Delta-T Flicker Noise Demonstrated with Molecular Junctions.

Electronic flicker noise is recognized as the most abundant noise in electronic conductors, either as an unwanted contribution or as a source of information on electron transport mechanisms and material properties. This noise is typically observed when a voltage difference is applied across a conductor or current is flowing through it. Here, we identify an unknown type of electronic flicker noise that is found when a temperature difference is applied across a nanoscale conductor in the absence of a net charge current or voltage bias. The revealed delta-T flicker noise is demonstrated in molecular junctions and characterized using quantum transport theory. This noise is expected to arise in nanoscale electronic conductors subjected to unintentional temperature gradients, where it can be a performance-limiting factor. On the positive side, delta-T flicker noise can detect temperature differences across a large variety of nanoscale conductors, down to atomic-scale junctions with no special setup requirements.

Electronic noise due to temperature differences in atomic-scale junctions.

Since the discovery a century ago1-3 of electronic thermal noise and shot noise, these forms of fundamental noise have had an enormous impact on science and technology research and applications. They can be used to probe quantum effects and thermodynamic quantities4-11, but they are also regarded as undesirable in electronic devices because they obscure the target signal. Electronic thermal noise is generated at equilibrium at finite (non-zero) temperature, whereas electronic shot noise is a non-equilibrium current noise that is generated by partial transmission and reflection (partition) of the incoming electrons8. Until now, shot noise has been stimulated by a voltage, either applied directly8 or activated by radiation12,13. Here we report measurements of a fundamental electronic noise that is generated by temperature differences across nanoscale conductors, which we term 'delta-T noise'. We experimentally demonstrate this noise in atomic and molecular junctions, and analyse it theoretically using the Landauer formalism8,14. Our findings show that delta-T noise is distinct from thermal noise and voltage-activated shot noise8. Like thermal noise, it has a purely thermal origin, but delta-T noise is generated only out of equilibrium. Delta-T noise and standard shot noise have the same partition origin, but are activated by different stimuli. We infer that delta-T noise in combination with thermal noise can be used to detect temperature differences across nanoscale conductors without the need to fabricate sophisticated local probes. Thus it can greatly facilitate the study of heat transport at the nanoscale. In the context of modern electronics, temperature differences are often generated unintentionally across electronic components. Taking into account the contribution of delta-T noise in these cases is likely to be essential for the design of efficient nanoscale electronics at the quantum limit.

Challenges in molecular dynamics simulations of heat exchange statistics.

We study heat exchange in temperature-biased metal-molecule-metal molecular junctions by employing the molecular dynamics simulator LAMMPS. Generating the nonequilibrium steady state with Langevin thermostats at the boundaries of the junction, we show that the average heat current across a gold-alkanedithiol-gold nanojunction behaves physically, with the thermal conductance value matching the literature. In contrast, the full probability distribution function for heat exchange, as generated by the simulator, violates the fundamental fluctuation symmetry for entropy production. We trace this failure back to the implementation of the thermostats and the expression used to calculate the heat exchange. To rectify this issue and produce the correct statistics, we introduce single-atom thermostats as an alternative to conventional many-atom thermostats. Once averaging heat exchange over the hot and cold thermostats, this approach successfully generates the correct probability distribution function, which we use to study the behavior of both the average heat current and its noise. We further examine the thermodynamic uncertainty relation in the molecular junction and show that it holds, albeit demonstrating nontrivial trends. Our study points to the need to carefully implement nonequilibrium molecular dynamics solvers in atomistic simulation software tools for future investigations of noise phenomena in thermal transport.

Bath-induced interactions and transient dynamics in open quantum systems at strong coupling: Effective Hamiltonian approach.

