In bulk systems, molecules are routinely identified by their vibrational spectrum using Raman or infrared spectroscopy. In recent years, vibrational excitation lines have been observed in low-temperature conductance measurements on single molecule junctions and they can provide a similar means of identification. We present a method to efficiently calculate these excitation lines in weakly coupled, gateable single-molecule junctions, using a combination of ab initio density functional theory and rate equations. Our method takes transitions from excited to excited vibrational state into account by evaluating the Franck-Condon factors for an arbitrary number of vibrational quanta, and is therefore able to predict qualitatively different behaviour from calculations limited to transitions from ground state to excited vibrational state. We find that the vibrational spectrum is sensitive to the molecular contact geometry and the charge state, and that it is generally necessary to take more than one vibrational quantum into account. Quantitative comparison to previously reported measurements on pi-conjugated molecules reveals that our method is able to characterize the vibrational excitations and can be used to identify single molecules in a junction. The method is computationally feasible on commodity hardware.
We analyze various limits of vibrationally coupled resonant electron transport in single-molecule junctions. Based on a master equation approach, we discuss analytic and numerical results for junctions under a high bias voltage or weak electronic-vibrational coupling. It is shown that in these limits the vibrational excitation of the molecular bridge increases indefinitely, i.e. the junction exhibits a vibrational instability. Moreover, our analysis provides analytic results for the vibrational distribution function and reveals that these vibrational instabilities are related to electron-hole …
We observe large, reversible, bias driven changes in the vibrational energies of PCBM, based on simultaneous transport and surface-enhanced Raman spectroscopy (SERS) measurements on PCBM-gold junctions. A combination of linear and quadratic shifts in vibrational energies with voltage is analyzed and compared with similar measurements involving C60-gold junctions. A theoretical model based on density functional theory (DFT) calculations suggests that both a vibrational Stark effect and bias-induced charging of the junction contribute to the shifts in vibrational energies. In the PCBM case, a li…
The influence of multiple vibrational modes on current fluctuations in electron transport through single-molecule junctions is investigated. Our analysis is based on a generic model of a molecular junction, which comprises a single electronic state on the molecular bridge coupled to multiple vibrational modes and fermionic leads, and employs a master equation approach. The results reveal that in molecular junctions with multiple vibrational modes already weak to moderate electronic-vibrational coupling may result in high noise levels, especially at the onset of resonant transport, in accordanc…
The interaction between electronic and vibrational degrees of freedom in single-molecule junctions may result from the dependence of the electronic energies or the electronic states of the molecular bridge on the nuclear displacement. The latter mechanism leads to a direct coupling between different electronic states and is referred to as nonadiabatic electronic-vibrational coupling. Employing a perturbative nonequilibrium Green's function approach, we study the influence of nonadiabatic electronic-vibrational coupling in model molecular junctions. Thereby we distinguish between systems with w…
In a nanoscale molecular junction at finite bias voltage,the intra-molecular distribution of vibrational energy can strongly deviate from the thermal equilibrium distribution and specific vibrational modes can be selectively excited in a controllable way,regardless of the corresponding mode frequency. This is demonstrated for generic models of asymmetric molecular junctions with localized electronic states, employing a master equation as well as a nonequilibrium Green's function approach. It is shown that the applied bias voltage controls the excitation of specific vibrational modes coupled to…
In nanoscale junctions the interaction between charge carriers and the local vibrations results in renormalization, damping and heating of the vibrational modes. We here formulate a nonequilibrium Green's functions based theory to describe such effects. Studying a generic junction model with an off-resonant electronic level, we find a strong bias dependence of the frequency renormalization and vibrational damping accompanied by pronounced nonlinear vibrational heating in junctions with intermediate values of the coupling to the leads. Combining our theory with ab-initio calculations we further…
Single-molecule junctions are found to show anomalous spikes in dI/dV spectra. The position in energy of the spikes are related to local vibration mode energies. A model of vibrationally induced two-level systems reproduces the data very well. This mechanism is expected to be quite general for single-molecule junctions. It acts as an intrinsic amplification mechanism for local vibration mode features and may be exploited as a new spectroscopic tool.
