Green's function formalism coupled with Gaussian broadening of discrete states for quantum transport: application to atomic and molecular wires

Does Kirchhoff's Law Work in Molecular-Scale Structures?

This study aims to theoretically and comprehensively investigate the single-molecule electrical conductance of symmetric and asymmetric alkane cyclic (SAC and AAC) molecules and their corresponding linear chains with three different terminal end groups including thiol (-SH), direct carbon (-C), and amine (-NH 2). Here, we examine the validity of Kirchhoff's law concerning sigma nonconjugated molecules at the nanoscale level. Counterintuitively, the electrical conductance (G) of symmetric and asymmetric alkane cyclic molecules with two parallel conductance paths is lower than that of their corresponding single chains with only one conductance path. This completely contradicts classical rules for combining conductances in parallel, regardless of the anchor group type, in light of this study's use of symmetric and asymmetric cyclic molecules. A comparison of the DFT prediction trends with scanning tunneling microscopy measurements indicates that they are well-supported. The results of this investigation demonstrate an excellent correlation between our simulations and experimental measurements, for both SAC and AAC structures of different cavity size n,m = 3,3; 4,4; 5,5···10,10 and n,m = 3,5; 4,6; 5,7; 6,8; 7,9; 8,10; and 9,11 and for three different terminal end groups.

Single Molecule Junction of Polynuclear Alkynylplatinum-Based Organometallic Molecular Wires: A Theoretical Study of the Long-Range Transport Effect of Linkage to Gold Surfaces

Density functional theory (DFT) and nonequilibrium Green's function (NEGF) calculations were employed to elucidate the single-molecule conductance properties of a series of polynuclear alkynylplatinum-based organometallic wires, AcS[[(p-tol3P)2PtC≡C-C≡C-C≡C-C≡C]mPt(Pp-tol3)2SAc (m = 1-3), featuring platinum centers incorporated within polyyne chains. This investigation aimed to determine the influence of the embedded platinum centers on conductance compared to all-carbon polyynes and organometallic wires with terminal metal entities. While the structural and electronic properties of the individual molecular units are minimally affected by the chain extension, DFT-NEGF calculations on the corresponding Au-wire-Au junctions reveal an exponential decay in conductance with increasing molecular length, consistent with a tunneling-dominated transport mechanism. However, for longer wires (m = 3), the markedly low conductance and partial localization of frontier orbitals indicate that hopping transport could contribute to the overall transport mechanism. These findings provide valuable insights into the factors governing charge transport, including the interplay of length, platinum incorporation, and orbital localization in polynuclear alkynylplatinum-based organometallic wires, which may have implications for the future design of molecular wires.

Theoretical Study on the Open-Shell Electronic Structure and Electron Conductivity of [18]Annulene as a Molecular Parallel Circuit Model

Herein, the electron conductivities of [18]annulene and its derivatives are theoretically examined as a molecular parallel circuit model consisting of two linear polyenes. Their electron conductivities are estimated by elastic scattering Green's function (ESGF) theory and density functional theory (DFT) methods. The calculated conductivity of the [18]annulene does not follow the classical conductivity, i.e., Ohm's law, suggesting the importance of a quantum interference effect in single molecules. By introducing electron-withdrawing groups into the annulene framework, on the other hand, a spin-polarized electronic structure appears, and the quantum interference effect is significantly suppressed. In addition, the total current is affected by the spin polarization because of the asymmetry in the coupling constant between the molecule and electrodes. From these results, it is suggested that the electron conductivity as well as the quantum interference effect of π-conjugated molecular systems can be designed using their open-shell nature, which is chemically controlled by the substituents.

High Seebeck Coefficient Achieved by Multinuclear Organometallic Molecular Junctions

This paper describes the thermoelectric properties of molecular junctions incorporating multinuclear ruthenium alkynyl complexes that comprise Ru(dppe)₂ [dppe = 1,2-bis(diphenylphosphino)ethane] fragments and diethylnyl aromatic bridging ligands with thioether anchors. Using the liquid metal technique, the Seebeck coefficient was examined as a function of metal nuclearity, oxidation state, and substituent on the organic ligand backbone. High Seebeck coefficients up to 73 μV/K and appreciable thermal stability with thermovoltage up to ∼3.3 mV at a heating temperature of 423 K were observed. An unusually high proximity of the highest occupied molecular orbital (HOMO) energy level to the Fermi level was revealed to give the remarkable thermoelectric performance as suggested by combined experiments and calculations. This work offers important insights into the development of molecular-scale devices for efficient thermoregulation and heat-to-electricity conversion.

