Simple dynamical models can produce intricate behaviors in large networks. These behaviors can often be observed in a wide variety of physical systems captured by the network of interactions. Here we describe a phenomenon where the increase of dimensions self-consistently generates a force field due to dynamical instabilities. This can be understood as an unstable ("rumbling") tunneling mechanism between minima in an effective potential. We dub this collective and nonperturbative effect a "Lyapunov force" which steers the system towards the global minimum of the potential function, even if the full system has a constellation of equilibrium points growing exponentially with the system size. The system we study has a simple mapping to a flow network, equivalent to current-driven memristors. The mechanism is appealing for its physical relevance in nanoscale physics, and to possible applications in optimization, novel Monte Carlo schemes and machine learning.
Based on the dielectric continuum model, we calculated the phonon assisted tunneling (PAT) current of general double barrier resonant tunneling structures (DBRTSs) including both symmetric and antisymmetric ones. The results indicate that the four higher frequency interface phonon modes (especially the one which peaks at either interface of the emitter barrier) dominate the PAT processes, which increase the valley current and decrease the PVR of the DBRTSs. We show that an asymmetric structure can lead to improved performance.
We investigate the conductance and zero-frequency shot noise of interacting, multi-level quantum dots coupled to leads. We observe that co-tunneling assisted sequential tunneling (CAST) processes play a dominant role in the transition region from Coulomb blockade to sequential tunneling. We analyze for intermediate coupling strength the dependence of the conductance due to CAST processes on temperature, coupling constant, and gate voltage. Remarkably, the width of the CAST transport feature scales only with temperature, but not with the coupling constant. While the onset of inelastic co-tunnel…
Magnetization measurements of a truly axial symmetry Mn12-tBuAc molecular nanomagnet with a spin ground state of S = 10 show resonance tunneling. This compound has the same magnetic anisotropy as Mn12-Ac but the molecules are better isolated and the crystals have less disorder and a higher symmetry. Hysteresis loop measurements at several temperatures reveal a well-resolved step fine-structure which is due to level crossings of excited states. All step positions can be modeled by a simple spin Hamiltonian. The crossover between thermally assisted and pure quantum tunneling can be investigated …
Tunneling between two classically disconnected regular regions can be strongly affected by the presence of a chaotic sea in between. This phenomenon, known as chaos-assisted tunneling, gives rise to large fluctuations of the tunneling rate. Here we study chaos-assisted tunneling in the presence of Anderson localization effects in the chaotic sea. Our results show that the standard tunneling rate distribution is strongly modified by localization, going from the known Cauchy distribution in the ergodic regime to a log-normal distribution in the strongly localized case. We develop an analytical s…
We present time-resolved measurements of spin-flip photon-assisted tunneling and spin-flip relaxation in a doubly occupied double quantum dot. The photon-assisted excitation rate as a function of magnetic field indicates that spin-orbit coupling is the dominant mechanism behind the spin-flip under the present conditions. We are able to extract the resulting effective `spin-flip tunneling' energy, which is found to be three orders of magnitude smaller than the regular spin-conserving tunneling energy. We also measure the relaxation and dephasing times of a qubit formed out of two two-electron s…
A theory for boson-assisted tunneling via randomly distributed resonant states in a layered metals is developed. As particular examples, we consider the electron-phonon interaction and the interaction between localized and conduction electrons. The theory is applied to explain a non-monotonic variation of the out-plane resistivity with temperature observed in quasi-two-dimensional metals.
We review experiments on single electron transport through single quantum dots in the presence of a microwave signal. In the case of a small dot with well-resolved discrete energy states, the applied high-frequency signal allows for inelastic tunnel events that involve the exchange of photons with the microwave field. These photon assisted tunneling (PAT) processes give rise to sideband resonances in addition to the main resonance. Photon absorption can also lead to tunneling via excited states instead of tunneling via the ground state of the quantum dot.
