Important issues facing model-based approaches to tunneling transport in molecular junctions

CP-AFM Molecular Tunnel Junctions with Alkyl Backbones Anchored Using Alkynyl and Thiol Groups: Microscopically Different Despite Phenomenological Similarity

In this paper, we report results on the electronic structure and transport properties of molecular junctions fabricated via conducting probe atomic force microscopy (CP-AFM) using self-assembled monolayers (SAMs) of n-alkyl chains anchored with acetylene groups (CnA; n = 8, 9, 10, and 12) on Ag, Au, and Pt electrodes. We found that the current-voltage (I-V) characteristics of CnA CP-AFM junctions can be very accurately reproduced by the same off-resonant single-level model (orSLM) successfully utilized previously for many other junctions. We demonstrate that important insight into the energy-level alignment can be gained from experimental data of transport (processed via the orSLM) and ultraviolet photoelectron spectroscopy combined with ab initio quantum chemical information based on the many-body outer valence Green's function method. Measured conductance GAg < GAu < GPt is found to follow the same ordering as the metal work function ΦAu < ΦAu < ΦPt, a fact that points toward a transport mediated by an occupied molecular orbital (MO). Still, careful data analysis surprisingly revealed that transport is not dominated by the ubiquitous HOMO but rather by the HOMO-1. This is an important difference from other molecular tunnel junctions with p-type HOMO-mediated conduction investigated in the past, including the alkyl thiols (CnT) to which we refer in view of some similarities. Furthermore, unlike in CnT and other junctions anchored with thiol groups investigated in the past, the AFM tip causes in CnA an additional MO shift, whose independence of size (n) rules out significant image charge effects. Along with the prevalence of the HOMO-1 over the HOMO, the impact of the "second" (tip) electrode on the energy level alignment is another important finding that makes the CnA and CnT junctions different. What ultimately makes CnA unique at the microscopic level is a salient difference never reported previously, namely, that CnA's alkyne functional group gives rise to two energetically close (HOMO and HOMO-1) orbitals. This distinguishes the present CnA from the CnT, whose HOMO stemming from its thiol group is well separated energetically from the other MOs.

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.

Controlling Charge Transport in Molecular Wires through Transannular π-π Interaction

This paper describes the influence of the transannular π-π interaction in controlling the carrier transport in molecular wires by employing the STM break junction technique. Five pentaphenylene-based molecular wires that contained [2.2]paracyclophane-1,9-dienes (PCD) as the building block were prepared as model compounds. Functional substituents with different electronic properties, ranging from strong acceptors to strong donors, were attached to the top parallel aromatic ring and used as a gate. It was found that the carrier transport features of these molecular wires, such as single-molecule conductance and a charge-tunneling barrier, can be systematically controlled through the transannular π-π interaction.

Redox-controlled conductance of polyoxometalate molecular junctions

We demonstrate the reversible in situ photoreduction of molecular junctions of a phosphomolybdate [PMo₁₂O₄₀]^3- monolayer self-assembled on flat gold electrodes, connected by the tip of a conductive atomic force microscope. The conductance of the one electron reduced [PMo₁₂O₄₀]^4- molecular junction is increased by ∼10, and this open-shell state is stable in the junction in air at room temperature. The analysis of a large current-voltage dataset by unsupervised machine learning and clustering algorithms reveals that the electron transport in the pristine phosphomolybdate junctions leads to symmetric current-voltage curves, controlled by the lowest unoccupied molecular orbital (LUMO) at 0.6-0.7 eV above the Fermi energy with ∼25% of the junctions having a better electronic coupling to the electrodes than the main part of the dataset. This analysis also shows that a small fraction (∼18% of the dataset) of the molecules is already reduced. The UV light in situ photoreduced phosphomolybdate junctions systematically feature slightly asymmetric current-voltage behaviors, which is ascribed to the electron transport mediated by the single occupied molecular orbital (SOMO) nearly at resonance with the Fermi energy of the electrodes and by a closely located single unoccupied molecular orbital (SUMO) at ∼0.3 eV above the SOMO with a weak electronic coupling to the electrodes (∼50% of the dataset) or at ∼0.4 eV but with a better electrode coupling (∼50% of the dataset). These results shed light on the electronic properties of reversible switchable redox polyoxometalates, a key point for potential applications in nanoelectronic devices.