Control of dominant conduction orbitals by peripheral substituents in paddle-wheel diruthenium alkynyl molecular junctions

Control of charge carriers that transport through the molecular junctions is essential for thermoelectric materials. In general, the charge carrier depends on the dominant conduction orbitals and is dominantly determined by the terminal anchor groups. The present study discloses the synthesis, physical properties in solution, and single-molecule conductance of paddle-wheel diruthenium complexes 1R having diarylformamidinato supporting ligands (DArF: p-R-C6H4-NCHN-C6H4-R-p) and two axial thioanisylethynyl conducting anchor groups, revealing unique substituent effects with respect to the conduction orbitals. The complexes 1R with a few different aryl substituents (R = OMe, H, Cl, and CF3) were fully characterized by spectroscopic and crystallographic analyses. The single-molecule conductance determined by the scanning tunneling microscope break junction (STM-BJ) technique was in the 10-5 to 10-4 G 0 region, and the order of conductance was 1OMe > 1CF3 ≫ 1H ∼ 1Cl, which was not consistent with the Hammett substituent constants σ of R. Cyclic voltammetry revealed the narrow HOMO-LUMO gaps of 1R originating from the diruthenium motif, as further supported by the DFT study. The DFT-NEGF analysis of this unique result revealed that the dominant conductance routes changed from HOMO conductance (for 1OMe) to LUMO conductance (for 1CF3). The drastic change in the conductance properties originates from the intrinsic narrow HOMO-LUMO gaps.

Organometallic Molecular Wires with Thioacetylene Backbones, trans-{RS-(C≡C)n }2 Ru(phosphine)4 : High Conductance through Non-Aromatic Bridging Linkers

In this work, the design, synthesis, and single-molecule conductance of ethynyl- and butadiynyl-ruthenium molecular wires with thioether anchor groups [RS=n-C₆ H₁₃ S, p-tert-Bu-C₆ H₄ S), trans-{RS-(C≡C)_n }₂ Ru(dppe)₂ (n=1 (1^R ), 2 (2^R ); dppe: 1,2-bis(diphenylphosphino)ethane) and trans-(n-C₆ H₁₃ S-C≡C)₂ Ru{P(OMe)₃ }₄ 3^hex ] are reported. Scanning tunneling microscope break-junction study has revealed conductance of the organometallic molecular wires with the thioacetylene backbones higher than that of the related organometallic wires having arylethynylruthenium linkages with the sulfur anchor groups, trans-{p-MeS-C₆ H₄ -(C≡C)_n }₂ Ru(phosphine)₄ 4ⁿ (n=1, 2) and trans-(Th-C≡C)₂ Ru(phosphine)₄ 5 (Th=3-thienyl). It should be noted that the molecular junctions constructed from the butadiynyl wire 2^R , trans-{Au-RS-(C≡C)₂ }₂ Ru(dppe)₂ (Au: gold metal electrode), show conductance comparable to that of the covalently linked polyynyl wire with the similar molecular length, trans-{Au-(C≡C)₃ }₂ Ru(dppe)₂ 6³ . The DFT non-equilibrium Green's function (NEGF) study supports the highly conducting nature of the thioacetylene molecular wires through HOMO orbitals.

Single-molecule junctions of multinuclear organometallic wires: long-range carrier transport brought about by metal-metal interaction

Here, we report multinuclear organometallic molecular wires having (2,5-diethynylthiophene)diyl-Ru(dppe)₂ repeating units. Despite the molecular dimensions of 2–4 nm the multinuclear wires show high conductance (up to 10⁻² to 10⁻³G₀) at the single-molecule level with small attenuation factors (β) as revealed by STM-break junction measurements. The high performance can be attributed to the efficient energy alignment between the Fermi level of the metal electrodes and the HOMO levels of the multinuclear molecular wires as revealed by DFT–NEGF calculations. Electrochemical and DFT studies reveal that the strong Ru–Ru interaction through the bridging ligands raises the HOMO levels to access the Fermi level, leading to high conductance and small β values.