We present the first experimental observation of resonance-assisted tunneling, a wave phenomenon, where regular-to-chaotic tunneling is strongly enhanced by the presence of a classical nonlinear resonance chain. For this we use a microwave cavity made of oxygen free copper with the shape of a desymmetrized cosine billiard designed with a large nonlinear resonance chain in the regular region. It is opened in a region, where only chaotic dynamics takes place, such that the tunneling rate of a regular mode to the chaotic region increases the line width of the mode. Resonance-assisted tunneling is…
The transfer Hamiltonian tunneling current is derived in a time-dependent density matrix formulation and is used to examine photon-assisted tunneling. Bardeen's tunneling expression arises as the result of first order perturbation theory in a mean-field expansion of the density matrix. Photon-assisted tunneling from confined electromagnetic fields in the forbidden barrier region occurs due to time-varying polarization and wavefunction overlap in the gap, which leads to a non-zero tunneling current in asymmetric device structures, even in an unbiased state. The photon energy is seen to act as a…
We report on electron transport through an artificial molecule formed by two tunnel coupled quantum dots, which are laterally confined in a two-dimensional electron system of an Al$_x$Ga$_{1-x}$As/GaAs heterostructure. Coherent molecular states in the coupled dots are probed by photon-assisted tunneling (PAT). Above 10 GHz, we observe clear PAT as a result of the resonance between the microwave photons and the molecular states. Below 8 GHz, a pronounced superposition of phonon- and photon-assisted tunneling is observed. Coherent superposition of molecular states persists under excitation of ac…
Josephson oscillations generated by a SQUID were used to measure photon assisted tunneling in a superconducting single electron tunneling (SET) transistor. The SQUID was fabricated only a few microns from the SET transistor. The close proximity of the generator to the SET transistor allowed for a continuous sweep of the excitation frequency from 10 GHz to 190 GHz. The amplitude of the Josephson oscillations were tuned by adjusting the flux through the SQUID loop while the frequency was independently adjusted by controlling the bias voltage across the SQUID. Current peaks which could be attribu…
We report both photon- and phonon-assisted tunneling transitions in a linear array of three quantum dots, which can only be understood by considering the full three-dimensionality of the charge stability diagram. Such tunneling transitions potentially contribute to leakage of qubits defined in this system. A detailed understanding of these transitions is important as they become more abundant and complex to analyze as quantum dot arrays are scaled up.
We measure electron tunneling in single-molecule transistors made from C_{140}, a molecule with a mass-spring-mass geometry chosen as a model system to study electron-vibration coupling. We observe vibration-assisted tunneling at an energy corresponding to the stretching mode of C_{140}. Molecular modeling provides explanations for why this mode couples more strongly to electron tunneling than the other internal modes of the molecule. We make comparisons between the observed tunneling rates and those expected from the Franck-Condon model.
We present a TCAD compatible multiscale model of phonon-assisted band-to-band tunneling (BTBT) in semiconductors, that incorporates the non-parabolic nature of complex bands within the bandgap of the material. This model is shown capture the measured current-voltage data in silicon, for current transport along the $[100]$, $[110]$ and $[111]$ directions. Our model will be useful to predict band-to-band tunneling phenomena to quantify on and off currents in Tunnel FETs and in small geometry MOSFETs and FINFETs.
We perform the investigations of the resonant tunneling via impurities embedded in the AlAs barrier of a single GaAs/AlGaAs heterostructure. In the $I(V)$ characteristics measured at 30mK, the contribution of individual donors is resolved and the fingerprints of phonon assistance in the tunneling process are seen. The latter is confirmed by detailed analysis of the tunneling rates and the modeling of the resonant tunneling contribution to the current. Moreover, fluctuations of the local structure of the DOS (LDOS) and Fermi edge singularities are observed.
Tunneling across superconducting junctions proceeds by a rich variety of processes, which transfer single electrons, Cooper pairs, or even larger numbers of electrons by multiple Andreev reflections. Photon-assisted tunneling combined with the venerable Tien-Gordon model has long been a powerful tool to identify tunneling processes between superconductors. Here, we probe superconducting tunnel junctions including an impurity-induced Yu-Shiba-Rusinov (YSR) state by exposing a scanning tunneling microscope with a superconducting tip to microwave radiation. We find that a simple Tien-Gordon descr…
We develop a theory of electron tunneling accompanied by carrier-carrier scattering in graphene - insulator - graphene heterostructures. Due to the dynamic screening of Coulomb interaction, the scattering-aided tunneling is resonantly enhanced if the transferred energy and momentum correspond to those of surface plasmons. We reveal the possible experimental manifestations of such plasmon-assisted tunneling in current-voltage curves and plasmon emission spectra of graphene-based tunnel junctions. We find that inelastic current and plasmon emission rates have sharp peaks at voltages providing eq…
Two main mechanisms dictate the tunneling process in a double quantum dot: overlap of excited wave functions, effectively described as a tunneling rate, and phonon-assisted tunneling. In this paper, we study different regimes of tunneling that arise from the competition between these two mechanisms in a double quantum dot molecule immersed in a unimodal optical cavity. We show how such regimes affect the mean number of excitations in each quantum dot and in the cavity, the spectroscopic resolution and emission peaks of the photoluminescence spectrum, and the second-order coherence function whi…
A theoretical and numerical investigations of the quantum tunneling of the domain walls in ferromagnets and weak ferromagnets was performed taking into account the interaction between walls and thermal excitations of a crystal. The numerical method for calculations of the probability of a thermally stimulated quantum depinning as the function of temperature at the wide temperature range has been evolved.
