Volume 37, Issue 10, October 1996
 SPECIAL ISSUE: MESOSCOPIC PHYSICS



Mesoscopic phenomena in semiconductor nanostructures by quantum design
View Description Hide DescriptionThe energy levels, wave functions, optical matrix elements, and scattering rates of electrons can be tailored at will using semiconductor nanostructures as building blocks for practically any kind of potential. This allows the design and experimental realization of new artificial materials and devices, with interesting optical and transport properties arising from quantum confinement, tunneling, and quan‐ tum coherence on a mesoscopic scale ranging typically from 1 to 100 nm. This approach is illustrated with a number of recent examples based on experiments and calculations, such as resonant tunneling through double barriers, quantum interference phenomena in transport and optical absorption, bound states in the continuum, quantum well ‘‘pseudomolecules’’ with giant nonlinear optical susceptibilities, and quantum cascade lasers.

Dynamic conductance and quantum noise in mesoscopic conductors
View Description Hide DescriptionWe present results for the dc conductance, the ac conductance, and the current–current fluctuation spectra of mesoscopic, phase‐coherent conductors based on a second quantization approach to scattering and a self‐consistent potential approach. A second quantization approach permits an investigation of statistical effects due to the symmetry of the wave functions under exchange of particles. A self‐consistent approach is needed to enforce overall charge conservation and to obtain current conserving expressions for frequency‐dependent conductances and fluctuation spectra. For the particular example of a mesoscopic capacitor we present microscopic expressions for the electrochemicalcapacitance and the charge relaxation resistance.

Transport properties in resonant tunneling heterostructures
View Description Hide DescriptionAn adiabatic approximation in terms of instantaneous resonances to study the steady‐state and time‐dependent transport properties of interacting electrons in biased resonant tunnelingheterostructures is used. This approach leads, in a natural way, to a transport model of large applicability consisting of reservoirs coupled to regions where the system is described by a nonlinear Schrödinger equation. From the mathematical point of view, this work is nonrigorous but may offer some fresh and interesting problems involving semiclassical approximation, adiabatic theory, nonlinear Schrödinger equations, and dynamical systems.

Electron counting statistics and coherent states of electric current
View Description Hide DescriptionA theory of electron counting statistics in quantum transport is presented. It involves an idealized scheme of current measurement using a spin 1/2 coupled to the current so that it precesses at the rate proportional to the current. Within such an approach, counting charge without breaking the circuit is possible. As an application, we derive the counting statistics in a single channel conductor at finite temperature and bias. For a perfectly transmitting channel the counting distribution is Gaussian, both for zero‐point fluctuations and at finite temperature. At constant bias and low temperature the distribution is binomial, i.e., it arises from Bernoulli statistics. Another application considered is the noise due to short current pulses that involve few electrons. We find the time‐dependence of the driving potential that produces coherent noise‐minimizing current pulses, and display analogies of such current states with quantum‐mechanical coherent states.

Bound states and scattering in quantum waveguides coupled laterally through a boundary window
View Description Hide DescriptionWe consider a pair of parallel straight quantum waveguides coupled laterally through a window of a width l in the common boundary. We show that such a system has at least one bound state for any l≳0. We find the corresponding eigenvalues and eigenfunctions numerically using the mode‐matching method, and discuss their behavior in several situations. We also discuss the scattering problem in this setup, in particular, the turbulent behavior of the probability flow associated with resonances. The level and phase‐shift spacing statistics shows that in distinction to closed pseudo‐integrable billiards, the present system is essentially nonchaotic. Finally, we illustrate time evolution of wave packets in the present model.

S‐matrix, resonances, and wave functions for transport through billiards with leads
View Description Hide DescriptionFor a simple model describing the S‐matrices of open resonators the statistical properties of the resonances are investigated, as well as the wave functions inside the resonator.

Diagrammatic method of integration over the unitary group, with applications to quantum transport in mesoscopic systems
View Description Hide DescriptionA diagrammatic method is presented for averaging over the circular ensemble of random‐matrix theory. The method is applied to phase‐coherent conduction through a chaotic cavity (a ‘‘quantum dot’’) and through the interface between a normal metal and a superconductor.

