Breakdown of Ohm's Law in Molecular Junctions with Electrodes of Single-Layer Graphene

For sufficiently low biases, Ohm's law, the cornerstone of electricity, stating that current I and voltage V are proportional, is satisfied at low biases for all known systems ranging from macroscopic conductors to nanojunctions. In this study, we predict theoretically and demonstrate experimentally that in single-molecule junctions fabricated with single-layer graphene as electrodes the current at low V scales as the cube of V, thereby invalidating Ohm's law. The absence of the ohmic regime is a direct consequence of the unique band structure of the single-layer graphene, whose vanishing density of states at the Dirac points precludes electron transfer from and to the electrodes at low biases.

Comment on "A single level tunneling model for molecular junctions: evaluating the simulation methods" by E. M. Opodi, X. Song, X. Yu and W. Hu, Phys. Chem. Chem. Phys., 2022, , 11958"

The present Comment demonstrates important flaws of the paper Opodi et al. Phys. Chem. Chem. Phys., 2022, 24, 11958 Their crown result ("applicability map") aims at indicating parameter ranges wherein two approximate methods (called method 2 and 3) apply. My calculations reveal that the applicability map is a factual error. Deviations of I2 from the exact current I1 do not exceed 3% for model parameters where Opodi et al. claimed that method 2 is inapplicable. As for method 3, the parameter range of the applicability map is beyond its scope, as stated in papers cited by Opodi et al. themselves.

Can tunneling current in molecular junctions be so strongly temperature dependent to challenge a hopping mechanism? Analytical formulas answer this question and provide important insight into large area junctions

Analytical equations like Richardson-Dushman's or Shockley's provided a general, if simplified conceptual background, which was widely accepted in conventional electronics and made a fundamental contribution to advances in the field. In the attempt to develop a (highly desirable, but so far missing) counterpart for molecular electronics, in this work, we deduce a general analytical formula for the tunneling current through molecular junctions mediated by a single level that is valid for any bias voltage and temperature. Starting from this expression, which is exact and obviates cumbersome numerical integration, in the low and high temperature limits we also provide analytical formulas expressing the current in terms of elementary functions. They are accurate for broad model parameter ranges relevant for real molecular junctions. Within this theoretical framework we show that: (i) by varying the temperature, the tunneling current can vary by several orders of magnitude, thus debunking the myth that a strong temperature dependence of the current is evidence for a hopping mechanism, (ii) real molecular junctions can undergo a gradual (Sommerfeld-Arrhenius) transition from a weakly temperature dependent to a strongly ("exponential") temperature dependent current that can be tuned by the applied bias, and (iii) important insight into large area molecular junctions with eutectic gallium indium alloy (EGaIn) top electrodes can be gained. E.g., merely based on transport data, we estimate that the current carrying molecules represent only a fraction of f ≈ 4 × 10-4 out of the total number of molecules in a large area Au-S-(CH2)13-CH3/EGaIn junction.

Seebeck Effect in Molecular Wires Facilitating Long-Range Transport

The study of molecular wires facilitating long-range charge transport is of fundamental interest for the development of various technologies in (bio)organic and molecular electronics. Defining the nature of long-range charge transport is challenging as electrical characterization does not offer the ability to distinguish a tunneling mechanism from the other. Here, we show that investigation of the Seebeck effect provides the ability. We examine the length dependence of the Seebeck coefficient in electrografted bis-terpyridine Ru(II) complex films. The Seebeck coefficient ranges from 307 to 1027 μV/K, with an increasing rate of 95.7 μV/(K nm) as the film thickness increases to 10 nm. Quantum-chemical calculations unveil that the nearly overlapped molecular-orbital energy level of the Ru complex with the Fermi level accounts for the giant thermopower. Landauer-Büttiker probe simulations indicate that the significant length dependence evinces the Seebeck effect dominated by coherent near-resonant tunneling rather than thermal hopping. This study enhances our comprehension of long-range charge transport, paving the way for efficient electronic and thermoelectric materials.

Can room-temperature data for tunneling molecular junctions be analyzed within a theoretical framework assuming zero temperature?

