian. The relation between the conductance and the local behaviour of the quantal trajectories is discussed. The vortex-like structure of the de Broglie-Bohm trajectories in the vicinity of wavefunction nodes is studied.

DOI
10.1088/0953-8984/10/25/010

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Ballistic and dissipative electron transport through a two-dimensional geometry is studied in the de Broglie-Bohm quantal trajectory model. The dissipative effect, incorporated to simulate inelastic scattering, is introduced via an imaginary potential term in the Hamiltonian. The relation between the conductance and the local behaviour of the quantal trajectories is discussed. The vortex-like structure of the de Broglie-Bohm trajectories in the vicinity of wavefunction nodes is studied.
We have previously shown that the flow in open quantum dots may be described in terms of meandering open orbits which separate regions of trapped flows around vortices. Furthermore the topology of the flow follows from a simple geometrical construction in which the classical ballistic trajectories are first computed and then all crossings are replaced by hyperbolic flow separation and for which all trapped flows form quantised closed orbits. In the present paper we extend this picture to a computational methodology for developing quantum hydrodynamic descriptions of flows in the presence of quantised vortices by extension from classical hydrodynamics. For coherent transport we again encounter the Bohm quantum potential but the singularity structure is predetermined by applying a fictitious repulsion which prevents trajectories and hence streamlines crossing. This leads to vortex producing terms in the hydrodynamic equations. The discussion is completed by including the effects of non-hermitian potentials into the Hamiltonian in order to simulate dissipation and phase breaking flows.
We investigate the phenomenon of the diffraction of charged particles by thin material targets using the method of the de Broglie–Bohm quantum trajectories. The particle wave function can be modeled as a sum of two terms ψ = ψingoing+ ψoutgoing. A thin separator exists between the domains of prevalence of the ingoing and outgoing wavefunction terms. The structure of the quantum-mechanical currents in the neighborhood of the separator implies the formation of an array of quantum vortices. The flow structure around each vortex displays a characteristic pattern called "nodal point–X point complex". The X point gives rise to stable and unstable manifolds. We find the scaling laws characterizing a nodal point–X point complex by a local perturbation theory around the nodal point. We then analyze the dynamical role of vortices in the emergence of the diffraction pattern. In particular, we demonstrate the abrupt deflections, along the direction of the unstable manifold, of the quantum trajectories approaching an X-point along its stable manifold. Theoretical results are compared to numerical simulations of quantum trajectories. We finally calculate the times of flight of particles following quantum trajectories from the source to detectors placed at various scattering angles θ, and thereby propose an experimental test of the de Broglie–Bohm formalism.
Within the de Broglie-Bohm theory, we numerically study a generic two-dimensional anharmonic oscillator including cubic and quartic interactions in addition to a bilinear coupling term. Our analysis of the quantum velocity fields and trajectories reveals the emergence of dynamical vortices. In their vicinity, fingerprints of chaotic behavior such as unpredictability and sensitivity to initial conditions are detected. The simultaneous presence of the off-diagonal -kxy and nonlinear terms leads to robust quantum chaos very analogous to its classical version.
Despite the dynamical stability and absence of chaos in quantum wave functions, it is demonstrated that the quantal trajectories of the de Broglie--Bohm quantum theory of motion are capable of showing linear and exponential divergences of neighboring particle orbits for a general nonintegrable system. A quantal trajectory is influenced by the whole phase space and can be associated with a set of Lyapunov-like exponents which separate time scales of different stability \textcopyright{} 1996 The American Physical Society.
Electronic transport in Fermi liquids is usually Ohmic, because of momentum-relaxing scattering due to defects and phonons. These processes can become sufficiently weak in two-dimensional materials, giving rise to either ballistic or hydrodynamic transport, depending on the strength of electron-electron scattering. We show that the ballistic regime is a quantum critical point (QCP) on the regime boundary separating Ohmic and hydrodynamic transport. The QCP corresponds to a \emph{free} conformal field theory (CFT) with a dynamical scaling exponent z=1z = 1z=1. Its nontrivial aspects emerge in device geometries with shear, wherein the regime has an intrinsic universal dissipation, a nonlocal current-voltage relation, and exhibits the critical scaling of the underlying CFT. The Fermi surface has electron-hole pockets across all angular scales and the current flow has \emph{vortices} at all spatial scales. We image the fluctuations in high-definition and animate their emergence as experimental parameters are tuned to the QCP (movie links in comments). The vortices clearly demonstrate that Pauli exclusion alone can produce collective effects, with low-frequency AC transport mediated by vortex dynamics. The scale-invariant spatial structure is much richer than that of an interaction-dominated hydrodynamic regime, which only has a single vortex at the device scale. Our findings provide a theoretical framework for both interaction-free and interaction-dominated non-Ohmic transport in two-dimensional materials, as seen in several contemporary experiments.