Recently, it has been shown that magnetic tunnel junctions with thin MgO tunnel barriers exhibit extraordinarily high tunneling magnetoresistance (TMR) values at room temperature1, 2. However, the physics of spin dependent tunneling through MgO barriers is only beginning to be unravelled. Using planar magnetic tunnel junctions in which ultra-thin layers of magnetic metals are deposited in the middle of a MgO tunnel barrier here we demonstrate that the TMR is strongly modified when these layers are discontinuous and composed of small pancake shaped nanodots. At low temperatures, in the Coulomb …
Phonon-assisted tunneling rates are evaluated for a well isolated double dot system defined in a GaAs semiconductor heterostructure of finite thickness. A separable model for the confining potential allows accurate determinations of doublet electron wavefunctions and energies. It is found that at small doublet energies the piezoelectric rates due to flexural modes give the dominant contribution. For small slab thicknesses the predicted rates are up to two orders of magnitude higher than for very thick slabs.
Band-to-band tunneling (BTBT) devices have recently gained a lot of interest due to their potential for reducing power dissipation in integrated circuits. We have performed extensive simulations for the BTBT operation of carbon nanotube metal-oxide-semiconductor field-effect transistors (CNT-MOSFETs) using the non-equilibrium Green's functions formalism for both ballistic and dissipative quantum transport. In comparison with recently reported experimental data (Y. Lu et al, J. Am. Chem. Soc., v. 128, p. 3518-3519, 2006), we have obtained strong evidence that BTBT in CNT-MOSFETs is dominated by…
We focus on a recently experimentally realized scenario of normal-metal-insulator-superconductor tunnel junctions coupled to a superconducting resonator. We develop a first-principles theory to describe the effect of photon-assisted electron tunneling on the quantum state of the resonator. Our results are in very good quantitative agreement with the previous experiments on refrigeration and heating of the resonator using the photon-assisted tunneling, thus providing a stringent verification of the developed theory. Importantly, our results provide simple analytical estimates of the voltage-tun…
Injection of spins into semiconductors is essential for the integration of the spin functionality into conventional electronics. Insulating layers are often inserted between ferromagnetic metals and semiconductors for obtaining an efficient spin injection, and it is therefore crucial to distinguish between signatures of electrical spin injection and impurity-driven effects in the tunnel barrier. Here we demonstrate an impurity-assisted tunneling magnetoresistance effect in nonmagnetic-insulator-nonmagnetic and ferromagnetic-insulator-nonmagnetic tunnel barriers. In both cases, the effect refle…
The low temperature spin dynamics of a Fe8 Single-Molecule Magnet was studied under circularly polarized electromagnetic radiation allowing us to establish clearly photon-assisted tunneling. This effect, while linear at low power, becomes highly non-linear above a relatively low power threshold. This non-linearity is attributed to the nature of the coupling of the sample to the thermostat.These results are of great importance if such systems are to be used as quantum computers.
The electric conductance of a strip of undoped graphene increases in the presence of a disorder potential, which is smooth on atomic scales. The phenomenon is attributed to impurity-assisted resonant tunneling of massless Dirac fermions. Employing the transfer matrix approach we demonstrate the resonant character of the conductivity enhancement in the presence of a single impurity. We also calculate the two-terminal conductivity for the model with one-dimensional fluctuations of disorder potential by a mapping onto a problem of Anderson localization.
