Faraday first characterised the behaviour of a fluid in a container subjected to vertical periodic oscillations. His study pertaining to hydrodynamic instability, the Faraday instability, has catalysed a myriad of experimental, theoretical, and numerical studies shedding light on the mechanisms responsible for the transition of a system at rest to a new state of well-ordered vibrational patterns at fixed frequencies. Here we study dual strata in a shallow vessel containing distilled water and high-viscosity lubrication oil on top of it. At elevated driving power, beyond the Faraday instability, the top stratum is found to freeze into a rigid pattern with maxima and minima. At the same time there is a dynamic crossover into a new state in the form of a lattice of recirculating vortices in the lower layer containing the water. Instrumentation and the physics behind are analysed in a phenomenological way together with a basic heuristic modelling of the wave field. The study, which is based on relatively low-budget equipment, stems from related art projects that have evolved over the years. The study is of value within basic research as well as in education, especially as more advanced collective project work in e.g. engineering physics, where it invites further studies of pattern formation, the emergence of vortex lattices and complexity.

Effects of randomly distributed impurities on conductance, spin polarization and electronlocalization in realistic gated semiconductor quantum point contacts (QPCs) have beensimulated numerically. To this end density functional theory in the local spin-densityapproximation has been used. In the case when the donor layer is embedded far from thetwo-dimensional electron gas (2DEG) the electrostatic confinement potential exhibits theconventional parabolic form, and thus the usual ballistic transport phenomena take place bothin the devices with split gates alone and with an additional metallic gate on the top.In the opposite case, i.e. when the randomly distributed donors are placed not far away fromthe 2DEG layer, there are drastic changes like the localization of electrons in the vicinity ofconfinement potential minima which give rise to fluctuations in conductance and resonances.The conductance as a function of the voltage applied to the top gate for asymmetricallycharged split gates has been calculated. In this case resonances in conductance caused byrandomly distributed donors are shifted and decrease in amplitude while the anomaliescaused by interaction effects remain unmodified. It has been also shown that for a wide QPCthe polarization can appear in the form of stripes. The importance of partial ionization ofthe random donors and the possibility of short range order among the ionized donors areemphasized. The motivation for this work is to critically evaluate the nature of impurities andhow to guide the design of high-mobility devices.

We study effects of randomly distributed impurities on spin polarization and electronlocalization in realistic semiconductor quantum point contacts (QPCs). To this end we usedensity functional theory in local spin-density approximation (LSDA). Previous studies (as,for example, in [1]) have been restricted to the Thomas-Fermi approximation, and thus theeffects of electron correlation and realistic confinement potentials were beyond the subject.Our studies have been performed for the two geometries of the gates, the first one with onlysplit gates, and the other one with an additional top gate situated over the split gates. In thelatter case there is a possibility to vary electron density within a fixed confinement whichgives an opportunity to separate the effects on conductance caused by impurities and electronelectroninteractions in a more distinct way. In both cases we recover the conventionalfluctuation free parabolic electrostatic potential when the distance between the donor layerand the two-dimensional electron gas (2DEG) exceeds ~50 nm. In the opposite case, i.e.,when the randomly distributed donors are placed more close to the 2DEG layer, there aredrastic changes like the localization of electrons in the vicinity of the confinement potentialminima which gives rise to fluctuation in conductance and resonances. At the same time theusual conductance steps vanish. By charging asymmetrically the split gates voltage wecalculate the conductance as a function of the voltage applied to the top gate. In this way wefind that resonances in conductance caused by randomly distributed donors are shifted anddescreased in amplitude while the anomalies caused by interaction effects remain unmodified.Resonance peaks in the conductance derive from localized states within the QPC due torandom fluctuations. The nature of electron localization has been discussed in our previousstudy [2] where we stress the crucial role of the shape of confinement potential on theformation of electron localization. In the present study we have shown that electronlocalization may be caused by randomly distributed donors and play an important role inelectron transport, especially near the pinch-off regime. The results of our numericalsimulations agree qualitatively with experimental studies [3-4]. We have also shown that fora wide QPC spin polarization appears in the form of stripes. This finding may be interesting inview of experimental study in [5] where it has been shown that the structure of such kind canbe responsible for the anomalous behavior of the quantized conductance of a quantum wire inthe shallow confinement limit. We also discuss the diminished effect of partially ionizedrandom donors on the electronic potentials and the appearance of short-range order among thedonors. The results of the present study is important for applications. For example,homogeneity and order of an assembly of nanostructures are crucial for their use in largescaleelectronic and optical systems.[1] J.A. Nixon, J.H. Davies, and H.U. Baranger, Phys. Rev. B 43, 12638 (1991)[2] I. I. Yakimenko, V. S. Tsykunov and K.-F. Berggren, J. Phys. Condens. Matter 25, 072201 (2013)[3] L.W. Smith, K. J. Thomas, M. Pepper, D. A. Ritchie, I. Farrer, J.P. Griffiths, G.A.C. Jones, J. ofPhys.: Conf. Series 376, 012018, (2012)[4] L. W. Smith, H. Al-Taie, F. Sfigakis, P. See, A. A. J. Lesage, B. Xu, J. P. Griffiths, H. E. Beere, G.A. C. Jones, D. A. Ritchie, M. J. Kelly, and C. G. Smith, Phys. Rev. B 90, 045426 (2014).

