Tunnelling through randomly inhomogeneous barriers as a special problem of the scattering theory

The tunnelling of electrons through an inhomogeneous delta barrier is considered. The strength of the barrier is defined as a function oscillating around a constant mean value along a plane. Owing to deviations from this value, the tunnelling through such a delta barrier has to be interpreted as a scattering. A simple model is discussed when circular windows of a given radius b representing themselves delta barriers of a given strength γ₀ are embedded in a homogeneous delta barrier defined with another strength . When the centers of the windows are distributed randomly in the barrier plane, the potential energy of the electrons is a random function of two space coordinates. The perpendicular incidence is discussed with emphasis on the angular probability density of the tunnelled electrons. The derivation of the angular probability density proves that three basic quantum-mechanical phenomena can be described by one simple formula: tunnelling, diffraction and scattering.     

Fractal Dimension of Julia Set for Nonanalytic Maps

The Hausdorff dimensions of the Julia sets for nonanalytic maps f(z) = z ²+εz* and f(z)=z*²+ε are calculated perturbatively for small ε. It is shown that Ruelle's formula for the Hausdorff dimensions of analytic maps cannot be generalized to nonanalytic maps.     

Quantum mechanical barrier problems: II. Dissipative tunnelling at finite temperatures for the unbiased oscillator

The decay rate for an unbiased local quasi-equilibrium oscillator, leaking through a quartic potential energy barrier into free space and weakly coupled to an ohmic thermal environment, is calculated in the low temperature quantum tunnelling regime. The imaginary part of the free energy is determined by a bound kink-antikink pair. It is found that the limits of zero dissipation (λ → 0) and zero temperature (T → 0) do not commute. In the limit λ → 0 (for nonzero T) one recovers the finite nondissipative quantum tunnelling rate obtained in a previous paper (I), whereas in the limit T → 0 (for nonzero λ) a freezing phenomenon occurs.     

Rigorous analysis of singularities and absence of analytic continuation at first-order phase-transition points in lattice-spin models

We report about two new rigorous results on the nonanalytic properties of thermodynamic potentials at first-order phase transition. For lattice models (d>or=2) with arbitrary finite state space, finite-range interactions which have two ground states, we prove that the pressure has no analytic continuation at the first-order phase-transition point, under the only further assumptions that the Peierls condition is satisfied for the ground states and that the temperature is sufficiently low. For Ising models with Kac potentials J(gamma)(x)=gamma(d)phi(gammax), where 00) and analyticity in the mean field limit (gamma SE pointing arrow 0).     

Rotational tunnelling splitting of a chain of four coupled methyl groups

It is our goal to obtain a reliable prediction of the rotational tunnelling spectrum to be expected for a long chain of coupled one-dimensional quantum rotors. The problem is intractable by the simple methods used so far for up to three coupled methyl groups. Therefore, an efficient, nevertheless sufficiently precise method for solving the stationary Schrödinger equation of interacting methyl groups is developed first; it proves to be valid for a broad range of not too weak potential strengths. Then, three scenarios are investigated: they differ with respect to the relative strength of the single-rotor potential and the interaction potential. For each scenario, we illustrate the dependence of the energy level scheme on the number of coupled groups. For strong coupling and weak single-particle potential, the characteristic features of the energy level scheme of interacting methyls are most clearly observable: For as few as four coupled methyl groups we predict tunnelling spectra which are hardly distinguishable from single-methyl spectra. However, the collective behaviour is still important for the value of the tunnelling splitting. Therefore, the interpretation of such a spectrum in terms of single-methyl tunnelling is obvious but misleading with respect to the potential seen by a methyl group in the crystal.     

Conduction mechanisms in polyacetylene nanofibres

The current–voltage (I–V) characteristics of individual nanofibres of lightly-doped polyacetylene show very strong nonlinearities. At low temperatures the I–V characteristics are consistent with Zener-type tunnelling, and independent of temperature and magnetic field. We propose that this behaviour arises from tunnelling of a segment of the conjugated bond system in the presence of an electric field, in analogy to the soliton-pair creation mechanism proposed by Maki for conduction in charge-density-wave (CDW) materials. A comparison is made with analogous tunnelling conduction mechanisms reported in CDW and spin-density-wave systems at low temperatures. At higher temperatures the I–V characteristics deviate from Zener-type behaviour and are temperature dependent, so other conduction mechanisms are important.     