We perform photon-assisted-tunneling (PAT) experiments on a GaAs double quantum dot device under high microwave excitation power. Photon-assisted absorption of up to 14 photons is observed, when electron temperature (>1K) are far above the lattice temperature. Signatures of Landau-Zener-Stückelberg (LZS) interference are found even in this non-equilibrium PAT spectrum. In addition, the charge state relaxation time T_1~8ns measured in this out of thermal equilibrium double quantum dot is in agreement with other previous reports.
We have investigated theoretically the influence of an AC drive on heat transport in a hybrid normal metal - superconductor tunnel junction in the photon-assisted tunneling regime. We find that the useful heat flux out from the normal metal is always reduced as compared to its magnitude under the static and quasi-static drive conditions. Our results are useful to predict the operative conditions of AC driven superconducting electron refrigerators.
We propose a novel approach to the problem of a transition from quantum to classical behavior in mesoscopic spin systems. This paper is intended to demonstrate that main cause of such transitions is quantum decoherence which appear as a result of a thermal interaction between spin system and its environment.
At finite temperature we study the quantum tunneling of magnetization for a small ferromagnetic particle with the biaxial symmetry placed in a magnetic field at an arbitrary angle. We present numerical WKB exponent below the crossover temperature in which the quantum tunneling is affected by the thermal activation, and the approximate form of the WKB exponent around the crossover region. The effect of quantum fluctuations on the thermal activation rate beyond the crossover regime are discussed.
The crossover between thermally assisted and pure quantum tunneling has been studied in single crystals of high spin (S=10) uniaxial molecular magnet Mn12 using micro-Hall-effect magnetometry. Magnetic hysteresis and relaxation experiments have been used to investigate the energy levels that determine the magnetization reversal as a function of magnetic field and temperature. These experiments demonstrate that the crossover occurs in a narrow (0.1 K) or broad (1 K) temperature interval depending on the magnitude of the field transverse to the anisotropy axis.
A physics-based compact model for silicon gate-all-around (GAA) nanowire tunneling FETs (NW-tFETs) with good accuracy has been developed by considering Phonon-Assisted Tunneling (PAT) and transition from Quantum Capacitance Limit (QCL) to Classical Limit (CL) during the device-size scaling. The impact of PAT results in the broadening of a single electron-energy level to an energy band with density-of-states (DOS) distribution of Lorentzian shape. As a consequence, the tunneling probability at the edge of tunneling window no longer changes abruptly from zero to having a finite value. By adjusti…
We analyze dephasing in single and double quantum dot systems. The decoherence is introduced by the Büttiker model with current conserving fictitious voltage leads connected to the dots. By using the non-equilibrium Green function method, we investigate the dephasing effect on the tunneling current. It is shown that a finite dephasing rate leads to observable effects. The result can be used to measure dephasing rates in quantum dots.
Electron transport through a double quantum dot system is studied with taking into account electron-phonon interaction. The Keldysh nonequilibrium Green function formalism is used to compute the current and transmission coefficient of the system. The influence of the electron-phonon interaction, interdot tunneling, and temperature on the density of states and current is analyzed. Results show that although the electron-phonon interaction results in the appearance of side peaks in the conductance at low temperatures, they are disappeared in high temperatures.
Current-voltage characteristics of suspended single-wall carbon nanotube quantum dots show a series of steps equally spaced in voltage. The energy scale of this harmonic, low-energy excitation spectrum is consistent with that of the longitudinal low-k phonon mode (stretching mode) in the nanotube. Agreement is found with a Franck-Condon-based model in which the phonon-assisted tunneling process is modeled as a coupling of electronic levels to underdamped quantum harmonic oscillators. Comparison with this model indicates a rather strong electron-phonon coupling factor of order unity.
We analyze long-range transport through an ac driven triple quantum dot with one single electron. An effective model is proposed for the analysis of photoassisted cotunnel in order to account for the virtual processes which dominate the long-range transport, which takes place at n-photon resonances between the edge quantum dots. The AC field renormalizes the inter dot hopping, modifying the levels hybridization. It results in a non trivial behavior of the current with the frequency and intensity of the external ac field.
