S. Bujkiewicz

Chaotic electron diffusion through stochastic webs enhances current flow in superlattices.

Understanding how complex systems respond to change is of fundamental importance in the natural sciences. There is particular interest in systems whose classical newtonian motion becomes chaotic as an applied perturbation grows. The transition to chaos usually occurs by the gradual destruction of stable orbits in parameter space, in accordance with the Kolmogorov–Arnold–Moser (KAM) theorem—a cornerstone of nonlinear dynamics that explains, for example, gaps in the asteroid belt. By contrast, ‘non-KAM’ chaos switches on and off abruptly at critical values of the perturbation frequency. This type of dynamics has wide-ranging implications in the theory of plasma physics, tokamak fusion, turbulence, ion traps, and quasicrystals. Here we realize non-KAM chaos experimentally by exploiting the quantum properties of electrons in the periodic potential of a semiconductor superlattice with an applied voltage and magnetic field. The onset of chaos at discrete voltages is observed as a large increase in the current flow due to the creation of unbound electron orbits, which propagate through intricate web patterns in phase space. Non-KAM chaos therefore provides a mechanism for controlling the electrical conductivity of a condensed matter device: its extreme sensitivity could find applications in quantum electronics and photonics.

Using Stochastic Webs to Control the Quantum Transport of Electrons in Semiconductor Superlattices

We show that electrons in a semiconductor superlattice can be used to realize and exploit the unique dynamics of the driven harmonic oscillator that were discovered and explored by George Zaslavsky and colleagues. Under the action of an electric and tilted magnetic field, the semiclassical dynamics of electrons in an energy band of the superlattice exhibit non-KAM chaos, which strongly affects the electrical conductivity. At certain critical field parameters, the electron trajectories change abruptly from fully localized to completely unbounded, and map out intricate stochastic webs in phase space, which act as conduction channels for the electrons. Delocalization of the electron paths produces a series of strong resonant peaks in the electron drift velocity versus electric field curves. We use these drift velocity characteristics to make self-consistent drift-diffusion calculations of the current-voltage and differential conductance-voltage curves of the superlattices, which agree well with our experimental data and reveal strong resonant features originating from the sudden delocalization of the stochastic single-electron paths. We show that this delocalization has a pronounced effect on the distribution of space charge and electric field domains within the superlattices. Inter-miniband tunneling greatly reduces the amount of space-charge buildup, thus enhancing the domain structure and both the strength and number of the current resonances.

Observation of current resonances due to enhanced electron transport through stochastic webs in superlattices

We present experimental and theoretical studies of a unique type of stochastic electron motion in superlattices (SLs) with a magnetic field B tilted at an angle to the SL axis. The magnetic field couples electronic Bloch oscillations along the SL axis to cyclotron orbits in the plane of the layers. At discrete values of the applied voltage, this coupling transforms the localised Bloch trajectories into unbounded chaotic electron paths, which propagate through intricate “stochastic web”patterns in phase space. This abrupt metal‐insulator‐like transition reveals itself in our experiments as a large resonant increase in the current flow and conductance, observed at critical values of the applied voltage.

Chaotic quantum transport in superlattices

We report a new type of quantum chaotic system in which the classical Hamiltonian originates from the intrinsically quantum mechanical nature of the device. The system is a semiconductor superlattice in a magnetic field. The energy–momentum dispersion curves can be used to calculate semi- classical orbits for electrons confined to a single miniband. When a magnetic field is applied along the superlattice axis (x-direction), the electrons perform Bloch oscillations along the axis with cyclotron motion in the orthogonal plane. But when the magnetic field is tilted away from the x-direction, the orbits are chaotic, and have a spatial width along the superlattice axis, which is much larger than the amplitude of the Bloch oscillations. This is because the tilted field transfers momentum between the x- and z-directions, thereby delocalizing the electron path. This type of chaotic dynamics is fundamentally different to that identified in our previous studies of double–barrier resonant tunneling diodes. We investigate the relation between the orbits of the effective Hamiltonian, and the quantum states of the superlattice. In the regime of strong chaos, the wave functions have a highly diffuse structure which extends across many periods of the superlattice, just like the corresponding classical orbits. This chaos-induced delocalization increases the current flow through real devices. By contrast, in the stable domain the electron orbits remain localized along the paths of particular quasi-periodic orbits. We use theoretical and experimental current–voltage curves to show how the onset of chaos manifests itself in the transport properties of two- and three-terminal superlattice structures, and identify current oscillations associated with classical resonances. We also consider analogies with ultra-cold atoms in an optical lattice with a tilted harmonic trap.

2D chaotic quantum states in superlattices

Abstract The semiclassical motion of an electron along the axis of a superlattice may be calculated from the miniband dispersion curve. Under a weak electric field the electron executes Bloch oscillations which confines the motion along the superlattice axis. When a magnetic field, tilted with respect to the superlattice axis, is applied the electron orbits become chaotic. The onset of chaos is characterised by a complex mixed stable-chaotic phase space and an extension of the orbital trajectories along the superlattice axis. This delocalisation of the orbits is also reflected in the quantum eigenstates of the system some of which show well-defined patterns of high probability density whose shapes resemble certain semiclassical orbits. This suggests that the onset of chaos will be manifest in electron transport through a finite superlattice. We also propose that these phenomena may be observable in the motion of trapped ultra-cold atoms in an optically induced superlattice potential and magnetic quadrupole potential whose axis is tilted relative to the superlattice axis.

Quantum chaotic electron transport in superlattices

Abstract We introduce semiconductor superlattices as a new type of quantum chaotic system that is accessible to experiment. In contrast to previous physical systems that have been used to study quantum chaos, the classical Hamiltonian for the superlattices has an intrinsically quantum–mechanical origin. For low electric fields, superlattices have well-defined minibands which originate from the quantum–mechanical coupling of electron states in a periodic array of potential wells. The energy-wave-vector dispersion curves define an effective Hamiltonian that can be used to calculate semiclassical orbits for electrons confined to a single miniband. We show that applying a tilted magnetic field induces a controllable transition from stable regular motion to strong classical chaos. The onset of chaos delocalizes both the classical orbits and the corresponding quantum wave functions, and is therefore expected to produce a sharp increase in the current flow measured in electron transport experiments. We show that magnetically confined atoms in a periodic optical potential exhibit similar chaotic dynamics at ultra-low (μK) temperatures.

Chaos-induced orbit delocalization and complex Bloch oscillations in semiconductor superlattices

Abstract We show how semiconductor superlattices can be used to provide a new type of quantum chaotic system that is accessible to experiment. When a magnetic field is tilted relative to the superlattice axis, the semiclassical orbits of electrons confined to a single miniband undergo a controllable transition from stable Bloch oscillations to chaotic orbits. The onset of chaos delocalizes the electron paths and generates complex trajectories that extend across many periods of the superlattice. We show that the chaotic motion has a fundamentally different origin to that reported for previous semiconductor structures.