Tunnelling current in Schottky diodes containing InAs quantum dots

Current transport mechanism in Schottky diode containing InAs quantum dots (QDs) is investigated using temperature-varying current-voltage characteristics. We found that the tunnelling emission has obvious effects on the I-V characteristics. The I-V-T measurements revealed clear effects of QDs on the overall current flow. Field emission (FE, pure tunnelling effect) was observed at low temperature and low voltages bias region. The zero-bias barrier height decreases and the ideality factor increases with decreasing temperature, and the ideality factor was found to follow the T₀-effect. When the reverse bias is varied, the ideality factors of Schottky barriers exhibit oscillations due to the tunnelling of electrons through discrete levels in quantum dots. The traps distributed within InAlAs layer can also act as a transition step for reverse bias defect-assisted tunnelling current which can phenomenologically explain the decrease of the effective barrier height with measurement temperature.     

Phase transitions from saddles of the potential energy landscape

The relation between saddle points of the potential of a classical many-particle system and the analyticity properties of its thermodynamic functions is studied. For finite systems, each saddle point is found to cause a nonanalyticity in the Boltzmann entropy, and the functional form of this nonanalytic term is derived. For large systems, the order of the nonanalytic term increases unboundedly, leading to an increasing differentiability of the entropy. Analyzing the contribution of the saddle points to the density of states in the thermodynamic limit, our results provide an explanation of how, and under which circumstances, saddle points of the potential energy landscape may (or may not) be at the origin of a phase transition in the thermodynamic limit. As an application, the puzzling observations by Risau-Gusman et al. [Phys. Rev. Lett. 95, 145702 (2005)] on topological signatures of the spherical model are elucidated.     

Thermal and field emission effects of laser radiation on metal whisker diodes: Application to infrared detection devices

In a series of recent experiments, research groups have made absolute frequency measurements with laser beams in the infrared region of the spectrum (λ ? 10 μm) using a metal point contact diode for generation, frequency mixing and detection. It has been postulated that the mechanism for the nonlinear current-voltage characteristic of the diode is tunnelling of electrons through an intermediate oxide film from the whisker into the metal base, i.e., the configuration is considered to be a metal-oxide-metal (MOM) tunnelling junction. Several features of the diode's operation create considerable doubt concerning the applicability of the MOM tunnelling mechanism. Analysis of the available experimental data led us to postulate an alternate solid state mechanism, namely a thermally enhanced field emission process. Such emission would be a consequence of the immersion of the whisker tip in the laser radiation resulting in (1) conduction heating which induces thermionic emission and (2) generation of an electric field at the tip necessary for electron tunnelling by field emission. In this paper we calculate rigorously the power absorbed in the metal whisker from the incident radiation. From the power absorbed, the heat conduction equation is solved for model geometries to obtain the laser induced temperature distribution at the whisker surface. Estimates of the electric field are obtained and combined with temperature calculations to obtain the nonlinear I–V characteristics of the thermally enhanced field emission model. Finally some simple experiments are proposed to test the thermal field emission hypothesis as a possible mechanism to explain the nonlinear characteristics of the metal whisker point contact diode.     

On the calculation of the tunnelling probability in a heavily-doped p-n junction

Abstract

The one-dimensional Schrodinger's equation for a triangular potential barrier (appropriate to tunnel diodes) is solved directly to obtain an expression for the probability of tunnelling of an electron through it. The result has been compared with that of Kane and the W.K.B. method. It is concluded that the results based on this calculation predict the same functional dependence of the tunnelling probability on m?, E_g and F as that predicted by earlier methods.     

Variation with temperature of the I–V characteristics of polyacetylene nanofibres

The current–voltage (I–V) characteristics of individual nanofibres of doped polyacetylene show a dramatic change from very strong nonlinearities for lightly-doped samples at low temperatures, to nearly ohmic behaviour for higher temperatures and doping levels. At low temperatures (below 10–30 K), the I–V characteristics are independent of temperature and follow the expression for Zener-type tunnelling, as predicted for field-induced tunnelling of the conjugated bond system. At higher temperatures, the I–V characteristics deviate from Zener-type behaviour and the current increases with temperature as thermally-assisted conduction mechanisms become important. The I–V characteristics for the most conductive sample are consistent with our calculations of fluctuation-induced tunnelling.     

Water dimer vibration-rotation tunnelling levels from vibrationally averaged monomer wavefunctions

The vibration-rotation tunnelling (VRT) spectra for the water dimer obtained by vibrationally averaging the dimer potential over accurate water monomer wavefunctions is reported. The vibrational averaging requires evaluation of the 12D dimer potential energy surface at more than 10¹² distinct geometries. The resulting vibrational spectra of the low-lying dimer states are presented and compared with both less computationally expensive methods based on fixed nuclei approximations and the recent (6+6)d adiabatic calculations of Leforestier et al. [38] (2009). The procedure gives some modest improvement in the agreement with experimental values for the vibration-rotation tunnelling (VRT) states of (H₂O)₂ and (D₂O)₂. This approach can be extended to treat dimer states involving monomer overtone excitations, which is important in obtaining water dimer spectra at infrared and visible wavelengths at atmospheric temperatures, and in characterizing the dimer contribution to the so-called water continuum absorption at these wavelengths.     