Thermopower in Transition from Tunneling to Hopping

The Seebeck effect of a molecular junction in a hopping regime or tunneling-to-hopping transition remains uncertain. This paper describes the Seebeck effect in molecular epitaxy films (OPI_n where n = 1-9) based on imine condensation between an aryl amine and aldehyde and investigates how the Seebeck coefficient (S, μV/K) varies at the crossover region. The S value of OPI_n linearly increased with increasing the molecular length (d, nm), ranging from 7.2 to 38.0 μV/K. The increasing rate changed from 0.99 to 0.38 μV·K^-1 Å^-1 at d = 3.4 nm (OPI₄). Combined experimental and theoretical studies indicated that such a change stems from a tunneling-to-hopping transition, and the small but detectable length-dependence of thermopower in the long molecules originates from the gradual reduction of the tunneling contribution to the broadening of molecular orbital energy level, rather than its relative position to the Fermi level. Our work helps to bridge the gap between bulk and nanoscale thermoelectric systems.

What Can We Learn from the Time Evolution of COVID-19 Epidemic in Slovenia?

A recent work indicates that temporarily splitting larger populations into smaller groups can efficiently mitigate the spread of SARS‐CoV‐2 virus. The fact that, soon afterward, the two million people Slovenia was the first European country proclaiming the end of COVID‐19 epidemic within national borders may be relevant from this perspective. Motivated by this evolution, in this paper the time dynamics of coronavirus cases in Slovenia is investigated with emphasis on how efficient various containment measures act to diminish the number of COVID‐19 infections. Noteworthily, the present analysis does not rely on any speculative theoretical assumption; it is solely based on raw epidemiological data. Out of the results presented here, the most important one is perhaps the finding that, while imposing drastic curfews and travel restrictions reduce the infection rate κ by a factor of four with respect to the unrestricted state, they only improve the κ‐value by ≈15% as compared to the much bearable state of social and economic life wherein wearing face masks and social distancing rules are enforced/followed. Significantly for behavioral and social science, our analysis may reveal an interesting self‐protection instinct of the population, which became manifest even before the official lockdown enforcement.

Tuning surface d bands with bimetallic electrodes to facilitate electron transport across molecular junctions

Understanding chemical bonding and conductivity at the electrode-molecule interface is key for the operation of single-molecule junctions. Here we apply the d-band theory that describes interfacial interactions between adsorbates and transition metal surfaces to study electron transport across these devices. We realized bimetallic Au electrodes modified with a monoatomic Ag adlayer to connect α,ω-alkanoic acids (HO₂C(CH₂)_nCO₂H). The force required to break the molecule-electrode binding and the contact conductance G_n=0 are 1.1 nN and 0.29 G₀ (the conductance quantum, 1 G₀ = 2e²/h ≈ 77.5 μS), which makes these junctions, respectively, 1.3-1.8 times stronger and 40-60-fold more conductive than junctions with bare Au or Ag electrodes. A similar performance was found for Au electrodes modified by Cu monolayers. By integrating the Newns-Anderson model with the Hammer-Nørskov d-band model, we explain how the surface d bands strengthen the adsorption and promote interfacial electron transport, which provides an alternative avenue for the optimization of molecular electronic devices.

