Statistical scattering theory, the supersymmetry method and universal conductance fluctuations

This paper describes a novel analytical approach to the problem of conductance fluctuations in mesoscopic systems which, in particular, gives account of the influence of the coupling to external leads. We consider the case of a linear disordered sample in the metallic regime, which is coupled to two ideally conducting external leads. Using the many-channel approximation to Landauer's formula, we relate the conductance to the total transmission probability through the sample. The microscopic Hamiltonian of the quasi-one-dimensional disordered sample is formulated in terms of a random matrix, and the elements of the associated scattering matrix which determine the transmission are constructed from statistical scattering theory. We show that in addition to the Thouless energy, E_c, and the mean level spacing, d, there exists in the theory, a third energy scale, Γ, determined by the number of channels in the leads and the strength of the coupling between disordered sample and leads. Related to Γ, is a new length scale, L₀. We find that for sample lengths L >L₀, the properties of the conductance depend only weakly on the coupling to the external leads and, for very large L, become identical with those of quasi-one-dimensional conductors in the weak localization limit. On the other hand, for L < L₀, the coupling to the leads strongly affects the behaviour of both the average and the variance of the conductance. The magnitude of L₀ is typically several magnitudes of ten times the elastic mean free path and thus comparable to the sizes of experimental devices. A further novel aspect of our work is the demonstration that the assumption of GOE statistics for the Hamiltonian is sufficient to yield universal conductance fluctuations.     

Rotational tunnelling: A computer simulation at finite temperatures

At low temperatures sharp tunnelling lines are observed from crystals containing XH₃ or XH₄ groups. These lines shift and broaden when fluctuations of the rotational potential become important with increasing temperature. We have modelled the shaking term by a classical oscillator which interacts with the tunnelling particle. The oscillator is irreversibly coupled to a bath at a temperatureT. An analysis of the correlation function relevant for neutron scattering leads to a Lorentzian line which shifts to lower frequencies and broadens with increasing amplitude of the thermal fluctuations.     

Electrical transport in open and closed systems

Electrical conductance is typically calculated by approaches which view the electrical field as a causative source and the motion of carriers as a response. An alternative viewpoint, which starts from the flux of carriers maintained at the edges of a sample, and then calculates how charges build up and fields develop, has gained acceptance in the treatment of disordered systems, the solid state Aharanov-Bohm effect, and universal fluctuations. We analyze some of the less appreciated concomitants of this viewpoint, emphasizing both the generality and limitations of the viewpoint. Particular emphasis is given to the Residual Resistivity Dipole; localized scatterers in metallic conductivity are accompanied by highly localized transport fields. Closed Hamiltonian systems, e.g. a metallic ring with elastic scattering and driven by a time-dependent magnetic flux, are conservative. They cannot exhibit dissipation, under our conventionally accepted forms of physics. It is suggested that the limited precision available,in principle, in calculating the behavior of physical systems limits our ability to retrieve energy from supposedly conservative systems. This can be regarded as the ultimate source of dissipative processes.Dedicated to Professor Harry Thomas on the occasion of his 60th birthday     

Sequential tunneling through molecular spin rings

We consider electrical transport through molecules with Heisenberg-coupled spins arranged in a ring structure in the presence of an easy-axis anisotropy. The molecules are coupled to two metallic leads and a gate. In the charged state of the ring, a Zener double-exchange mechanism links transport properties to the underlying spin structure. This leads to a remarkable contact-site dependence of the current, which for an antiferromagnetic coupling of the spins can lead to a total suppression of the zero-bias conductance when the molecule is contacted at adjacent sites.     

Dissipative contributions of internal multiplicative noise: I. mechanical oscillator

Langevin equations for closed systems with multiplicative fluctuations must also include appropriate dissipative terms that ensure eventual equilibration of the system. We consider an oscillator coupled to a heat bath and show that a particular nonlinear coupling to a harmonic heat bath leads to a fluctuating frequency and to nonlinear dissipative terms. We also analyze the effects of the multiplicative fluctuations and of the corresponding nonlinear dissipation on the temporal evolution of the average oscillator energy. We find that the rate of equilibration of this system can be significantly different from that of an oscillator with only additive fluctuations and linear dissipation.     