A finite free energy density inSU(2) pure Yang-Mills thermodynamics

We show how monopoles, which are stabilized by quantum fluctuations and have a massM～T, contribute to yield an infrared finite expansion for the free energy density of Yang-Mills thermodynamics. The leading behaviour of this expansion is consistent with the perturbative one, but in higher orders it does involve both contributions, which are nonanalytic atg ²=0, and additional analytic terms, which are of nonperturbative origin.     

Resonant tunneling of a few-body cluster through repulsive barriers

A model for quantum tunnelling of a cluster comprised of A identical particles, interacting via oscillator-type potential, through short-range repulsive barrier potentials is introduced for the first time in symmetrized-coordinate representation and numerically studied in the s-wave approximation. A constructive method for symmetrizing or antisymmetrizing the (A ? 1)-dimensional harmonic oscillator basis functions in the new symmetrized coordinates with respect to permutations of coordinates of A identical particles is described. The effect of quantum transparency, manifesting itself in nonmonotonic resonance-type dependence of the transmission coefficient upon the energy of the particles, their number A = 2, 3, 4 and the type of their symmetry, is analyzed. It is shown that the total transmission coefficient demonstrates the resonance behavior due to the existence of barrier quasi-stationary states, embedded in the continuum.     

Tunnelling across a potential barrier

Generalized expressions for the tunnelling probability spectrum and the tunnelling time are derived for a potential barrier of arbitrary shape independent of the use of the perturbation theory. A method to obtain the tunnelling current density by solving the continuity equation in the barrier region is proposed.     

Quantum dissipation: Dynamics of a particle in a symmetric double well in the presence of non-ohmic friction

We investigate the equilibrium value and the approach to equilibrium of a particle in a symmetric double-well potential and subjected to dissipation, within the model of Caldeira and Leggett, Ann. Phys. (N.Y.) 149 (1983) 374, but without the ohmic assumption. For subohmic friction, the critical value of the coupling with the bath above which the particle localizes is found to be dependent on the ratio (tunnelling frequency)/(bandwidth of the bath); for an infinite bandwidth, the results given by Leggett et al., Rev. Mod. Phys. 59 (1987) 1, are recovered. For superohmic dissipation, the particle never localizes and its coordinate always goes to zero by following, at large times, a power law in time of the form t^−ν; the exponent ν is shown to have a non-intuitive variation as a function of the non-ohmicity.     

Transport damping by fluctuations and cluster-type expansions in kinetics

A mechanism of transport damping by fluctuations is proposed and analyzed in detail for the thermal diffusivity of a one-dimensional gas in the case when the externally induced temperature gradient is weaker than fluctuating gradients. Thermal diffusivity is calculated as a function of parameters of a one-dimensional gas characterized by a homogeneous potential of interaction between particles: for the potential V(x)=Q/|x|^k, thermal diffusivity is χ ∼ (Qn ^k/T)^1/(k−1)ν_T/n. The nonanalytic form of thermal diffusivity as a function of concentration is elucidated. The nonanalytic behavior of thermal diffusivity as a function of concentration is explained by a nontrivial symmetry of the problem under analysis at long times, after the initial fluctuations have dissipated. It is shown that spontaneous generation of macroscopic structures developing through selective growth of fluctuations of a certain type in a medium with an externally induced temperature gradient controls the transport properties of the medium when k≈1.     

s-State potts model on a Cayley tree

The Potts model is shown to exhibit a phase transition of continuous order on a Cayley tree. The leading nonanalytic part of the free energy ∣L∣ ^$\text{K}$_s involves a critical exponent $\text{K}$_s going from one to infinity as the coupling goes f rom infinity to K_BP, the Bethe-Peierls critical coupling.     

Nonanalytic dependence of the transition temperature of the homogeneous dilute Bose gas on scattering length

We show that the shift in the transition temperature of the dilute homogeneous Bose gas is nonanalytic in the scattering amplitude a. The first correction beyond the positive linear shift in a is negative and of order a(2)lna. This nonuniversal nonanalytic structure indicates how the discrepancies between numerical calculations at finite a can be reconciled with calculations of the limit a-->0, since the linearity is apparent only for anomalously small a.     

Coherent Tunnelling in Coupled Quantum Wells under a Uniform Magnetic Field

Coherent tunnelling is studied in the framework of the effective mass approximation for an asymmetric coupled quantum well. The Hartree potential due to the electron-electron interaction is considered in our calculation. The effects of the longitudinal and transverse magnetic field on coherent tunnelling characteristics are discussed. It has been found that the external field plays an important role in modulating the electron states.