We analyze quantum interference and decoherence effects in single-molecule junctions both experimentally and theoretically by means of the mechanically controlled break junction technique and density-functional theory. We consider the case where interference is provided by overlapping quasi-degenerate states. Decoherence mechanisms arising from the electronic-vibrational coupling strongly affect the electrical current flowing through a single-molecule contact and can be controlled by temperature variation. Our findings underline the all-important relevance of vibrations for understanding charg…
Vibrationally coupled electron transport through single-molecule junctions is considered. Reviewing our recent theoretical work, we show that electron-hole pair creation processes represent the key to understand the vibrational excitation characteristic of a single-molecule contact. Moreover, these processes can lead to a number of interesting transport phenomena such as, for example, negative differential resistance, rectification, mode-selective vibrational excitation and a pronounced temperature dependence of the electrical current. Thus, electron-hole pair creation processes are crucial to…
We investigate the role of electronic-vibrational coupling in resonant electron transport through single-molecule junctions, taking into account that the corresponding coupling strengths may depend on the charge and excitation state of the molecular bridge. In the presence of multiple electronic states, this requires to extend the commonly used model and include vibrationally dependent electron-electron interaction. We use Born-Markov master equation methods and consider selected models to exemplify the effect of the additional interaction on the transport characteristics of a single-molecule …
Vibrational modes of molecules are fundamental properties determined by intramolecular bonding, atomic masses, and molecular geometry, and often serve as important channels for dissipation in nanoscale processes. Although single-molecule junctions have been employed to manipulate electronic structure and related functional properties of molecules, electrical control of vibrational mode energies has remained elusive. Here we use simultaneous transport and surface-enhanced Raman spectroscopy measurements to demonstrate large, reversible, voltage-driven shifts of vibrational mode energies of C60 …
We present a detailed study of the non-linear thermoelectric properties of a molecular junction, represented by a dissipative Anderson-Holstein model. A single orbital level with strong Coulomb interaction is coupled to a localized vibrational mode and we account for both electron and phonon exchange with both electrodes, investigating how these contribute to the heat and charge transport. We calculate the efficiency and power output of the device operated as a heat to electric power converter and identify the optimal operating conditions, which are found to be qualitatively changed by the pre…
We investigate the interplay of quantum interference effects and electronic-vibrational coupling in electron transport through single-molecule junctions, employing a nonequilibrium Green's function approach. Our findings show that inelastic processes lead, in general, to a quenching of quantum interference effects. This quenching is more pronounced for increasing bias voltages and levels of vibrational excitation. As a result of this vibrationally induced decoherence, vibrational signatures in the transport characteristics of a molecular contact may strongly deviate from a simple Franck-Condon…
We investigate the nonequilibrium population of a vibrational mode in the steady state of a biased molecular junction, using a rate equation approach. We focus on the limit of weak electronic-vibrational coupling and show that, in the resonant transport regime and for sufficiently high bias voltages, the level of vibrational excitation increases with decreasing coupling strength, assuming a finite and non-zero value. An analytic behavior with respect to the electronic-vibrational coupling strength is only observed if the influence of environmental degrees of freedom is explicitly taken into ac…
We theoretically examine the effect of the coupling of the transport electrons to a vibrational mode of the molecule on the ac linear-response conductance of molecular junctions. Representing the molecule by a single electronic state, we find that at very low temperatures the frequency-dependent conductance is mainly enhanced (suppressed) by the electron-vibration interaction when the chemical potential is below (above) the energy of that state. The vertex corrections of the electron-vibration interaction induce an additional peak structure in the conductance, which can be observed by tuning t…
We consider an open molecular junction in which electrons are coupled to multi-mode phonons. Using a master equation approach for equilibrated phonons, we show that nearly-degenerate phonon modes can be mapped into an effective single mode, which can result in a noticeable reduction of computational costs. As an illustration, we apply our theory to the OPV-5 molecule and show that we can obtain the consistent conductance spectra while reducing the computational time by orders of magnitude
Vibronic effects in resonant electron transport through single-molecule junctions are analyzed. The study is based on generic models for molecular junctions, which include electronic states on the molecular bridge that are vibrationally coupled and exhibit Coulomb interaction. The transport calculations employ a master equation approach. The results, obtained for a series of models with increasing complexity, show a multitude of interesting transport phenomena, including vibrational excitation, rectification, negative differential resistance (NDR) as well as local cooling. While some of these …