Suppression of Groups Intermingling as an Appealing Option for Flattening and Delaying the Epidemiological Curve While Allowing Economic and Social Life at a Bearable Level during the COVID-19 Pandemic

The COVID‐19 pandemic in a population modelled as a network wherein infection can propagate both via intra‐ and inter‐group interactions is simulated. The results emphasize the importance of diminishing the inter‐group infections in the effort of substantial flattening/delaying of the epi(demiologic) curve with concomitant mitigation of disastrous economy and social consequences. To exemplify, splitting a population into m (say, 5 or 10) noninteracting groups while keeping intra‐group interaction unchanged yields a stretched epidemiological curve having the maximum number of daily infections reduced and postponed in time by the same factor m (5 or 10). More generally, the study suggests a practical approach to fight against SARS‐ CoV‐ 2 virus spread based on population splitting into groups and minimizing intermingling between them. This strategy can be pursued by large‐scale infrastructure reorganization of activity at different levels in big logistic units (e.g., large productive networks, factories, enterprises, warehouses, schools, (seasonal) harvest work). Importantly, unlike total lockdown, the proposed approach prevents economic ruin and keeps social life at a more bearable level than distancing everyone from anyone. The declaration for the first time in Europe that COVID‐19 epidemic ended in the two‐million population Slovenia may be taken as support for the strategy proposed here.

Evidence That Molecules in Molecular Junctions May Not Be Subject to the Entire External Perturbation Applied to Electrodes

Whether molecules forming molecular junctions are really subject to the entire external perturbation applied to electrodes is an important issue, but so far, it has not received adequate consideration in the literature. In this paper, we demonstrate that, out of the temperature difference ΔT_electr between electrodes applied in thermopower measurements, molecules only feel a significantly smaller temperature difference (ΔT_molec < ΔT_electr). Rephrasing, temperature drops at metal-molecule interfaces are substantial. Our theoretical analysis to address this problem of fundamental importance for surface science is based on experimental data collected via ultraviolet photoelectron spectroscopy, transition voltage spectroscopy, and Seebeck coefficient measurements. An important practical consequence of the presently reported finding is that the energetic alignment of the frontier molecular orbital (HOMO or LUMO) of the embedded molecules with respect to the metallic Fermi level position deduced from thermopower data-and this is frequently the case in current studies of molecular electronics-is substantially overestimated. Another important result presented here is that, unlike the exponential length dependence characterizing electric conduction (which is a fingerprint for quantum tunneling), thermal conduction through the molecules considered (oligophenylene thiols and alkane thiols) exhibits a length dependence compatible with classical physics.

Why one can expect large rectification in molecular junctions based on alkane monothiols and why rectification is so modest

Many attempts to obtain high current rectification ratios (RRs) in molecular electronics are triggered by a potentiometer rule argument, which predicts that a strongly asymmetric location of the dominant molecular orbital yields large RR-values. Invoking this argument, molecular junctions based on alkane monothiols (CnT) can be expected to exhibit high RRs; the HOMO of these molecules is localized on the thiol terminal group bonded to one electrode. The extensive current-voltage (I-V) results for CP-AFM (conducting probe atomic force microscope) CnT junctions of various molecular lengths (n = 7, 8, 9, 10, and 12) and different metallic contacts (Ag, Au, and Pt) are consistent with conduction dominated by the HOMO, but the measured RR ∼ 1.5 is much smaller than that predicted by the potentiometer rule framework. Further, the linear shift in the HOMO position with applied bias, γ, which gives rise to rectification, is also smaller than expected, and critically, γ has the opposite sign from potentiometer rule predictions. Companion ab initio OVGF (outer valence Green's function) quantum chemical calculations provide important insight. Namely, a linear Stark shift γm is calculated for the HOMO of CnT molecules for electric field strengths (106-107 V cm-1) typical of molecular junctions, and the sign of γm matches the sign of the experimental γ for junctions derived from transport measurements, suggesting that the Stark effect plays an important role. However, the magnitude of the measured γ is only 10-15% of the computed value γm. We propose that this implies that the contacts are far from optimal; they substantially screen the effect of the applied bias, possibly via molecule-electrode interface states. We predict that, with optimized contacts, the rectification ratios in CnT-based junctions can reach reasonably high values (RR ≈ 500). We believe that Stark shifts and limited current rectification due to non-ideal contacts discussed here for the specific case of alkane monothiol junctions are issues of general interest for molecular electronics that deserve further consideration.