Multinuclear organometallic molecular wires having (diethynylthiophene)diyl-Ru(dppe)₂ repeating units show high conductance with small attenuation factors. The strong Ru–Ru interaction is the key for the long-range carrier transport.

Single-Molecule Junction of a Cationic Rh(III) Polyyne Molecular Wire

Single-molecule conductance studies on metal-containing inorganic and organometallic molecular wires are relatively less explored compared to those on organic molecular wires. Furthermore, conductance and transmission profiles of the metal-containing wires insensitive to the metal centers often hinder rational design for high performance wires. Here, synthesis and single-molecule conductance measurements of the bis(butadiynyl)rhodium wires with tetracarbene ligands 1^H and 1^Au are reported as rare examples for Rh(III) diacetylide molecular wires. The rhodium wires derived from the terminal acetylene and gold-functionalized precursors show comparable, high single-molecule conductance ((6-7) × 10^-3 G₀) as determined by the STM break-junction measurements, suggesting formation of virtually the same covalently linked metal electrode-molecule-metal electrode junctions. The values for the metallapolyynes are larger than those of the organic polyyne wires having the similar molecular lengths. The hybrid DFT-NEGF calculations of the model systems suggest that profiles of transmission spectra are highly sensitive to the presence and species of the metal fragments doped into the polyyne molecular wire because the conductance orbitals of the metallapolyynes molecular junctions carry significant metal fragment characters. Thus, the metallapolyyne junctions turn out to be suitable platforms for rationally designed molecular wires.

Achieving Predictive Description of Molecular Conductance by Using a Range-Separated Hybrid Functional

The conductance of molecular bridges tends to be overestimated by computational studies in comparison to measured values. While this well-established trend may be related to difficulties for achieving robust bridges, the employed computational scheme can also contribute to this tendency. In particular, caveats of the traditional functionals employed in first-principles-based calculations can lead to discrepancies reflected in exaggerated conductance. Here, we show that by employing a range-separated hybrid functional the calculated values are within the same order as the measured conductance for all four considered cases. On the other hand, with B3LYP, which is a widely used functional, the calculated values greatly overestimate the conductance (by about 1-2 orders of magnitude). The improved description of the conductance with a RSH functional builds on achieving a physically meaningful treatment of the quasi particles associated with the frontier orbitals.

Organometallic molecular wires as versatile modules for energy-level alignment of the metal–molecule–metal junction

Organometallic Ru molecular wires show significantly higher conductance compared to their organic counterpart due to high-lying HOMOs.

Signature of snaking states in the conductance of core-shell nanowires

We model a core-shell nanowire (CSN) by a cylindrical surface of finite length. A uniform magnetic field perpendicular to the axis of the cylinder forms electron states along the lines of zero radial field projection, which can classically be described as snaking states. In a strong field, these states converge pairwise to quasidegenerate levels, which are situated at the bottom of the energy spectrum. We calculate the conductance of the CSN by coupling it to leads and predict that the snaking states govern transport at low chemical potential, forming isolated peaks, each of which may be split in two by applying a transverse electric field. If the contacts with the leads do not completely surround the CSN, as is usually the case in experiments, the amplitude of the snaking peaks changes when the magnetic field is rotated, determined by the overlap of the contacts with the snaking states.

Molecular design of electron transport with orbital rule: toward conductance-decay free molecular junctions

In this study, we report our viewpoint of single molecular conductance in terms of frontier orbitals.

Single-molecule conductance of π-conjugated rotaxane: new method for measuring stipulated electric conductance of π-conjugated molecular wire using STM break junction

An electronic conductance with small fluctuations, which is stipulated in single-molecule junctions, is necessary for the precise control of single-molecule devices. However, the suppression of conductance fluctuations in conventional molecular junctions is intrinsically difficult because the fluctuations are related to the contact fluctuations and molecular motion. In the present study involving experimental and theoretical investigations, it is found that covering a single π-conjugated wire with an α-cyclodextrin molecule is a promising technique for suppressing conductance fluctuations. The conductance histogram of the covered molecular junction measured with the scanning tunneling microscope break-junction technique shows that the conductance peak for the covered junction is sharper than that of the uncovered junction. The covering technique thus has two prominent effects: the suppression of intramolecular motion, and the elimination of intermolecular interactions. Theoretical calculations of electronic conductance clearly support these experimental observations.