Understanding the dynamics of dissipative quantum systems, particularly beyond the weak coupling approximation, is central to various quantum applications. While numerically exact methods provide accurate solutions, they often lack the analytical insight provided by theoretical approaches. In this study, we employ the recently developed method dubbed the effective Hamiltonian theory to understand the dynamics of system-bath configurations without resorting to a perturbative description of the system-bath coupling energy. Through a combination of mapping steps and truncation, the effective Hamiltonian theory offers both analytical insights into signatures of strong couplings in open quantum systems and a straightforward path for numerical simulations. To validate the accuracy of the method, we apply it to two canonical models: a single spin immersed in a bosonic bath and two noninteracting spins in a common bath. In both cases, we study the transient regime and the steady state limit at nonzero temperature and spanning system-bath interactions from the weak to the strong regime. By comparing the results of the effective Hamiltonian theory with numerically exact simulations, we show that although the former overlooks non-Markovian features in the transient equilibration dynamics, it correctly captures non-perturbative bath-generated couplings between otherwise non-interacting spins, as observed in their synchronization dynamics and correlations. Altogether, the effective Hamiltonian theory offers a powerful approach for understanding strong coupling dynamics and thermodynamics, capturing the signatures of such interactions in both relaxation dynamics and in the steady state limit.

Quantum Flicker Noise in Atomic and Molecular Junctions.

We report on a quantum form of electronic flicker noise in nanoscale conductors that contains valuable information on quantum transport. This noise is experimentally identified in atomic and molecular junctions and theoretically analyzed by considering quantum interference due to fluctuating scatterers. Using conductance, shot-noise, and flicker-noise measurements, we show that the revealed quantum flicker noise uniquely depends on the distribution of transmission channels, a key characteristic of quantum conductors. This dependence opens the door for the application of flicker noise as a diagnostic probe for fundamental properties of quantum conductors and many-body quantum effects, a role that up to now has been performed by the experimentally less-accessible shot noise.

Long-range charge transport in homogeneous and alternating-rigidity chains.

We study the interplay of intrinsic-electronic and environmental factors in long-range charge transport across molecular chains with up to N ∼ 80 monomers. We describe the molecular electronic structure of the chain with a tight-binding Hamiltonian. Thermal effects in the form of electron decoherence and inelastic scattering are incorporated with the Landauer-Büttiker probe method. In short chains of up to ten units, we observe the crossover between coherent (tunneling, ballistic) motion and thermally-assisted conduction, with thermal effects enhancing the current beyond the quantum coherent limit. We further show that unconventional (nonmonotonic with size) transport behavior emerges when monomer-to-monomer electronic coupling is made large. In long chains, we identify a different behavior, with thermal effects suppressing the conductance below the coherent-ballistic limit. With the goal to identify a minimal model for molecular chains displaying unconventional and effective long-range transport, we simulate a modular polymer with alternating regions of high and low rigidity. Simulations show that, surprisingly, while charge correlations are significantly affected by structuring environmental conditions, reflecting charge delocalization, the electrical resistance displays an averaging effect, and it is not sensitive to this patterning. We conclude by arguing that efficient long-range charge transport requires engineering both internal electronic parameters and environmental conditions.

Quantum thermal transport beyond second order with the reaction coordinate mapping.

Standard quantum master equation techniques, such as the Redfield or Lindblad equations, are perturbative to second order in the microscopic system-reservoir coupling parameter λ. As a result, the characteristics of dissipative systems, which are beyond second order in λ, are not captured by such tools. Moreover, if the leading order in the studied effect is higher-than-quadratic in λ, a second-order description fundamentally fails even at weak coupling. Here, using the reaction coordinate (RC) quantum master equation framework, we are able to investigate and classify higher-than-second-order transport mechanisms. This technique, which relies on the redefinition of the system-environment boundary, allows for the effects of system-bath coupling to be included to high orders. We study steady-state heat current beyond second-order in two models: The generalized spin-boson model with non-commuting system-bath operators and a three-level ladder system. In the latter model, heat enters in one transition and is extracted from a different one. Crucially, we identify two transport pathways: (i) System's current, where heat conduction is mediated by transitions in the system, with the heat current scaling as jq ∝ λ2 to the lowest order in λ. (ii) Inter-bath current, with the thermal baths directly exchanging energy between them, facilitated by the bridging quantum system. To the lowest order in λ, this current scales as jq ∝ λ4. These mechanisms are uncovered and examined using numerical and analytical tools. We contend that the RC mapping brings, already at the level of the mapped Hamiltonian, much insight into transport characteristics.