We study vibrational statistics in current-carrying model molecular junctions using master equation approach. Especially, we concentrate on the validity of using an effective temperature $T_{\rm eff}$ to characterize the nonequilibrium steady state of a vibrational mode. We identify cases where a single $T_{\rm eff}$ can not fully describe one vibrational state. In such cases, the probability distribution among different vibrational states does not follow the Boltzmann type. Consequently, the actual entropy (free energy) of the vibrational mode is lower (higher) than the corresponding thermal …
We report on inelastic electron tunneling spectroscopy measurements carried out on single molecules incorporated into a mechanically controllable break-junction of Au and Pt electrodes at low temperature. Here we establish a correlation between the molecular conformation and conduction properties of a single-molecule junction. We demonstrate that the conductance through single molecules crucially depends on the contact material and configuration by virtue of their mechanical and electrical properties. Our findings prove that the charge transport via single molecules can be manipulated by varyi…
By using a propagation scheme for current matrices and an auxiliary mode expansion method, we investigate the transient dynamics of a single molecular junction coupled with a vibrational mode. Our approach is based on the Anderson-Holstein model and the dressed tunneling approximation for the electronic self-energy in the polaronic regime. The time-dependent currents after a sudden switching on the tunneling to leads and an abrupt upward step bias pulse are calculated. We show that the strong electron-phonon interaction greatly influences the nonlinear response properties of the system, and gi…
We show that individual vibrational modes in single-molecule junctions with asymmetric molecule-lead coupling can be selectively excited by applying an external bias voltage. Thereby, a non-statistical distribution of vibrational energy can be generated, that is, a mode with a high frequency can be stronger excited than a mode with a lower frequency. This is of particular interest in the context of mode-selective chemistry, where one aims to break specific (not necessarily the weakest) chemical bond in a molecule. Such mode-selective vibrational excitation is demonstrated for two generic model…
Vibrational nonequilibrium effects in charge transport through single-molecule junctions are investigated. Focusing on molecular bridges with multiple electronic states, it is shown that electronic-vibrational coupling triggers a variety of vibronic emission and absorption processes, which influence the conductance properties and mechanical stability of single-molecule junctions profoundly. Employing a master equation and a nonequilibrium Green's function approach, these processes are analyzed in detail for a generic model of a molecular junction and for benzenedibutanethiolate bound to gold e…
Understanding and controlling the flow of heat is a major challenge in nanoelectronics. When a junction is driven out of equilibrium by light or the flow of electric charge, the vibrational and electronic degrees of freedom are, in general, no longer described by a single temperature[1-6]. Moreover, characterizing the steady-state vibrational and electronic distributions {\it in situ} is extremely challenging. Here we show that surface-enhanced Raman emission may be used to determine the effective temperatures for both the vibrational modes and the flowing electrons in a biased metallic nanosc…
Phase relaxation of electrons transferring through an electromechanical transistor is studied using the Aharonov-Bohm interferometer. With the approach of quantum master equation, the phase properties of an electron are numerically analyzed based on the interference fringes. Coherence of electron is partially destroyed by its scattering on excited levels of the local nanomechanical oscillator. Transmission amplitudes with respect to two adjacent mechanical vibrational levels have a phase difference of $π$. The character of phase shift by $π$ depends on the oscillator frequency only and is robu…
The interaction between electronic and nuclear degrees of freedom in single-molecule junctions is an essential mechanism, which may result in the current-induced rupture of chemical bonds. As such, it is fundamental for the stability of molecular junctions and for the applicability of molecular electronic devices. In this publication, we study current-induced bond rupture in molecular junctions using a numerically exact scheme, which is based on the hierarchical quantum master equation (HQME) method in combination with a discrete variable representation for the nuclear degrees of freedom. Empl…
Quantum interference effects and decoherence mechanisms in single-molecule junctions are analyzed employing a nonequilibrium Green's function approach. Electrons tunneling through quasi-degenerate states of a nanoscale molecular junction exhibit interference effects. We show that electronic-vibrational coupling, inherent to any molecular junction, strongly quenches such interference effects. As a result, the electrical current can be significantly larger than without electronic-vibrational coupling. The analysis reveals that the quenching of quantum interference is particularly pronounced if t…
The multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) theory within second quantization representation of the Fock space, a novel numerically exact methodology to treat many-body quantum dynamics for systems containing identical particles, is applied to study the effect of vibrational motion on electron transport in a generic model for single-molecule junctions. The results demonstrate the importance of electronic-vibrational coupling for the transport characteristics. For situations where the energy of the bridge state is located close to the Fermi energy, the simulations show t…
Electronic devices composed of single molecules constitute the ultimate limit in the continued downscaling of electronic components. A key challenge for single-molecule electronics is to control the temperature of these junctions. Controlling heating and cooling effects in individual vibrational modes, can in principle, be utilized to increase stability of single-molecule junctions under bias, to pump energy into particular vibrational modes to perform current-induced reactions or to increase the resolution in inelastic electron tunneling spectroscopy by controlling the life-times of phonons i…
The influence of vibrational motion on electron conduction through single molecules bound to metal electrodes is investigated employing first-principles electronic-structure calculations and projection-operator Green's function methods. Considering molecular junctions where a central phenyl ring is coupled via (alkane)thiol-bridges to gold electrodes, it is shown that -- depending on the distance between the electronic $π$-system and the metal -- electronic-vibrational coupling may result in pronounced vibrational substructures in the transmittance, a significantly reduced current as well as a…
We propose a mechanism which allows one to control the transmission of single electrons through a molecular junction. The principle utilizes the emergence of transmission sidebands when molecular vibrational modes are coupled to the electronic state mediating the transmission. We will show that if a molecule-metal junction is biased just below a molecular resonance one may induce the transmission of a single electron by externally exciting a vibrational mode of the molecule. The analysis is quite general but requires that the molecular orbital does not hybridize strongly with the metallic stat…
We develop a classical mapping approach suitable to describe vibrationally coupled charge transport in molecular junctions based on the Cartesian mapping for many-electron systems [J. Chem. Phys. 137, 154107 (2012)]. To properly describe vibrational quantum effects in the transport characteristics, we introduce a simple transformation rewriting the Hamiltonian in terms of occupation numbers and use a binning function to facilitate quantization. The approach provides accurate results for the nonequilibrium Holstein model for a range of bias voltages, vibrational frequencies and temperatures. It…
Problems of heat transport are ubiquitous to various technologies such as power generation, cooling, electronics, and thermoelectrics. In this paper we advocate for the application of the quantum self-consistent reservoir method, which is based on the generalized quantum Langevin equation, to study phononic thermal conduction in molecular junctions. The method emulates phonon-phonon scattering processes while taking into account quantum effects and far-from-equilibrium (large temperature difference) conditions. We test the applicability of the method by simulating the thermal conductance of mo…
The role of multimode vibrational dynamics in electron transport through single molecule junctions is investigated. The study is based on a generic model, which describes charge transport through a single molecule that is attached to metal leads. To address vibrationally-coupled electron transport, we employ a nonequilibrium Green's function approach that extends a method recently proposed by Galperin et al. [Phys. Rev. B 73, 045314 (2006)] to multiple vibrational modes. The methodology is applied to two systems: a generic model with two vibrational degrees of freedom and benzenedibutanethiola…
We study current-induced vibrational cooling, heating, and instability in a donor-acceptor rectifying molecular junction using a full counting statistics approach. In our model, electron-hole pair excitations are coupled to a given molecular vibrational mode which is either harmonic or highly anharmonic. This mode may be further coupled to a dissipative thermal environment. Adopting a master equation approach, we confirm the charge and heat exchange fluctuation theorem in the steady-state limit, for both harmonic and anharmonic models. Using simple analytical expressions, we calculate the char…
We review studies of vibrational energy transfer in a molecular junction geometry, consisting of a molecule bridging two heat reservoirs, solids or large chemical compounds. This setup is of interest for applications in molecular electronics, thermoelectrics, and nanophononics, and for addressing basic questions in the theory of classical and quantum transport. Calculations show that system size, disorder, structure, dimensionality, internal anharmonicities, contact interaction, and quantum coherent effects, are factors that interplay to determine the predominant mechanism (ballistic/diffusive…
