The chemical stability of graphene and other free-standing two-dimensional crystals means that they can be stacked in different combinations to produce a new class of functional materials, designed for specific device applications. Here we report resonant tunnelling of Dirac fermions through a boron nitride barrier, a few atomic layers thick, sandwiched between two graphene electrodes. The resonance occurs when the electronic spectra of the two electrodes are aligned. The resulting negative differential conductance in the device characteristics persists up to room temperature and is gate voltage-tuneable due to graphene’s unique Dirac-like spectrum. Although conventional resonant tunnelling devices comprising a quantum well sandwiched between two tunnel barriers are tens of nanometres thick, the tunnelling carriers in our devices cross only a few atomic layers, offering the prospect of ultra-fast transit times. This feature, combined with the multi-valued form of the device characteristics, has potential for applications in high-frequency and logic devices.

Recent developments in the technology of van der Waals heterostructures made from two-dimensional atomic crystals have already led to the observation of new physical phenomena, such as the metal-insulator transition and Coulomb drag, and to the realization of functional devices, such as tunnel diodes, tunnel transistors and photovoltaic sensors. An unprecedented degree of control of the electronic properties is available not only by means of the selection of materials in the stack, but also through the additional fine-tuning achievable by adjusting the built-in strain and relative orientation of the component layers. Here we demonstrate how careful alignment of the crystallographic orientation of two graphene electrodes separated by a layer of hexagonal boron nitride in a transistor device can achieve resonant tunnelling with conservation of electron energy, momentum and, potentially, chirality. We show how the resonance peak and negative differential conductance in the device characteristics induce a tunable radiofrequency oscillatory current that has potential for future high-frequency technology.

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.

We present theoretical and experimental studies of the dynamics of Bose-Einstein condensates in a one-dimensional optical lattice and a three-dimensional harmonic trap. For low atom densities and inertial forces, the condensate performs regular Bloch oscillations, and its center-of-mass motion closely follows semiclassical single-particle trajectories, shaped by the lowest-energy band. But in other regimes, the center-of-mass motion disrupts the internal structure of the condensate by generating solitons and vortex rings, which can trigger explosive expansion of the atom cloud. We use images of the atom cloud to provide experimental evidence for this internal disruption, and find that the process occurs most readily in high-density condensates undergoing slow Bloch oscillations

We show that a tilted magnetic field transforms the structure and THz dynamics of charge domains in a biased semiconductor superlattice. At critical field values, strong coupling between the Bloch and cyclotron motion of a miniband electron triggers chaotic delocalization of the electron orbits, causing strong resonant enhancement of their drift velocity. This dramatically affects the collective electron behavior by inducing multiple propagating charge domains and GHz-THz current oscillations with frequencies ten times higher than with no tilted field.Superlattices ͑SLs͒, comprising alternating layers of different semiconductor materials, provide a flexible environment for studying quantum transport in periodic potentials and for generating, detecting, mixing, and amplifying highfrequency electromagnetic radiation. [1][2][3][4][5][6][7][8][9][10][11] Due to the formation of energy "minibands," electrons perform THz Bloch oscillations when a sufficiently high electric field, F, is applied along the SL. The Bloch orbits become more localized as F increases, thus producing negative differential velocity ͑NDV͒ in the electron drift velocity, v d , versus F characteristic. 12,13 This single-particle NDV also strongly influences the collective behavior of the electrons, causing them to slow and accumulate in high-density charge domains. 1-3 For sufficiently high F, the domains are unstable and propagate through the SL, generating current oscillations at GHz-THz frequencies determined by the form of v d ͑F͒ and the SL length. 1,14-17 In a given SL with fixed v d ͑F͒, the frequency of domain dynamics can be tuned, over a limited range, by changing the applied voltage. 6,14 Here, we show that both the spatial profile of charge domains and their oscillation frequency can be flexibly controlled by using a tilted magnetic field, B, to engineer the v d ͑F͒ characteristic of the SL. At F values for which the Bloch frequency equals the cyclotron frequency corresponding to the B component along the SL axis, the semiclassical electron motion changes abruptly from localized stable trajectories to unbounded chaotic paths, which propagate rapidly through the SL. [18][19][20][21][22] This delocalization creates a series of sharp resonant peaks in v d ͑F͒, which were detected in previous dc current-voltage measurements, 19-21 relate to mode coupling in Josephson junctions, 23 and can stabilize the SL Bloch gain profile in the vicinity of Stark-cyclotron resonances. 24 We show that these v d peaks create multiple propagating charge domains, shaped by B and , and thereby generate ac currents whose magnitude and frequencies are far higher than when B = 0. Chaos-assisted single-electron transport induced by the interplay between cyclotron and Bloch motion therefore provides a mechanism for controlling the collective dynamics of the miniband electrons, thus increasing the power and frequency of the resulting current oscillations by an order of magnitude.We consider the GaAs/AlAs/InAs SL used in recent experiments. 19,20 14 unit cells, each...

We show that resonant electron transport in semiconductor superlattices with an applied electric and tilted magnetic field can, surprisingly, become more pronounced as the lattice and conduction electron temperature increases from 4.2 K to room temperature and beyond. It has previously been demonstrated that at certain critical field parameters, the semiclassical trajectories of electrons in the lowest miniband of the superlattice change abruptly from fully localised to completely unbounded. The unbounded electron orbits propagate through intricate web patterns, known as stochastic webs, in phase space, which act as conduction channels for the electrons and produce a series of resonant peaks in the electron drift velocity versus electric field curves. Here, we show that increasing the lattice temperature strengthens these resonant peaks due to a subtle interplay between thermal population of the conduction channels and transport along them. This enhances both the electron drift velocity and the influence of the stochastic webs on the current-voltage characteristics, which we calculate by making self-consistent solutions of the coupled electron transport and Poisson equations throughout the superlattice. These solutions reveal that increasing the temperature also transforms the collective electron dynamics by changing both the threshold voltage required for the onset of self-sustained current oscillations, produced by propagating charge domains, and the oscillation frequency.

We investigate the effect of interatomic interactions on the quantum-mechanical reflection of Bose-Einstein condensates from regions of rapid potential variation. The reflection process depends critically on the density and incident velocity of the condensate. For low densities and high velocities, the atom cloud has almost the same form before and after reflection. Conversely, at high densities and low velocities, the reflection process generates solitons and vortex rings that fragment the condensate. We show that this fragmentation can explain the anomalously low reflection probabilities recently measured for low-velocity condensates incident on a silicon surface.

Resonant tunneling spectroscopy is used to study the energy-level spectrum of a new chaotic dynamical system, an electron in a trapezoidal potential well in the presence of a high magnetic field tilted relative to the confining barriers. Distinct series of quasiperiodic resonances are observed in the currentvoltage characteristics which change dramatically with tilt angle. These resonances are related to unstable closed orbits within the chaotic domain. The experimental results are explained by identifying and studying the properties of periodic orbits accessible to the tunneling electrons.