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.

Effect of Heteroatom Substitution on Transport in Alkanedithiol-Based Molecular Tunnel Junctions: Evidence for Universal Behavior

The transport properties of molecular junctions based on alkanedithiols with three different methylene chain lengths were compared with junctions based on similar chains wherein every third -CH₂- was replaced with O or S, that is, following the general formula HS(CH₂CH₂X)_nCH₂CH₂SH, where X = CH₂, O, or S and n = 1, 2, or 3. Conducting probe atomic force microscopy revealed that the low bias resistance of the chains increased upon substitution in the order CH₂ < O < S. This change in resistance is ascribed to the observed identical trend in contact resistance, R_c, whereas the exponential prefactor β (length sensitivity) was essentially the same for all chains. Using an established, analytical single-level model, we computed the effective energy offset ε_h (i.e., Fermi level relative to the effective HOMO level) and the electronic coupling strength Γ from the current-voltage (I-V) data. The ε_h values were only weakly affected by heteroatom substitution, whereas the interface coupling strength Γ varied by over an order of magnitude. Consequently, we ascribe the strong variation in R_c to the systematic change in Γ. Quantum chemical calculations reveal that the HOMO density shifts from the terminal SH groups for the alkanedithiols to the heteroatoms in the substituted chains, which provides a plausible explanation for the marked decrease in Γ for the dithiols with electron-rich heteroatoms. The results indicate that the electronic coupling and thus the resistance of alkanedithiols can be tuned by substitution of even a single atom in the middle of the molecule. Importantly, when appropriately normalized, the experimental I-V curves were accurately simulated over the full bias range (±1.5 V) using the single-level model with no adjustable parameters. The data could be collapsed to a single universal curve predicted by the model, providing clear evidence that the essential physics is captured by this analytical approach and supporting its utility for molecular electronics.

Protocol for disentangling the thermally activated contribution to the tunneling-assisted charge transport. Analytical results and experimental relevance

In this paper we present results demonstrating that the charge transport by tunneling in molecular junctions can exhibit a substantial temperature dependence.

Revealing tunnelling details by normalized differential conductance analysis of transport across molecular junctions

A new analysis tool reveals hidden fingerprints of tunnelling mechanisms and quantifies the bias response via a scaling bias parameter.

Counterintuitive issues in the charge transport through molecular junctions

Whether at phenomenological or microscopic levels, most theoretical approaches to charge transport through molecular junctions postulate or attempt to justify microscopically the existence of a dominant molecular orbital (MO). Within such single level descriptions, experimental current-voltage I-V curves are sometimes/often analyzed by using analytical formulas expressing the current as a cubic expansion in terms of the applied voltage V, and the possible V-driven shifts of the level energy offset relative to the metallic Fermi energy ε0 are related to the asymmetry of molecule-electrode couplings or an asymmetric location of the "center of gravity" of the MO with respect to electrodes. In this paper, we present results demonstrating the failure of these intuitive expectations. For example, we show how typical data processing based on cubic expansions yields a value of ε0 underestimated by a typical factor of about two. When compared to theoretical results of DFT approaches, which typically underestimate the HOMO-LUMO gap by a similar factor, this may create the false impression of "agreement" with experiments in situations where this is actually not the case. Furthermore, such cubic expansions yield model parameter values dependent on the bias range width employed for fitting, which is unacceptable physically. Finally, we present an example demonstrating that, counter-intuitively, the bias-induced change in the energy of an MO located much closer to an electrode can occur in a direction that is opposite to the change in the Fermi energy of that electrode. This is contrary to what one expects based on a "lever rule" argument, according to which the MO "feels" the local value of the electric potential, which is assumed to vary linearly across the junction and is closer to the potential of the closer electrode. This example emphasizes the fact that screening effects in molecular junctions can have a subtle character, contradicting common intuition.