Statistics of wave functions in mesoscopic systems
View Description Hide DescriptionWe review the results of a recent study of fluctuations of wave functions in confined chaotic systems. The fluctuations can be due to a random potential or be a consequence of a chaotic scattering by the walls. The entire distribution function of the local amplitudes of the wave functions,f _{1}, and the joint two‐point distribution are calculated in various situations. The computation is performed using the supersymmetry technique and employs the studies of a reduced version of the non‐linear supersymmetric σ‐model developed especially for investigating the properties of a single eigenstate in a discrete spectrum of a chaotic quantum system. For not very large amplitudes, the complete description can be achieved using the zero‐dimensional approximation of the σ‐model. The distribution function calculated in the limit of various symmetry classes shows the universal behavior known as the Porter‐Thomas statistics, and fluctuations at distant points do not correlate. In the crossover regime between the ensembles, the distribution of local amplitudes shows a somewhat more sophisticated behavior: the fluctuations in this case are correlated over distances exceeding the mean free path. For large amplitudes generated by the states the most affected by the localization (we call them prelocalized), the zero‐dimensional approximation is no longer valid. Instead, the statistics of their wave functions is determined by nontrivial vacua of the reduced σ‐model which is quite similar to the Liouville model known in conformal field theory. In particular, the vacuum state of the reduced σ‐model obeys the Liouville equation, which indicates that in two dimensions the prelocalized states have nearly critical properties: we prove their multifractality and power‐law statistically averaged envelope φ(r)^{2}∝r ^{−2μ} at the intermediate range of distances below the localization length with a spectrum of exponents μ<1, as well as obtain a logarithmically‐normal tail of the distribution functionf _{1}. We also find an evidence of prelocalized states in quasi‐one‐dimensional wires with the length shorter than the localization length: their statistically averaged envelope has power‐law asymptotics, φ(x)^{2}∝x ^{−2}, and the tail of the distribution function is similar to that describing localized states in the infinite wires.

Energy level correlations in disordered metals: Beyond universality
View Description Hide DescriptionThis short review is concerned with energy level statistics in weakly disordered metallic grains. In particular, we concentrate on how these statistics deviate from the universal ones. Using a nonperturbative approach we evaluate the large frequency asymptotics of the two‐point correlator of the density of states. This allows us to describe the behavior of the system at arbitrary times.

Riemannian symmetric superspaces and their origin in random‐matrix theory
View Description Hide DescriptionGaussian random‐matrix ensembles defined over the tangent spaces of the large families of Cartan’s symmetric spaces are considered. Such ensembles play a central role in mesoscopic physics, as they describe the universal ergodic limit of disordered and chaotic single‐particle systems. The generating function for the spectral correlations of each ensemble is reduced to an integral over a Riemannian symmetric superspace in the limit of large matrix dimension. Such a space is defined as a pair (G/H,M _{ r }), where G/H is a complex‐analytic graded manifold homogeneous with respect to the action of a complex Lie supergroup G, and M _{ r } is a maximal Riemannian submanifold of the support of G/H.

Singularities in the spectra of random matrices
View Description Hide DescriptionWe consider singularities of the set of energy levels E _{ n }(X) of a quantum Hamiltonian, obtained by varying a set of d parameters X=(X _{1},..,X _{ d }). Singularities such as minima, degeneracies, branch points, and avoided crossings can play an important role in physical applications. We discuss a general method for counting these singularities, and apply it to a random matrix model for the parameter dependence of energy levels. We also show how the density of avoided crossing singularities is related to a non‐analyticity of a correlation function describing the energy levels.

Asymptotic properties of large random matrices with independent entries
View Description Hide DescriptionWe study the normalized trace g _{ n }(z)=n ^{−1} tr(H−zI)^{−1} of the resolvent of n×n real symmetric matrices H=[(1+δ_{ jk })W _{ jk }√n]_{ j,k=1} ^{ n } assuming that their entries are independent but not necessarily identically distributed random variables. We develop a rigorous method of asymptotic analysis of moments of g _{ n }(z) for  Iz≥η_{0} where η_{0} is determined by the second moment of W _{ jk }. By using this method we find the asymptotic form of the expectation E{g _{ n }(z)} and of the connected correlator E{g _{ n }(z _{1})g _{ n }(z _{2})}−E{g _{ n }(z _{1})}E{g _{ n } (z _{2})}. We also prove that the centralized trace ng _{ n }(z)−E{ng _{ n }(z)} has the Gaussian distribution in the limit n=∞. Based on these results we present heuristic arguments supporting the universality property of the local eigenvalue statistics for this class of random matrix ensembles.

Fictitious level dynamics: A novel approach to spectral statistics in disordered conductors
View Description Hide DescriptionWe establish a new approach to calculating spectral statistics in disordered conductors, by considering how energy levels move in response to changes in the impurity potential. We use this fictitious dynamics to calculate the spectral form factor in two ways. First, describing the dynamics using a Fokker–Planck equation, we make a physically motivated decoupling, obtaining the spectral correlations in terms of the quantum return probability. Second, from an identity which we derive between two‐ and three‐particle correlation functions, we make a mathematically controlled decoupling to obtain the same result. We also calculate weak localization corrections to this result, and show for two dimensional systems (which are of most interest) that corrections vanish to three‐loop order.

Integrability and disorder in mesoscopic systems: Application to orbital magnetism
View Description Hide DescriptionWe present a semiclassical theory of weak disorder effects in small structures and apply it to the magnetic response of non‐interacting electrons confined in integrable geometries. We discuss the various averaging procedures describing different experimental situations in terms of one‐ and two‐particle Green functions. We demonstrate that the anomalously large zero‐field susceptibilitycharacteristic of clean integrable structures is only weakly suppressed by disorder. This damping depends on the ratio of the typical size of the structure with the two characteristic length scales describing the disorder (elastic mean‐free‐path and correlation length of the potential) in a power‐law form for the experimentally relevant parameter region. We establish the comparison with the available experimental data and we extend the study of the interplay between disorder and integrability to finite magnetic fields.