Simulations of heat transport in single-molecule junctions: Investigations of the thermal diode effect.

With the objective of understanding microscopic principles governing thermal energy flow in nanojunctions, we study phononic heat transport through metal-molecule-metal junctions using classical molecular dynamics (MD) simulations. Considering a single-molecule gold-alkanedithiol-gold junction, we first focus on aspects of method development and compare two techniques for calculating thermal conductance: (i) The Reverse Nonequilibrium MD (RNEMD) method, where heat is inputted and extracted at a constant rate from opposite metals. In this case, the thermal conductance is calculated from the nonequilibrium temperature profile that is created at the junction. (ii) The Approach-to-Equilibrium MD (AEMD) method, with the thermal conductance of the junction obtained from the equilibration dynamics of the metals. In both methods, simulations of alkane chains of a growing size display an approximate length-independence of the thermal conductance, with calculated values matching computational and experimental studies. The RNEMD and AEMD methods offer different insights, and we discuss their benefits and shortcomings. Assessing the potential application of molecular junctions as thermal diodes, alkane junctions are made spatially asymmetric by modifying their contact regions with the bulk, either by using distinct endgroups or by replacing one of the Au contacts with Ag. Anharmonicity is built into the system within the molecular force-field. We find that, while the temperature profile strongly varies (compared with the gold-alkanedithiol-gold junctions) due to these structural modifications, the thermal diode effect is inconsequential in these systems-unless one goes to very large thermal biases. This finding suggests that one should seek molecules with considerable internal anharmonic effects for developing nonlinear thermal devices.

Periodically Driven Quantum Thermal Machines from Warming up to Limit Cycle.

Theoretical treatments of periodically driven quantum thermal machines (PD-QTMs) are largely focused on the limit-cycle stage of operation characterized by a periodic state of the system. Yet, this regime is not immediately accessible for experimental verification. Here, we present a general thermodynamic framework that handles the performance of PD-QTMs both before and during the limit-cycle stage of operation. It is achieved by observing that periodicity may break down at the ensemble average level, even in the limit-cycle phase. With this observation, and using conventional thermodynamic expressions for work and heat, we find that a complete description of the first law of thermodynamics for PD-QTMs requires a new contribution, which vanishes only in the limit-cycle phase under rather weak system-bath couplings. Significantly, this contribution is substantial at strong couplings even at limit cycle, thus largely affecting the behavior of the thermodynamic efficiency. We demonstrate our framework by simulating a quantum Otto engine building upon a driven resonant level model. Our results provide new insights towards a complete description of PD-QTMs, from turn-on to the limit-cycle stage and, particularly, shed light on the development of quantum thermodynamics at strong coupling.

Universal Bounds on Fluctuations in Continuous Thermal Machines.

We study bounds on ratios of fluctuations in steady-state time-reversal energy conversion devices. In the linear response regime, we prove that the relative fluctuations (precision) of the output current (power) is always lower bounded by the relative fluctuations of the input current (heat current absorbed from the hot bath). As a consequence, the ratio between the fluctuations of the output and input currents are bounded both from above and below, where the lower (upper) bound is determined by the square of the averaged efficiency (square of the Carnot efficiency) of the engine. The saturation of the lower bound is achieved in the tight-coupling limit when the determinant of the Onsager response matrix vanishes. Our analysis can be applied to different operational regimes, including engines, refrigerators, and heat pumps. We illustrate our findings in two types of continuous engines: two-terminal coherent thermoelectric junctions and three-terminal quantum absorption refrigerators. Numerical simulations in the far-from-equilibrium regime suggest that these bounds apply more broadly, beyond linear response.