Invariance of molecular charge transport upon changes of extended molecule size and several related issues

As a sanity test for the theoretical method employed, studies on (steady-state) charge transport through molecular devices usually confine themselves to check whether the method in question satisfies the charge conservation. Another important test of the theory's correctness is to check that the computed current does not depend on the choice of the central region (also referred to as the "extended molecule"). This work addresses this issue and demonstrates that the relevant transport and transport-related properties are indeed invariant upon changing the size of the extended molecule, when the embedded molecule can be described within a general single-particle picture (namely, a second-quantized Hamiltonian bilinear in the creation and annihilation operators). It is also demonstrates that the invariance of nonequilibrium properties is exhibited by the exact results but not by those computed approximately within ubiquitous wide- and flat-band limits (WBL and FBL, respectively). To exemplify the limitations of the latter, the phenomenon of negative differential resistance (NDR) is considered. It is shown that the exactly computed current may exhibit a substantial NDR, while the NDR effect is absent or drastically suppressed within the WBL and FBL approximations. The analysis done in conjunction with the WBLs and FBLs reveals why general studies on nonequilibrium properties require a more elaborate theoretical than studies on linear response properties (e.g., ohmic conductance and thermopower) at zero temperature. Furthermore, examples are presented that demonstrate that treating parts of electrodes adjacent to the embedded molecule and the remaining semi-infinite electrodes at different levels of theory (which is exactly what most NEGF-DFT approaches do) is a procedure that yields spurious structures in nonlinear ranges of current-voltage curves.

Fractional molecular charge studied via molecular vibrational properties. Specific aspects in Jahn–Teller active molecular species

Jahn–Teller active and inactive vibrational modes are affected in a different manner by charge removal.

Experimental and Theoretical Analysis of Nanotransport in Oligophenylene Dithiol Junctions as a Function of Molecular Length and Contact Work Function

We report the results of an extensive investigation of metal-molecule-metal tunnel junctions based on oligophenylene dithiols (OPDs) bound to several types of electrodes (M1-S-(C6H4)n-S-M2, with 1 ≤ n ≤ 4 and M1,2 = Ag, Au, Pt) to examine the impact of molecular length (n) and metal work function (Φ) on junction properties. Our investigation includes (1) measurements by scanning Kelvin probe microscopy of electrode work function changes (ΔΦ = ΦSAM - Φ) caused by chemisorption of OPD self-assembled monolayers (SAMs), (2) measurements of junction current-voltage (I-V) characteristics by conducting probe atomic force microscopy in the linear and nonlinear bias ranges, and (3) direct quantitative analysis of the full I-V curves. Further, we employ transition voltage spectroscopy (TVS) to estimate the energetic alignment εh = EF - EHOMO of the dominant molecular orbital (HOMO) relative to the Fermi energy EF of the junction. Where photoelectron spectroscopy data are available, the εh values agree very well with those determined by TVS. Using a single-level model, which we justify via ab initio quantum chemical calculations at post-density functional theory level and additional UV-visible absorption measurements, we are able to quantitatively reproduce the I-V measurements in the whole bias range investigated (∼1.0-1.5 V) and to understand the behavior of εh and Γ (contact coupling strength) extracted from experiment. We find that Fermi level pinning induced by the strong dipole of the metal-S bond causes a significant shift of the HOMO energy of an adsorbed molecule, resulting in εh exhibiting a weak dependence with the work function Φ. Both of these parameters play a key role in determining the tunneling attenuation factor (β) and junction resistance (R). Correlation among Φ, ΔΦ, R, transition voltage (Vt), and εh and accurate simulation provide a remarkably complete picture of tunneling transport in these prototypical molecular junctions.