We introduce a scheme that combines photon-assisted tunneling by a moving optical lattice with strong Hubbard interactions, and allows for the quantum simulation of paradigmatic quantum many-body models. We show that, in a certain regime, this quantum simulator yields an effective Hubbard Hamiltonian with tunable bond-charge interactions, a model studied in the context of strongly-correlated electrons. In a different regime, we show how to exploit a correlated destruction of tunneling to explore Nagaoka ferromagnetism at finite Hubbard repulsion. By changing the photon-assisted tunneling param…
We report the first experimental observation of the resonance-assisted dynamical tunneling (RADT) in the inter-mode interaction in an asymmetric-deformed microcavity. A selection rule for strong inter-mode coupling induced by RADT was observed on angular mode numbers as predicted by the RADT theory. In addition, the coupling strength was measured to be proportional to the square of the phase-space area associated with the nonlinear resonance involved in RADT. The proportionality constant was found to depend only on the nonlinear resonance, supporting the semiclassical nature of RADT.
We study the influence of the phonon environment on the electron dynamics in a doped quantum dot molecule. A non-perturbative quantum kinetic theory based on correlation expansion is used in order to describe both diagonal and off-diagonal electron-phonon couplings representing real and virtual processes with relevant acoustic phonons. We show that the relaxation is dominated by phonon-assisted electron tunneling between constituent quantum dots and occurs on a picosecond time scale. The dependence of the time evolution of the quantum dot occupation probabilities on the energy mismatch between…
Electronic transport in a carbon nanotube (CNT) metal-oxide-semiconductor field effect transistor (MOSFET) is simulated using the non-equilibrium Green's functions method with the account of electron-phonon scattering. For MOSFETs, ambipolar conduction is explained via phonon-assisted band-to-band (Landau-Zener) tunneling. In comparison to the ballistic case, we show that the phonon scattering shifts the onset of ambipolar conduction to more positive gate voltage (thereby increasing the off current). It is found that the subthreshold swing in ambipolar conduction can be made as steep as 40mV/d…
We investigate the charge fluctuations of a grain (large dot) coupled to a lead via a small quantum dot in the Kondo regime. We show that the strong entanglement of charge and spin flips in this setup can result in a stable SU(4) Kondo fixed point, which considerably smears out the Coulomb staircase behavior already in the weak tunneling limit. This behavior is robust enough to be experimentally observable.
We theoretically studied the thermally assisted spin torque switching of a synthetic free layer and showed that the switching time is minimized if the condition H_J=|H_s|/(2 alpha) is satisfied, where H_J, H_s and alpha are the coupling field of two ferromagnetic layers, the amplitude of the spin torque, and the Gilbert damping constant. We also showed that the coupling field of the synthetic free layer can be determined from the resonance frequencies of the spin-torque diode effect.
We study the effect of electron-phonon interaction on current and zero-frequency shot noise in resonant tunneling through a series triple-quantum-dot coupling to a local phonon mode by means of a nonperturbative mapping technique along with the Green function formulation. By fixing the energy difference between the first two quantum dots to be equal to phonon frequency and sweeping the level of the third quantum dot, we find a largely enhanced current spectrum due to phonon effect, and in particular we predict current peaks corresponding to phonon-absorption and -emission assisted resonant tun…
We study phonon-assisted electron tunneling in semiconductor quantum dot molecules. In particular, singlet-singlet relaxation in a two-electron doped structure is considered. The influence of Coulomb interaction is discussed via comparison with a single electron system. We find that the relaxation rate reaches similar values in the two cases but the Coulomb interaction shifts the maximum rates towards larger separations between the dots. The difference in electron-phonon interaction between deformation potential and piezoelectric coupling is investigated. We show that the phonon-induced tunnel…
We propose a theory for photon-assisted macroscopic quantum tunneling (MQT) in a stack of capacitively-coupled intrinsic Josephson junctions in which the longitudinal Josephson plasma, i.e., longitudinal collective phase oscillation modes, is excited. The scheme of energy-level quantization in the collective oscillatory states is clarified in the $N$-junction system. When the MQT occurs from the single-plasmon states excited by microwave irradiation in the multi-photon process to the uniform voltage state, our theory predicts that the escape rate is proportional to $N^2$. This result is consis…
Ultrafast photon-assisted tunneling is shown to lead to large nonlinear dynamic currents that can be used to generate dc rectified current as well as higher order harmonic response from a metasurface-coupled tunnel diode. This artificial nonlinear media can have tailored resonances in the infrared around the longitudinal optical phonon modes of various compatible restrahlen oxides leading to enhanced electric field confinement in a metal-oxide-semiconductor tunnel diode. This provides a new avenue for the examination and design of nonlinear phenomena extending the concept of nonlinear suscepti…
Pauli exclusion principle can lead to a supression of transport through double quantum dots, even if both quantum dots are connected by a resonant driving ac field. In this work, we study how this effect, known as spin blockade, can be removed by means of the influence of the driving field on the contacts to the leads.