Routinely, experiments on tunneling molecular junctions report values of conductances (G_RT) and currents (I_RT) measured at room temperature. On the other hand, theoretical approaches based on simplified models provide analytic formulas for the conductance (G_0K) and current (I_0K) valid at zero temperature. Therefore, interrogating the applicability of the theoretical results deduced in the zero-temperature limit to real experimental situations at room temperature (i.e., G_RT ≈ G_0K and I_RT ≈ I_0K) is a relevant aspect. Quantifying the pertaining temperature impact on the transport properties computed within the ubiquitous single-level model with Lorentzian transmission is the specific aim of the present work. Comprehensive results are presented for broad ranges of the relevant parameters (level's energy offset ε₀ and width Γ_a, and applied bias V) that safely cover values characterizing currently fabricated junctions. They demonstrate that the strongest thermal effects occur at biases below resonance (2|ε₀| - δε₀ - 0.3 ≲ |eV| - 0.3 ≲ 2|ε₀|). At fixed V, they affect an ε₀-range whose largest width δε₀ is about nine times larger than the thermal energy (δε₀ ≈ 3πk_BT) at Γ_a → 0. The numerous figures included aim to convey a quick overview on the applicability of the zero-temperature limit to a specific real junction. In quantitative terms, the conditions of applicability are expressed as mathematical inequalities involving elementary functions. They constitute the basis of a proposed interactive data-fitting procedure, which aims to guide experimentalists interested in data processing in a specific case.

Estimating the Number of Molecules in Molecular Junctions Merely Based on the Low Bias Tunneling Conductance at Variable Temperature

Temperature (T) dependent conductance G=G(T) data measured in molecular junctions are routinely taken as evidence for a two-step hopping mechanism. The present paper emphasizes that this is not necessarily the case. A curve of lnG versus 1/T decreasing almost linearly (Arrhenius-like regime) and eventually switching to a nearly horizontal plateau (Sommerfeld regime), or possessing a slope gradually decreasing with increasing 1/T is fully compatible with a single-step tunneling mechanism. The results for the dependence of G on T presented include both analytical exact and accurate approximate formulas and numerical simulations. These theoretical results are general, also in the sense that they are not limited, e.g., to the (single molecule electromigrated (SET) or large area EGaIn) fabrication platforms, which are chosen for exemplification merely in view of the available experimental data needed for analysis. To be specific, we examine in detail transport measurements for molecular junctions based on ferrocene (Fc). As a particularly important finding, we show how the present analytic formulas for G=G(T) can be utilized to compute the ratio f=Aeff/An between the effective and nominal areas of large area Fc-based junctions with an EGaIn top electrode. Our estimate of f≈0.6×10-4 is comparable with previously reported values based on completely different methods for related large area molecular junctions.

Temperature-Dependent Coherent Tunneling across Graphene-Ferritin Biomolecular Junctions

Understanding the mechanisms of charge transport (CT) across biomolecules in solid-state devices is imperative to realize biomolecular electronic devices in a predictive manner. Although it is well-accepted that biomolecule-electrode interactions play an essential role, it is often overlooked. This paper reveals the prominent role of graphene interfaces with Fe-storing proteins in the net CT across their tunnel junctions. Here, ferritin (AfFtn-AA) is adsorbed on the graphene by noncovalent amine-graphene interactions confirmed with Raman spectroscopy. In contrast to junctions with metal electrodes, graphene has a vanishing density of states toward its intrinsic Fermi level ("Dirac point"), which increases away from the Fermi level. Therefore, the amount of charge carriers is highly sensitive to temperature and electrostatic charging (induced doping), as deduced from a detailed analysis of CT as a function of temperature and iron loading. Remarkably, the temperature dependence can be fully explained within the coherent tunneling regime due to excitation of hot carriers. Graphene is not only demonstrated as an alternative platform to study CT across biomolecular tunnel junctions, but it also opens rich possibilities in employing interface electrostatics in tuning CT behavior.