Ergodic properties of infinite quantum harmonic crystals: An analytic approach
View Description Hide DescriptionWe prove that the quantum dynamics of a class of infinite harmonic crystals becomes ergodic and mixing in the following sense: if H _{ m } is the m‐particle Schrödinger operator, ω_{β,m }(A)=Tr(A exp−βH _{ m })/Tr(exp−βH _{ m }) the corresponding quantum Gibbs distribution over the observables A,B,ψ_{ m,λ} the coherent states in the mth particle Hilbert space,g _{ m },λ=(exp−βH _{ m })ψ_{ m,λ} then lim_{ t→∞} lim_{ n→∞} lim_{ m→∞}(1/T)∫^{ T } _{0}〈e ^{ iH } _{ n } _{ t } Ae ^{−iH } _{ n } _{ t }ψ_{ m,λ},ψ_{m,λ}〉dt=lim_{ m→∞} ω_{β,m }(A) if the classical infinite dynamics is ergodic, and lim_{ t→∞} lim_{ n→∞} lim_{ m→∞} ω_{β,m }(e ^{ /iiH nt Ae −iH nt B)=lim m→∞ ωβ,m }(A)lim_{ m→∞}ω_{β,m }(B) if it is in addition mixing. The classical ergodicity and mixing properties are recovered as ℏ→0, and lim_{ m→∞} ω_{β,m }(A) turns out to be the average over a classical Gibbs measure of the symbol generating A under Weyl quantization.

Ergodic properties of the quantum ideal gas in the Maxwell–Boltzmann statistics
View Description Hide DescriptionIt is proved that the q uantization of the Volkovyski–Sinai model of ideal gas (in the Maxwell–Boltzmann statistics) enjoys at the thermodynamical limit the property of quantum mixing in the following sense: lim_{t→∞} lim_{ m/L→ρ m,L→∞ } ω_{β ,L} ^{ m }(e ^{ iH m t/ℏ}×Ae ^{−iH mt / mC} B)=lim_{ m }/L→ρ^{ m,L→∞} ω_{β,L } ^{ m }( A)⋅lim_{ m/L→ρ m,L→∞ } ω_{β,L } ^{ m }(B ). Here H _{ m } is the Schrödinger operator of m free particles moving on a circle of length L; A and B are the Weyl quantization of two classical observables a and b; ω^{ m } _{β,L }(A) is the corresponding quantum Gibbs state. Moreover, one has lim_{ m }/L→ρ^{ m,L→∞} ω_{β,m }(A)=P _{ρ,β (a)}, where P _{ρ,β}(a) is the classical Gibbs measure. The consequent notion of quantum ergodicity is also independently proven.

A mean field theory for arrays of Josephson junctions
View Description Hide DescriptionI present here some results on the mean field theory approach to the statistical mechanics of a D‐dimensional array of Josephson junctions in the presence of a magnetic field. The mean field theory equations are obtained by computing the thermodynamicalproperties. In the high temperature region in the limit D→∞, where the problem is simplified, this limit defines the mean field approximation. Close to the transition point the system behaves very similar to a particular form of spin glasses, i.e., to gauge glasses. We have noticed that in this limit the evaluation of the coefficients of the high temperature expansion may be mapped onto the computation of some matrix elements for the q‐deformed harmonic oscillator. The same arguments can be used to predict the thermodynamicalproperties in the mean field limit. These results can be extended to the low temperature phase using a conjecture on the equivalence of some system without disorder with appropriate random systems.

Spectral properties of a charged particle in antidot array: A limiting case of quantum billiard
View Description Hide DescriptionA model of the periodic array of quantum antidots in the presence of a uniform magnetic field is suggested. The model can be conceived as a periodic lattice of resonators(curvilinear triangles)connected through ‘‘infinitely small’’ openings at the vertices of the triangles. The model Hamiltonian is obtained by means of operator extension theory in indefinite metric spaces. In the case of rational magnetic flux through an elementary cell of the lattice, the dispersion equation is found in an explicit form with the help of harmonic analysis on the magnetic translation group. It is proved, at least in the case of integer flux, that the spectrum of the model Hamiltonian consists of three parts: (1) Landau levels (they correspond to the classical orbits lying between antidots); (2) extended states that correspond to the classical propagation trajectories; and (3) bound states satisfying the dispersion equation; they correspond to the classical chaotic orbits rotating around single antidots. Among other things, methods of finding the Green’s function for some planar domains with curvilinear boundaries are derived.

Singular continuous spectra and discrete wave packet dynamics
View Description Hide DescriptionAsymptotic estimates which relate the diffusion of wave packets on discrete lattices to Hausdorff dimensions of the local density of states are discussed.