We consider the photon-assisted tunneling (PAT) and the Landau-Zener-Stueckelberg (LZS) interference for double quantum dots induced electrostatically along a semiconducting carbon nanotube. An atomistic tight-binding approach and the time-dependent configuration interaction method are employed for description of the systems of a few confined electrons and holes. We reproduce the patterns of the LZS interference recently observed for the quantum double dots describing transport across hole-localized states. Moreover, we indicate that for charge configurations for which the ground-state is Paul…
Applying time-periodic modulations is routinely used to control and design synthetic matter in quantum-engineered settings. In lattice systems, this approach is explored to engineer band structures with non-trivial topological properties, but also to generate exotic interaction processes. A prime example is density-assisted tunneling, by which the hopping amplitude of a particle between neighboring sites explicitly depends on their respective occupations. Here, we show how density-assisted tunneling can be tailored in view of simulating the effects of strain in synthetic graphene-type systems.…
We present a theoretical analysis of the phonon-assisted relaxation in a system composed of two self-assembled vertically stacked quantum dots. We construct realistic model, which takes into account the geometry and strain distribution in the system. We calculate phonon-assisted relaxation rates between the two lowest states (in one- and two-electron cases). The relaxation rates and energy levels are studied as a function of external (axial) electric field and geometry of the structure (dot sizes). We show that the relaxation times can be as low as 1~ps but efficent relaxation occurs only for …
We study the optical rectification in a metallic tunnel junction. We consider a planar junction in a Kretschmann configuration and measure the photon-assisted tunneling under infrared illumination at $λ= 1.5\, μ\mathrm{m}$. To address the microscopic mechanism at the origin of the optical rectification, we compare the photon assisted current and the current-voltage characteristics of the junction measured on a voltage range much greater than $V_0=\frac{hc}{eλ}=0.825 \, \mathrm{V}$. The experimental results do not agree with the Tucker theory based on the exchange of energy quanta between elect…
We present electronic transport measurements of GaAs nanowires grown by catalyst-free metal-organic chemical vapor deposition. Despite the nanowires being doped with a relatively high concentration of substitutional impurities, we find them inordinately resistive. By measuring sufficiently high aspect-ratio nanowires individually in situ, we decouple the role of the contacts and show that this semi-insulating electrical behavior is the result of trap-mediated carrier transport. We observe Poole-Frenkel transport that crosses over to phonon-assisted tunneling at higher fields, with a tunneling …
We analyze quantum charge pumping in an open ring with a dot embedded in one of its arms. We show that cyclic driving of the dot levels by a \textit{single} parameter leads to a pumped current when a static magnetic flux is simultaneously applied to the ring. Based on the computation of the Floquet-Green's functions, we show that for low driving frequencies $ω_0$, the interplay between the spatial interference through the ring plus photon-assisted tunneling gives an average direct current (dc) which is proportional to $ω_0^{2}$. The direction of the pumped current can be reversed by changing t…
Based on exactly mapping of a many-body electron-phonon interaction problem onto a one-body problem, we apply the well-established nonequilibrium Green function technique to solve the time-dependent phonon-assisted tunneling at low temperature through a single-molecular quantum dot connected to two leads, which is subject to a microwave irradiation field. It is found that in the presence of the electron-phonon interaction and the microwave irradiation field, the time-average transmission and the nonlinear differential conductance display additional peaks due to pure photon absorption or emissi…
The problem of resonant transport of strongly interacting electrons through a one-dimensional single-level vibrating quantum dot is being considered. In this paper, we generalize the Komnik and Gogolin model [Phys. Rev. Lett., 90, 246403, (2003)] for the single-electron transistor with g=1/2 Luttinger liquid leads to the case of a strong electron-vibron interaction in a quantum dot. The effective transmission coefficient and differential conductance of the system has been derived for the general case of asymmetric tunnel barriers. The main result obtained is that, in the zero-temperature limit…