Sharp Negative Differential Resistance from Vibrational Mode Softening in Molecular Junctions.

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 physical parameters for a nitro-substituted oligo(phenylene ethynylene) junction, agree very well with the measurements. Our results establish that vibrational mode softening is a crucial effect at high voltage, underlying NDR, a substantial diode effect, and the breakdown of current-carrying molecular junctions.

On the definitions and simulations of vibrational heat transport in nanojunctions.

Thermal transport through nanosystems is central to numerous processes in chemistry, material sciences, and electrical and mechanical engineering, with classical molecular dynamics as the key simulation tool. Here, we focus on thermal junctions with a molecule bridging two solids that are maintained at different temperatures. The classical steady state heat current in this system can be simulated in different ways, either at the interfaces with the solids, which are represented by thermostats, or between atoms within the conducting molecule. We show that while the latter, intramolecular definition feasibly converges to the correct limit, the molecule-thermostat interface definition is more challenging to converge to the correct result. The problem with the interface definition is demonstrated by simulating heat transport in harmonic and anharmonic one-dimensional chains illustrating unphysical effects such as thermal rectification in harmonic junctions.

Phononic heat transport in molecular junctions: Quantum effects and vibrational mismatch.

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 molecular junctions with one-dimensional molecules sandwiched between solid surfaces. Our results satisfy the expected behavior of the thermal conductance in anharmonic chains as a function of length, phonon scattering rate, and temperature, thus validating the computational scheme. Moreover, we examine the effects of vibrational mismatch between the solids' phonon spectra on the heat transfer characteristics in molecular junctions. Here, we reveal the dual role of vibrational anharmonicity: It raises the resistance of the junction due to multiple scattering processes, yet it promotes energy transport across a vibrational mismatch by enabling phonon recombination and decay processes.

Path-integral methodology and simulations of quantum thermal transport: Full counting statistics approach.

We develop and test a computational framework to study heat exchange in interacting, nonequilibrium open quantum systems. Our iterative full counting statistics path integral (iFCSPI) approach extends a previously well-established influence functional path integral method, by going beyond reduced system dynamics to provide the cumulant generating function of heat exchange. The method is straightforward; we implement it for the nonequilibrium spin boson model to calculate transient and long-time observables, focusing on the steady-state heat current flowing through the system under a temperature difference. Results are compared to perturbative treatments and demonstrate good agreement in the appropriate limits. The challenge of converging nonequilibrium quantities, currents and high order cumulants, is discussed in detail. The iFCSPI, a numerically exact technique, naturally captures strong system-bath coupling and non-Markovian effects of the environment. As such, it is a promising tool for probing fundamental questions in quantum transport and quantum thermodynamics.

Heat transfer statistics in mixed quantum-classical systems.

The modelling of quantum heat transfer processes at the nanoscale is crucial for the development of energy harvesting and molecular electronic devices. Herein, we adopt a mixed quantum-classical description of a device, in which the open subsystem of interest is treated quantum mechanically and the surrounding heat baths are treated in a classical-like fashion. By introducing such a mixed quantum-classical description of the composite system, one is able to study the heat transfer between the subsystem and bath from a closed system point of view, thereby avoiding simplifying assumptions related to the bath time scale and subsystem-bath coupling strength. In particular, we adopt the full counting statistics approach to derive a general expression for the moment generating function of heat in systems whose dynamics are described by the quantum-classical Liouville equation (QCLE). From this expression, one can deduce expressions for the dynamics of the average heat and heat current, which may be evaluated using numerical simulations. Due to the approximate nature of the QCLE, we also find that the steady state fluctuation symmetry holds up to order ℏ for systems whose subsystem-bath couplings and baths go beyond bilinear and harmonic, respectively. To demonstrate the approach, we consider the nonequilibrium spin boson model and simulate its time-dependent average heat and heat current under various conditions.

Vibrational Heat Transport in Molecular Junctions.