Phase behavior and collective excitations of the Morse ring chain

Using primarily numerical methods we study clustering processes and collective excitations in a one-dimensional ring chain. The ring chain is constituted by N identical point particles with next neighbors interacting via nonlinear Morse springs. If the system is coupled to a heat bath (Gaussian white noise and viscous friction), then depending on the particle density and the bath temperature different phase-like states can be distinguished. This will be illustrated by means of numerically calculated phase diagrams. In order to identify collective excitations activated by the heat bath we calculate the spectrum of the normalized dynamical structure factor (SDF). Our numerical results show that the transition regions between different phase-like states are typically characterized by a 1/f-type SDF spectrum, reflecting the fact that near critical points correlations on all length and time scales become important. In the last part of the paper we also discuss a non-equilibrium effect, which occurs if an additional nonlinearly velocity-dependent force is included in the equations of motions. In particular it will be shown that such additional dissipative effects may stabilize cluster configurations.Received: 27 June 2003, Published online: 2 October 2003PACS: 05.70.Fh Phase transitions: general studies - 05.70.Ln Non-equilibrium and irreversible processes - 05.40.-a Fluctuation phenomena, random processes, noise and Brownian motion     

Characteristics of quantum conductance in a quasi-one-dimension quantum wire containing cavity structures

Quantum-mechanical calculations of the conductance for model devices, consisting of dual semi-infinite quantum wires connected in series by a cavity, are carried out with use of the coupled-mode transfer method and mode matching technique. The effects of the mode-mode coupling and geometry-induced scattering on the quantum conductance are in detail studied by varying the geometric structure of the cavity. There are no traces of quantization conductance. The pattern of the conductance displays many peaks and dips. The threshold energy of the first onset of the conductance is lower than the normal value for opening the propagation channel of the lowest subband in the quantum wire. The overall character of the conductance exhibits heavy fluctuations around the classical conductance for the relevant point contact. The fluctuation amplitude is of order of 2e ²/h, similar to universal conductance fluctuations. The oscillatory structure becomes rich and dense as the scale of the cavity increases. There is a global trend for the conductance to rise as the cavity is compressed. The structures of resonant peaks and antiresonance dips in the conductance are originated from the mode coupling among the subbands in the cavity and quantum wires. The heavy conductance fluctuation may be caused by the quantum interference of the electron waves due to the multiple scattering (reflections) of electrons by the cavity boundaries.     

Poloidal Structure of the Turbulence and the Turbulent Flux During Edge Plasma Biasing in the Castor Tokamak

The poloidal structure of the turbulence and the turbulent particle flux in the scrape-off layer (SOL) of the CASTOR tokamak are analysed by means of a poloidal ring of 124 probes distributed uniformly along the whole poloidal circumference. Fluctuation measurements are performed in standard ohmic regime as well as in discharges when a biased electrode is inserted either into the SOL or deeper into the confinement region. It is found that in both cases a strongly sheared radial electric field is created in the SOL, which de-correlates the density and radial velocity fluctuations and reduces their levels. Consequently, the turbulent flux is reduced. However, no phase shift is observed between density and radial velocity fluctuations. When the electrode is localized in the confinement region, all the above effects are less pronounced because of a smaller shear. In addition, the increase of the E × B velocity at biasing leads in both cases to the formation of oscillations in the temporal correlation function. These oscillations are specifically associated to the poloidal mode (m = q), which is created for a limited time and rotates poloidally. This mode does not modify the phase between density and radial velocity fluctuations and has no further effect on the turbulent flux.     

Critical dynamics and multifractal exponents at the Anderson transition in 3d disordered systems

We investigate the dynamics of electrons in the vicinity of the Anderson transition in d = 3 dimensions. Using the exact eigenstates from a numerical diagonalization, a number of quantities related to the critical behavior of the diffusion function are obtained. The relation η = d ? D₂ between the correlation dimension D₂ of the multifractal eigenstates and the exponent η which enters into correlation functions is verified. Numerically, we have η ≈? 1.3. Implications of critical dynamics for experiments are predicted. We investigate the long-time behavior of the motion of a wave packet. Furthermore, electron-electron and electron-phonon scattering rates are calculated. For the latter, we predict a change of the temperature dependence for low T due to η. The electron-electron scattering rate is found to be linear in T and depends on the dimensionless conductance at the critical point.     

Multiple scattering theory in condensed materials

Second-order elliptic differential equations (such as the time-independent single particle Schrödinger equation) may be solved in a finite closed disjoint region of space independently of the rest of space. The solution in all space may then be determined by solving the equations in the exterior region together with boundary conditions at the junction of the two regions. These boundary conditions are determined by the previously found interior solution. This means that such regions may be taken as ‘black boxes’ whose exact details do not matter. The simplest example of this is phase-shift scattering theory from a single scatterer where all the scattering properties are described by the phase shifts, and the exact details of the scattering potential are unimportant. In a macroscopic condensed system, however, there are many core regions and one is really concerned with the multiple scattering which takes place between these different scattering centres. Much of this article is devoted to investigating the formal properties of scattering theory when there are many non-overlapping spherical regions of radius R _M, each of which is described by its own scattering matrix, or, equivalently for a spherically symmetric potential, by its phase shifts. Non-spherically symmetric and spin-dependent potentials are permitted, but for simplicity we assume initially that the interstitial region between each disjoint scattering region has zero potential. The generalization of the multiple scattering formalism for non-zero interstitial potential is also given at a later stage.