The interplay between nuclear and electronic degrees of freedom strongly influences molecular charge transport. Herein, we report on transport through a porphyrin dimer molecule, weakly coupled to graphene electrodes, that displays sequential tunneling within the Coulomb-blockade regime. The sequential transport is initiated by current-induced phonon absorption and proceeds by rapid sequential transport via a non-equilibrium vibrational distribution. We demonstrate this is possible only when the vibrational dissipation is slow relative to sequential tunneling rates, and obtain a lower bound fo…
We unravel the critical role of vibrational mode softening in single-molecule electronic devices at high bias. Our theoretical analysis is carried out with a minimal model for molecular junctions, with mode softening arising due to quadratic electron-vibration couplings, and by developing a mean-field approach. We discover that the negative sign of the quadratic electron-vibration coupling coefficient can realize at high voltage a sharp negative differential resistance (NDR) effect with a large peak-to-valley ratio. Calculated current-voltage characteristics, obtained based on ab initio parame…
Introduction (2) Experimental background: Test beds (8) Theoretical approaches: A microscopic model(10) The electron-phonon coupling(14)Time and energy scales(15) Theoretical methods(19)Numerical calculations(28) Incoherent vs. coherent transport (28) Inelastic tunneling spectroscopy: Experimental background(31) Theoretical considerations:the weak coupling limit(36) Theoretical considerations: moderately strong coupling(41)Comparison of approximation schemes(48)Asymmetry in IETS(51)The origin of dips in IETS signals(53)Computational approaches (56) Effects of electron-electron(e-e)interactions…
The realization of molecular-based electronic devices depends to a large extent on the ability to mechanically stabilize the involved molecular bonds, while making use of efficient resonant charge transport through the device. Resonant charge transport can induce vibrational instability of molecular bonds, leading to bond rupture under a bias voltage. In this work, we go beyond the wide-band approximation in order to study the phenomenon of vibrational instability in single molecule junctions and show that the energy-dependence of realistic molecule-leads couplings affects the mechanical stabi…
Vibrationally inelastic electron transport through a molecular bridge that is connected to two leads is investigated. The study is based on a generic model of vibrational excitation in resonant transmission of electrons through a molecular junction. Employing methods from electron-molecule scattering theory, the transmittance through the molecular bridge can be evaluated numerically exactly. The current through the junction is obtained approximately using a Landauer-type formula. Considering different parameter regimes, which include both the case of a molecular bridge that is weakly coupled t…
We propose a very accurate computational scheme for the dynamics of a classical oscillator coupled to a molecular junction driven by a finite bias, including the finite mass effect. We focus on two minimal models for the molecular junction: Anderson-Holstein (AH) and two-site Su-Schrieffer-Heeger (SSH) models. As concerns the oscillator dynamics, we are able to recover a Langevin equation confirming what found by other authors with different approaches and assessing that quantum effects come from the electronic subsystem only. Solving numerically the stochastic equation, we study the position …
We expand iterative numerically-exact influence functional path-integral tools and present a method capable of following the nonequilibrium time evolution of subsystems coupled to multiple bosonic and fermionic reservoirs simultaneously. Using this method, we study the real-time dynamics of charge transfer and vibrational mode excitation in an electron conducting molecular junction. We focus on nonequilibrium vibrational effects, particularly, the development of vibrational instability in a current-rectifying junction. Our simulations are performed by assuming large molecular vibrational anhar…
The interaction between electronic and vibrational degrees of freedom is an important mechanism in nonequilibrium charge transport through molecular nanojunctions. While adiabatic polaron-type coupling has been studied in great detail, new transport phenomena arise for nonadiabatic coupling scenarios corresponding to a breakdown of the Born-Oppenheimer approximation. Employing the numerically exact hierarchical equations of motion approach, we analyze the effect of nonadiabatic electronic-vibrational coupling on electron transport in molecular junctions considering a series of models with incr…
Within a generic model we analyze the Stokes linewidth in surface enhanced Raman scattering (SERS) from molecules embedded as bridges in molecular junctions. We identify four main contributions to the off-resonant Stokes signal and show that under zero voltage bias (a situation pertaining also to standard SERS experiments) and at low bias junctions only one of these contributions is pronounced. The linewidth of this component is determined by the molecular vibrational relaxation rate, which is dominated by interactions with the essentially bosonic thermal environment when the relevant molecula…
We have combined ab initio quantum chemistry calculations with a rate-equation formalism to analyze electroluminescence spectra in single-molecule junctions, measured recently by several groups in Scanning Tunneling Microscope setups. In our method, the entire vibrational spectrum is taken into account. Our method leads to good quantitative agreement with both the spectroscopic features of the measurements and their current and voltage dependence. Moreover, our method is able to explain several previously unexplained features. We show that in general, the quantum yield is expected to be suppre…