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 combine to determine the predominant mechanism (ballistic/diffusive), effectiveness (poor/good), and functionality (linear/nonlinear) of thermal conduction at the nanoscale. We review recent experiments and relevant calculations of quantum heat transfer in molecular junctions. We recount the Landauer approach, appropriate for the study of elastic (harmonic) phononic transport, and outline techniques that incorporate molecular anharmonicities. Theoretical methods are described along with examples illustrating the challenge of reaching control over vibrational heat conduction in molecules.

Controlling charge transport mechanisms in molecular junctions: Distilling thermally induced hopping from coherent-resonant conduction.

The electrical conductance of molecular junctions may depend strongly on the temperature and weakly on molecular length, under two distinct mechanisms: phase-coherent resonant conduction, with charges proceeding via delocalized molecular orbitals, and incoherent thermally assisted multi-step hopping. While in the case of coherent conduction, the temperature dependence arises from the broadening of the Fermi distribution in the metal electrodes, in the latter case it corresponds to electron-vibration interaction effects on the junction. With the objective to distill the thermally activated hopping component, thus exposing intrinsic electron-vibration interaction phenomena on the junction, we suggest the design of molecular junctions with "spacers," extended anchoring groups that act to filter out phase-coherent resonant electrons. Specifically, we study the electrical conductance of fixed-gap and variable-gap junctions that include a tunneling block, with spacers at the boundaries. Using numerical simulations and analytical considerations, we demonstrate that in our design, resonant conduction is suppressed. As a result, the electrical conductance is dominated by two (rather than three) mechanisms: superexchange (deep tunneling) and multi-step thermally induced hopping. We further exemplify our analysis on DNA junctions with an A:T block serving as a tunneling barrier. Here, we show that the electrical conductance is insensitive to the number of G:C base-pairs at the boundaries. This indicates that the tunneling-to-hopping crossover revealed in such sequences truly corresponds to the properties of the A:T barrier.

The Anderson impurity model out-of-equilibrium: Assessing the accuracy of simulation techniques with an exact current-occupation relation.

We study the interacting, symmetrically coupled single impurity Anderson model. By employing the nonequilibrium Green's function formalism, we reach an exact relationship between the steady-state charge current flowing through the impurity (dot) and its occupation. We argue that the steady-state current-occupation relation can be used to assess the consistency of simulation techniques and identify spurious transport phenomena. We test this relation in two different model variants: First, we study the Anderson-Holstein model in the strong electron-vibration coupling limit using the polaronic quantum master equation method. We find that the current-occupation relation is violated numerically in standard calculations, with simulations bringing up incorrect transport effects. Using a numerical procedure, we resolve the problem efficiently. Second, we simulate the Anderson model with electron-electron interaction on the dot using a deterministic numerically exact time-evolution scheme. Here, we observe that the current-occupation relation is satisfied in the steady-state limit-even before results converge to the exact limit.

Inelastic effects in molecular transport junctions: The probe technique at high bias.

We extend the Landauer-Büttiker probe formalism for conductances to the high bias regime and study the effects of environmentally induced elastic and inelastic scattering on charge current in single molecule junctions, focusing on high-bias effects. The probe technique phenomenologically incorporates incoherent elastic and inelastic effects to the fully coherent case, mimicking a rich physical environment at trivial cost. We further identify environmentally induced mechanisms which generate an asymmetry in the current, manifested as a weak diode behavior. This rectifying behavior, found in two types of molecular junction models, is absent in the coherent-elastic limit and is only active in the case with incoherent-inelastic scattering. Our work illustrates that in the low bias-linear response regime, the commonly used "dephasing probe" (mimicking only elastic decoherence effects) operates nearly indistinguishably from a "voltage probe" (admitting inelastic-dissipative effects). However, these probes realize fundamentally distinct I-V characteristics at high biases, reflecting the central roles of dissipation and inelastic scattering processes on molecular electronic transport far-from-equilibrium.