It is shown that in such a system a generalized T-matrix may be defined which describes the radiation from one of the core regions when another one has been excited. It is then a many channel T-matrix in which the channels are the different disjoint scattering regions. It is shown that the formal properties of this T matrix are the same as for a normal T matrix. In § 2 we review the properties of ordinary scattering theory, and then in § 3 we show that analogous properties for the generalized T matrix hold. An exact expression for the density of particle eigenstates is derived in terms of the positions and scattering matrices of the individual scattering centres. This expression reduces to the standard KKR band structure equation for the infinite regular lattice. We also consider how to construct the density of eigenstates and the charge density for such a system. These latter quantities may be approached in two different ways: the usual way is to consider the scattering material to occupy all space, but from a multiple scattering viewpoint one must consider the total volume of condensed material to be small compared with all space, even if both limit to infinity. It is not obvious that the latter method leads to the same results as the former (formally the density of eigenvalues is identical to the free electron density of eigenvalues in the latter method) and it is shown how the differences in the two approaches are resolved. We also discuss the expansion of some of these results for a perfect lattice. While the usual expansions are pseudo-potential expansions, a manifestly ‘on-energy shell’ expansion is derived which does not contain the arbitrary parameters of the pseudo-potential expansions. Finally, in § 4, we review the most significant contributions of other authors to the theory of multiple scattering.     

任意正多边形量子环自旋输运的普遍解

在有关偶数正多边形量子环对称连接特殊情形的自旋输运特性的研究基础上,进一步探讨了任意正多边形量子环的自旋输运性质.不仅解析地求解了相关电子散射问题,而且得到了 Landauer-Buttiker 电导的普遍公式,并讨论了它的圆环极限和 Aharonov-Casher 相位问题.结合数值计算,研究了正多边形量子环的Landauer-Buttiker 电导随多边形边数、引线连接方式、自旋轨道耦合强度以及电子波矢的周期变化特性和零点分布规律. 关键词： Rashba 自旋-轨道耦合 Aharonov-Casher 相位 量子网络 量子输运     

任意正多边形量子环自旋输运的普遍解

在有关偶数正多边形量子环对称连接特殊情形的自旋输运特性的研究基础上,进一步探讨了任意正多边形量子环的自旋输运性质.不仅解析地求解了相关电子散射问题,而且得到了 Landauer-Buttiker 电导的普遍公式,并讨论了它的圆环极限和 Aharonov-Casher 相位问题.结合数值计算,研究了正多边形量子环的Landauer-Buttiker 电导随多边形边数、引线连接方式、自旋轨道耦合强度以及电子波矢的周期变化特性和零点分布规律.     

无耗散介观电感耦合电路的量子效应

本文从无耗散的电感耦合电路的经典运动方程出发,分别研究了这一耦合电路在其任意的本征态下和压缩真空态下电路中电荷、电流的量子涨落,其结果表明,每个回路中的电荷、电流都存在着量子涨落,且两回路中的量子噪音是相互关联的。     

The controlled creation of a maximally entangled state between a SQUID ring and a single electromagnetic field mode

We consider a SQUID ring inductively coupled to an electromagnetic field mode, both treated quantum mechanically. We demonstrate a method for creating a maximally entangled state between the ring and the field mode. Our method utilises a non-adiabatic external magnetic flux pulse to move into and out of a transition region. Hence, our approach is fundamentally different to techniques based on Landau–Zener tunnelling that can also be used to achieve similar results. Our analysis is extended to include the effects of coupling the system to a dissipative environment. With this model we show that although such an environment makes a noticeable difference to the time evolution of the system, it need not destroy the entanglement of this coupled system over time scales required for quantum technologies.     

Nonlinear Dynamics and Fluctuations of Dissipative Toda Chains

The dynamics of a ring of masses including dissipative forces (passive or active friction) and Toda interactions between the masses is investigated. The characteristic attractor structure and the influence of noise by coupling to a heat bath are studied. The system may be driven from the thermodynamic equilibrium to far from equilibrium states by including negative friction. We show, that over-critical pumping with free energy may lead to a partition of the phase space into attractor regions corresponding to several types of collective motions including uniform rotations, one- and multiple soliton-like excitations and relative oscillations. The distribution functions in the phase space and the correlation functions of the forces and the spectra of nonlinear excitations are calculated. We show that a finite-size Toda ring with weak thermal coupling develops at intermediate temperatures a broadband colored noise spectrum with an 1/f tail at low frequencies.     