Empirical Parameter to Compare Molecule-Electrode Interfaces in Large-Area Molecular Junctions

This paper describes a simple model for comparing the degree of electronic coupling between molecules and electrodes across different large-area molecular junctions. The resulting coupling parameter can be obtained directly from current–voltage data or extracted from published data without fitting. We demonstrate the generalizability of this model by comparing over 40 different junctions comprising different molecules and measured by different laboratories. The results agree with existing models, reflect differences in mechanisms of charge transport and rectification, and are predictive in cases where experimental limitations preclude more sophisticated modeling. We also synthesized a series of conjugated molecular wires, in which embedded dipoles are varied systematically and at both molecule–electrode interfaces. The resulting current–voltage characteristics vary in nonintuitive ways that are not captured by existing models, but which produce trends using our simple model, providing insights that are otherwise difficult or impossible to explain. The utility of our model is its demonstrative generalizability, which is why simple observables like tunneling decay coefficients remain so widely used in molecular electronics despite the existence of much more sophisticated models. Our model is complementary, giving insights into molecule–electrode coupling across series of molecules that can guide synthetic chemists in the design of new molecular motifs, particularly in the context of devices comprising large-area molecular junctions.

Aerosol jet printed 3 omega sensors for thermal conductivity measurement

The 3 omega (3ω) method is a trusted technique for measuring thermal conductivity-a fundamental material property of critical importance in a broad range of applications. However, traditional 3ω sensor processing requires some form of physical vapor deposition, such as metal evaporation or sputtering. These 3ω sensor deposition techniques limit the materials and sample sizes applicable to the 3ω method. This work demonstrates an aerosol jet printing method to directly print silver 3ω sensors that yield accurate temperature-dependent measurement up to 300 °C on materials with thermal conductivity ranging from 1 to 150 W/m K. The interrelationship between printed sensor geometry, sensor thermal stability, and applicability to the 3ω method is examined. Thermal conductivity measurement with 3ω sensors conventionally sintered at 300 °C agrees to independent laser flash measurement within 4% from room temperature to 150 °C. An unconventional rapid high-temperature sintering method is shown to produce sensors that agree within 3% of the laser flash measurements from room temperature to 300 °C. The rapid sintering profiles also reduced the sensor-substrate thermal boundary resistance of the printed sensors by as much as 88%. The direct printing of 3ω sensors creates opportunities for measurement of thermal transport properties in applications previously inapplicable to the 3ω method.

Porphyrins as building blocks for single-molecule devices

Direct measurement of single-molecule electrical transparency by break junction experiments has become a major field of research over the two last decades. This review specifically and comprehensively highlights the use of porphyrins as molecular components and discusses their potential use for the construction of future devices. Throughout the review, the features provided by porphyrins, such as low level misalignments and very low attenuation factors, are shown with numerous examples, illustrating the potential and limitations of these molecular junctions, as well as differences emerging from applied integration/investigation techniques.

Temperature-Dependent Tunneling in Furan Oligomer Single-Molecule Junctions

Two commonly observed charge transport mechanisms in single-molecule junctions are coherent tunneling and incoherent hopping. It has been generally believed that tunneling processes yield temperature-independent conductance behavior and hopping processes exhibit increasing conductance with increasing temperature. However, it has recently been proposed that tunneling can also yield temperature-dependent transport due to the thermal broadening of the Fermi energy of the contacts. In this work, we examine a series of rigid, planar furan oligomers that are free from a rotational internal degree of freedom to examine the temperature dependence of tunneling transport directly over a wide temperature range (78-300 K). Our results demonstrate conductance transition from a temperature-independent regime to a temperature-dependent regime. By examining various hopping and tunneling models and the correlation between the temperature dependence of conductance and molecular orbital energy offset from the Fermi level, we conclude thermally assisted tunneling is the dominant cause for the onset of temperature-dependent conductance in these systems.

Gate-tunable electroluminescence in Aviram-Ratner-type molecules: Kinetic description

A theoretical study of the mechanisms of electroluminescence (EL) generation in photoactive molecules with donor and acceptor centers linked by saturated σ-bonds (molecules of the Aviram-Ratner-type) is presented. The approach is based on the kinetics of single-electron transitions between many-body molecular states. This study shows that the EL polarity arises due to asymmetric coupling of molecular orbitals of the photochromic part of the molecule to the electrodes. The gate voltage controls the power of the EL through the occupancy of the excited singlet state. The shifting of the orbital energies forms a resonant or a non-resonant path for the transmission of electrons through the molecule. The action of the gate voltage is reflected in specific critical voltages. An analytical dependence of the critical voltages on the energies of molecular states involved in the formation of EL, as well as on the gate voltage, was derived for both positive and negative polarities. Conditions under which the gate voltage lowers the absolute value of the bias voltage that is responsible for the activation of the resonance mechanism of EL formation were also established. This is an important factor in control of EL in molecular junctions.