Understanding current-induced bond rupture in single-molecule junctions is both of fundamental interest and a prerequisite for the design of molecular junctions, which are stable at higher bias voltages. In this work, we use a fully quantum mechanical method based on the hierarchical quantum master equation approach to analyze the dissociation mechanisms in molecular junctions. Considering a wide range of transport regimes, from off-resonant to resonant, non-adiabatic to adiabatic transport, and weak to strong vibronic coupling, our systematic study identifies three dissociation mechanisms. In…
The coupling of the charge carriers passing through a molecule bridging two bulky conductors with local vibrational modes of the molecule, gives rise to distinct features in the electronic transport properties on one hand, and to nonequilibrium features in the vibrations' properties, e.g., their population, on the other. Here we explore theoretically a generic model for a molecular junction biased by an arbitrary dc voltage in the weak-coupling regime. We analyze the signature of the electron-vibration interaction on the full-counting statistics of the current fluctuations (i.e., the cumulant …
We study inelastic vibration-assisted charge transfer effects in two-site molecular junctions, focusing on signatures of vibrational anharmonicity on the electrical characteristics and the thermoelectric response of the junction. We consider three types of oscillators: harmonic, anharmonic-Morse allowing bond dissociation, and harmonic-quartic, mimicking a confinement potential. Using a quantum master equation method which is perturbative in the electron-vibration interaction we find that the (inelastic) electrical and thermal conductances can be largely affected by the nature of the vibration…
The evolution of electron conductance in the presence of inelastic effects is studied as an atomic gold contact is formed evolving from a low-conductance regime (tunneling) to a high-conductance regime (contact). In order to characterize each regime, we perform density functional theory (DFT) calculations to study the geometric and electronic structures, together with the strength of the atomic bonds and the associated vibrational frequencies. The conductance is calculated by first evaluating the transmission of electrons through the system, and secondly by calculating the conductance change d…
Electronic-vibrational coupling in single-molecule junctions may result in current-induced bond rupture and is thus an important mechanism for the stability of molecular junctions. We use the hierarchical quantum master equation (HQME) method in combination with the quasi-classical Ehrenfest approach for the nuclear degrees of freedom to simulate current-induced bond rupture in single-molecule junctions. Employing generic models for molecular junctions with dissociative nuclear potentials, we analyze the underlying mechanisms. In particular, we investigate the dependence of the dissociation pr…
We study charge and energy transfer in two-site molecular electronic junctions in which electron transport is assisted by a vibrational mode. To understand the role of mode harmonicity/anharmonicity on transport behavior, we consider two limiting situations: (i) the mode is assumed harmonic, (ii) we truncate the mode spectrum to include only two vibrational levels, to represent an anharmonic molecular mode. Based on the models' cumulant generating functions, we analyze the linear-response and nonlinear performance of these junctions and demonstrate that while the electrical and thermal conduct…
Electronic current flowing in a molecular electronic junction dissipates significant amounts of energy to vibrational degrees of freedom, straining and rupturing chemical bonds and often quickly destroying the integrity of the molecular device. The infamous mechanical instability of molecular electronic junctions critically limits performance, lifespan, and raises questions as to the technological viability of single-molecule electronics. Here we propose a practical scheme for cooling the molecular vibrational temperature via application of an AC voltage over a large, static operational DC vol…
In the present work, we theoretically analyze the steady-state thermoelectric transport through a single-molecule junction with a vibrating bridge. Thermally induced charge current in the system is explored using a nonequilibrium Green's functions formalism. We study combined effects of Coulomb interactions between charge carriers on the bridge and electron-phonon interactions on the thermocurrent beyond the linear response regime. It is shown that electron-vibron interactions may significantly affect both magnitude and direction of the thermocurrent, and vibrational signatures may appear.
We study the conductance of three or more semi-infinite wires which meet at a junction. The electrons in the wires are taken to interact weakly with each other through a short-range density-density interaction, and they encounter a general scattering matrix at the junction. We derive the renormalization group equations satisfied by the S-matrix, and we identify its fixed points and their stabilities. The conductance between any pair of wires is then studied as a function of physical parameters such as temperature. We discuss the possibility of observing the effects of junctions in present day …