Coupled channel T-operator equations with connected kernels: (II). N coupled two-body channels

A time-independent theory of rearrangement collisions involving transitions between two-body states is presented. It is assumed that the system of interest consists of particles that may be partitioned into two-body systems in N ways, including interchanges of particle labels without changing the kind of channel. An infinite family of sets of N coupled T-operator equations is derived by use of the channel coupling array, as in previous work on the three-body problem. Specialization to the channel-permuting arrays guaranteeing connected (N?1)th iterates of the kernel of the coupled equations is made in the N-channel case (N > 3) and the nature of the solutions to the coupled equations is discussed. Various approximation schemes to be used with numerical calculations are suggested. Since the transition operators for all rearrangement channels are coupled together, no problems concerning non-orthogonality of the eigenstates of different channel Hamiltonians are encountered; also the presence of the outgoing wave boundary condition in all channels is made explicit. The close resemblance of the equations in matrix form to those of one-channel scattering is exploited by introducing Møller wave operators and associated channel scattering states, an optical potential formalism that leads to rearrangement channel optical potential operators, and a variational formulation of the coupled equations using a Schwinger-like variational principle. A brief comparison with other many-body formalisms is also given.     

Microscopic theory of brownian motion: III. The nonlinear Fokker-Planck equation

The nonlinear Fokker-Planck equation for the momentum distribution of a brownian particle of mass M in a bath of particles of mass m is derived. The contribution to this equation arising from initial deviation from bath equilibrium is analysed. This contribution is free of slow M-dependent decays and with certain restrictions leads to an effective shift in the initial value of the B particle momentum. The nonlinear Fokker-Planck equation for an initial bath equilibrium state is analyzed in terms of its predictions for momentum relaxation and mode coupling effects. It is found that in addition to nonlinear renormalization of the type previously found for the momentum correlation function, mode coupling leads to long-lived memory of the initial momentum state.     

A dynamic scattering approach for a gated interacting wire

A scattering approach for correlated one-dimensional systems is developed. The perfect contact to charge reservoirs is encoded in time-dependent boundary conditions. The conductance matrix for an arbitrary gated wire, respecting charge conservation, is expressed through a dynamic scattering matrix. Two applications are developed. First, it is shown that the dc conductance is equal to e ²/h for any model with conserved total left- and right-moving charges. Second, the ac conductance matrix is explicitly computated for the Tomonaga-Luttinger model (TLL). Received 31 August 1998     

Eigenvalue problem for arbitrary linear combinations of a boson annihilation and creation operator

The eigenvalue problem for arbitrary linear combinations kα + μα^? of a boson annihilation operator α and a boson creation operator α^? is solved. It is shown that these operators possess nondegenerate eigenstates to arbitrary complex eigenvalues. The expansion of these eigenstates into the basic set of number states | n >, (n = 0, 1, 2, …), is found. The eigenstates are normalizable and are therefore states of a Hilbert space for | ζ | < 1 with ζ ? μ/k and represent in this case squeezed coherent states of minimal uncertainty product. They can be considered as states of a rigged Hilbert space for | ζ | ? 1. A completeness relation for these states is derived that generalizes the completeness relation for the coherent states | α 〉. Furthermore, it is shown that there exists a dual orthogonality in the entire set of these states and a connected dual completeness of the eigenstates on widely arbitrary paths over the complex plane of eigenvalues. This duality goes over into a selfduality of the eigenstates of the hermitian operators kα + k* α^? to real eigenvalues. The usually as nonexistent considered eigenstates of the boson creation operator α^? are obtained by a limiting procedure. They belong to the most singular case among the considered general class of eigenstates with ζ ? μ/k as a parameter.     

Parametric interaction of space-charge waves in thin-film semiconductor structures

The general theory of parametric coupling between space-charge waves and drifting charge carriers in thin-film semiconductor structures has been worked out. This theory is applicable, in particular, to n-GaAs and n-InP semiconductors with negative differential conductance due to intervalley electron transitions under high electric fields. We started from the electrodynamic theory of waveguide excitation by extraneous currents, which was extended for arbitrary waveguide structures with composite active media. Our theory makes it possible to study parametric interaction between space-charge waves in semiconductor films with regard for boundary conditions, diffusion, the anisotropy and the frequency dispersion of the differential electron mobility, as well as the multifrequency and multimode nature of a wave process in thin-film structures.