A basic quantum-mechanical model for wave functions and current flow in open quantum dots or billiards is investigated. The model involves non-Hertmitian quantum mechanics, parity-time (PT) symmetry, and PT-symmetry breaking. Attached leads are represented by positive and negative imaginary potentials. Thus probability densities, currents flows, etc., for open quantum dots or billiards may be simulated in this way by solving the Schrödinger equation with a complex potential. Here we consider a nominally open ballistic quantum dot emulated by a planar microwave billiard. Results for probability distributions for densities, currents (Poynting vector), and stress tensor components are presented and compared with predictions based on Gaussian random wave theory. The results are also discussed in view of the corresponding measurements for the analogous microwave cavity. The model is of conceptual as well as of practical and educational interest.

We analyze the occurrence of local  magnetization and the effects of electron localization in different models of quantum point contacts (QPCs) using spin-relaxed density functional theory (DFT/LSDA) by means of numerical simulations. In the case of soft confinement potentials the degree of localization is weak and we therefore observe only traces of partial electron localization in the middle of the QPC. In the pinch-off regime there is, however, distinct accumulation at the QPC edges. At the other end, strong confinement potential, low-electron density in the leads and top or implant gates favor electron localization. In such cases one may create a variety of electron configurations from a single localized electron to more complex structures with multiple rows and Wigner lattices.

ABSTRACT We simulate effects of electron localization and quantum correlations in realistic GaAs/AlGaAs quantum point contacts (QPCs) in the presence of randomly distributed donors (1) using spin-relaxed density functional theory (DFT/LSDA). Two different configurations of gates defining the QPCs were studied: a split gate and a top gate in addition to the split gate. In both cases we recover the conventional fluctuation-free parabolic electrostatic potential when the distance between the donor layer and the 2D electron gas exceeds 70 nm.  Hence we also find ballistic phenomena such as integer conductance steps as well as the 0.7 anomaly. The electrostatic potential changes dramatically,however, when the random donors are placed closer to the 2D gas. Electron localization is then increased and conductance fluctuations and resonance peaks appear. At the same time the usual conductance steps vanish. By charging asymmetrically the split gates voltage we have found that conductance fluctuations caused by random donors are shifted while the anomalies caused by interaction effects may remain.  Resonance peaks in the conductance derive from localized states inside within the QPC. The nature of electron localization has been discussed in our previous study (2) where we stress the crucial role of confinement potential on the formation of electron localization. In the present study we have shown that electron localization may be caused by randomly distributed donors and play an important role in electron transport, especially near the pinch-off regime. The results of our numerical simulations agree with recent experimental studies (3). 

(1) J.A. Nixon, J.H. Davies, and H.U Baranger, Phys. Rev. B 43, 12638 (1991) 

(2) I. I. Yakimenko, V. S. Tsykunov and K.-F. Berggren, J. Phys. Condens. Matter 25, 072201 (2013)

(3) L.W. Smith, K. J. Thomas, M. Pepper, D. A. Ritchie, I. Farrer, J.P. Griffiths, G.A.C. Jones,  J. of Phys.: Conf. Series  376, 012018,  (2012)