Angus MacKinnon

Universality classes and fluctuations in disordered systems

Waves transmitted through disordered media show increasing fluctuations with thickness of material so that averages of different properties of the wavefield have very different scaling with thickness traversed. We have been able to classify these properties according to a scheme that is independent of the nature of the medium, such that members of a class have a universal scaling independent of the nature of the medium. We apply this result to trace (TL T†L)M, where TL is the amplitude transmission matrix. The eigenfunctions of TL T†L define a set of channels through which the current flows, and the eigenvalues are the corresponding transmission coefficients. We prove that these coefficients are either ≈ 0 or ≈ 1. As L increases more channels are shut down. This is the maximal fluctuation theorem: fluctuations cannot be greater than this. We expect that our classification scheme will prove of further value in proving theorems about limiting distributions. We show by numerical simulations that our theorem holds good for a wide variety of systems, in one, two and three dimensions.

The Anderson transition: New numerical results for the critical exponents

The results of our recent evaluation of the critical exponents of the localization length, v and the conductivity, s, at the Anderson transition for box and Gaussian distributions of the random potential are reported. Data are evaluated not only in the centre of the band but also near the band edge. Near the band centre we obtain s = v = 1.4±0.2, and s = v = 0.9±0.3 for the box and the Gaussian distribution, respectively. Near the band edge we found it impossible to determine accurately the exponents.

CONDUCTANCE AND CONDUCTANCE FLUCTUATIONS OF NARROW DISORDERED QUANTUM WIRES

In this paper we present and discuss our results for the conductance and conductance fluctuations of narrow quantum wires with two types of disorder: boundary roughness (hard wall confining potential) and islands of strongly scattering impurities within the bulk of the wire. We use a tight--binding Hamiltonian to describe the quantum wire, infinite perfect leads, a two--terminal Landauer--type formula for the conductance, and the recursive single-particle Greens function technique. We find that conductance quantization is easily destroyed by strong scattering. We also find that Anderson localization poses a serious restriction on the high carrier mobility predicted in quantum wires. Conductance fluctuations in narrow quantum wires are not, in general, universal (as in the metallic regime), but can be independent of the wire length over a short range of lengths.

Critical exponents for the metal-insulator transition

The critical exponents of the metal-insulator transition in disordered systems have been the subject of much published work containing often contradictory results. Values ranging between 1/2 and 2 can be found even in the recent literature. In this paper the results of a long-term study of the transition are presented. The data have been calculated with sufficient accuracy (0.2%) that the calculated exponent can be quoted as s= nu =1.54+or-0.08 with confidence. The reasons for the previous scatter of results are discussed.

Incoherent tunnelling through two quantum dots with Coulomb interaction

The coherent conductance and current is calculated through two quantum dots using the Hubbard model for a single level per spin. The occurrence of negative differential conductance is demonstrated. The ohmic conductance is calculated for dots with equally spaced levels. Transport is determined by matching energy levels, even when they do not occur at the charge degeneracy points.

Stark-Wannier states and Stark ladders in semiconductor superlattices

A two-band tight-binding Hamiltonian is used to investigate the interaction between the energy bands of a periodic system in the presence of an electric field. Information on the existence of Stark-Wannier states and Stark ladders in this two-band system is obtained. The possibility of using a semiconductor superlattice device to experimentally observe Stark ladders and Zener tunnelling between the energy bands is also discussed.

Transport via a quantum shuttle

We present preliminary results for a model of electron transport through an electromechanical system in the extreme quantum mechanical and Coulomb blockade limits. The system consists of three quantum dots in which the outer dots are fixed and the central dot is mounted on a quantum harmonic oscillator, thus forming a quantum shuttle. The current through the system as a function of the energy level shift between the outer dots has marked resonances which are associated with avoided level-crossings in the eigenvalue spectrum.

Quantum transport in a resonant tunnel junction coupled to a nanomechanical oscillator

We discuss the quantum transport of electrons through a resonant tunnel junction coupled to a nanomechanical oscillator at zero temperature. By using the Greens-function technique, we calculate the transport properties of electrons through a single dot strongly coupled to a single oscillator. We consider a finite chemical-potential difference between the right and left leads. In addition to the main resonant peak of electrons on the dot, we find satellite peaks due to the creation of phonons. These satellite peaks become sharper and more significant with increasing coupling strength between the electrons and the oscillator. We also consider the energy transferred from the electrons to the oscillator.

Finite Size Scaling Analysis of the Anderson Transition

This chapter describes the progress made during the past three decades in the finite size scaling analysis of the critical phenomena of the Anderson transition. The scaling theory of localization and the Anderson model of localization are briefly sketched. The finite size scaling method is described. Recent results for the critical exponents of the different symmetry classes are summarised. The importance of corrections to scaling are emphasised. A comparison with experiment is made, and a direction for future work is suggested.

On the conductivity of antidot lattices in magnetic fields

We present results for the density of states, and the band conductivity, of strongly modulated lateral-surface superlattices in a magnetic field. The density of states with an energy smoothing of 0.4 meV shows roughly periodic oscillations of the density of states as the magnetic field is varied, comparable in magnitude and period to the oscillations seen experimentally in the resistance. The oscillations are found to correspond to the perturbed, periodic, cyclotron orbits. The band conductivity shows periodic oscillations with a period of one flux quantum per unit cell of the periodic potential related to Hofstadters butterfly, but these are suppressed much more strongly than the density of states oscillations by inelastic scattering of the electrons.