Thermally activated charge transport in carbon atom chains

Charge transport through single molecules is at the heart of molecular electronics for realizing the practical use of the rich quantum characteristics of electrode-molecule-electrode systems. Despite the extensive studies reported in the past, little experimental efforts have been focused on the electron transport mechanism at a temperature higher than the ambient temperature. In this work, we have reported the observation of the subtle interplay between electron tunneling and charge hopping in carbon chains connected to two Au electrodes at elevated temperatures. We measured the single-molecule conductance of Au-alkanedithiol-Au molecular junctions at various temperatures from 300 K to 420 K in vacuum. The temperature dependence of conductance suggested substantial roles of superexchange with inter-chain charge hopping under elevated temperatures for alkane chains longer than heptane. This finding provides a guide to design functional molecular junctions under practical conditions.

Transition from direct to inverted charge transport Marcus regions in molecular junctions via molecular orbital gating

Solid-state molecular tunnel junctions are often assumed to operate in the Landauer regime, which describes essentially activationless coherent tunnelling processes. In solution, on the other hand, charge transfer is described by Marcus theory, which accounts for thermally activated processes. In practice, however, thermally activated transport phenomena are frequently observed also in solid-state molecular junctions but remain poorly understood. Here, we show experimentally the transition from the Marcus to the inverted Marcus region in a solid-state molecular tunnel junction by means of intra-molecular orbital gating that can be tuned via the chemical structure of the molecule and applied bias. In the inverted Marcus region, charge transport is incoherent, yet virtually independent of temperature. Our experimental results fit well to a theoretical model that combines Landauer and Marcus theories and may have implications for the interpretation of temperature-dependent charge transport measurements in molecular junctions.

Work function and temperature dependence of electron tunneling through an N-type perylene diimide molecular junction with isocyanide surface linkers

Conducting probe atomic force microscopy (CP-AFM) was employed to examine electron tunneling in self-assembled monolayer (SAM) junctions. A 2.3 nm long perylene tetracarboxylic acid diimide (PDI) acceptor molecule equipped with isocyanide linker groups was synthesized, adsorbed onto Ag, Au and Pt substrates, and the current-voltage (I-V) properties were measured by CP-AFM. The dependence of the low-bias resistance (R) on contact work function indicates that transport is LUMO-assisted ('n-type behavior'). A single-level tunneling model combined with transition voltage spectroscopy (TVS) was employed to analyze the experimental I-V curves and to extract the effective LUMO position ε_l = E_LUMO - E_F and the effective electronic coupling (Γ) between the PDI redox core and the contacts. This analysis revealed a strong Fermi level (E_F) pinning effect in all the junctions, likely due to interface dipoles that significantly increased with increasing contact work function, as revealed by scanning Kelvin probe microscopy (SKPM). Furthermore, the temperature (T) dependence of R was found to be substantial. For Pt/Pt junctions, R varied more than two orders of magnitude in the range 248 K < T < 338 K. Importantly, the R(T) data are consistent with a single step electron tunneling mechanism and allow independent determination of ε_l, giving values compatible with estimates of ε_l based on analysis of the full I-V data. Theoretical analysis revealed a general criterion to unambiguously rule out a two-step transport mechanism: namely, if measured resistance data exhibit a pronounced Arrhenius-type temperature dependence, a two-step electron transfer scenario should be excluded in cases where the activation energy depends on contact metallurgy. Overall, our results indicate (1) the generality of the Fermi level pinning phenomenon in molecular junctions, (2) the utility of employing the single level tunneling model for determining essential electronic structure parameters (ε_l and Γ), and (3) the importance of changing the nature of the contacts to verify transport mechanisms.