Critical comparison of electrode models in density functional theory based quantum transport calculations

We study the performance of two different electrode models in quantum transport calculations based on density functional theory: parametrized Bethe lattices and quasi-one-dimensional wires or nanowires. A detailed account of implementation details in both the cases is given. From the systematic study of nanocontacts made of representative metallic elements, we can conclude that the parametrized electrode models represent an excellent compromise between computational cost and electronic structure definition as long as the aim is to compare with experiments where the precise atomic structure of the electrodes is not relevant or defined with precision. The results obtained using parametrized Bethe lattices are essentially similar to the ones obtained with quasi-one-dimensional electrodes for large enough cross-sections of these, adding a natural smearing to the transmission curves that mimics the true nature of polycrystalline electrodes. The latter are more demanding from the computational point of view, but present the advantage of expanding the range of applicability of transport calculations to situations where the electrodes have a well-defined atomic structure, as is the case for carbon nanotubes, graphene nanoribbons, or semiconducting nanowires. All the analysis is done with the help of codes developed by the authors which can be found in the quantum transport toolbox ALACANT and are publicly available.

Exchange and correlation in molecular wire conductance: nonlocality is the key

We study real-time electron dynamics in a molecular junction with a variety of approximations to the electronic structure, toward the ultimate aim of determining what ingredients are crucial for the accurate prediction of charge transport. We begin with real-time, all electron simulations using some common density functionals that differ in how they treat long-range Hartree-Fock exchange. We find that the inclusion or exclusion of nonlocal exchange is the dominant factor determining the transport behavior, with all semilocal contributions having a smaller effect. In order to study nonlocal correlation, we first map our junction onto a simple Pariser-Parr-Pople (PPP) model Hamiltonian. The PPP dynamics are shown to faithfully reproduce the all electron results, and we demonstrate that nonlocal correlation can be readily included in the model space using the generator coordinate method (GCM). Our PPP-GCM simulations suggest that nonlocal correlation has a significant impact on the I-V character that is not captured even qualitatively by any of the common semilocal approximations to exchange and correlation. The implications of our results for transport calculations are discussed.

A combined DFT/Green's function study on electrical conductivity through DNA duplex between Au electrodes

Electrical conducting properties of DNA duplexes sandwiched between Au electrodes have been investigated by use of first-principles molecular simulation based on DFT and Green's function to elucidate the origin of their base sequence dependence. The theoretically simulated effects of DNA base sequence on the electrical conducting properties are in qualitative agreement with experiment. The HOMOs localized on Guanine bases have the major contribution to the electrical conductivity through DNA duplexes.

Nonequilibrium electronic transport of 4,4'-bipyridine molecular junction

The electronic transport properties of a 4,4'-bipyridine molecule sandwiched between two Au(111) surfaces are studied with a fully self-consistent nonequilibrium Green's-function method combined with the density-functional theory. The 4,4'-bipyridine molecule prefers to adsorb near the hollow site of the Au(111) surface and distorts slightly. The modifications on the electronic structure of the molecule due to the presence of the electrodes are described by the renormalized molecular orbitals, which correspond well to the calculated transmission peaks. The average Fermi level lies close to the lowest unoccupied renormalized molecular orbital, which determines the electronic transport property of the molecular junction under a small bias voltage. The total transmission is contributed by a single channel. The transmission peaks shift with the applied bias voltage, and this behavior depends on the spatial distribution of the renormalized molecular orbitals and the voltage drop along the molecular junction. The shape of the calculated conductance curve of the equilibrium geometric configuration reproduces the main feature of the experimental results, but the value is larger than the measured data by about 6 times. Good agreement with the experimental measurements can be obtained by elongating the molecular junction. The electronic transport behaviors depend strongly on the interface configuration.

Analysis on the contribution of molecular orbitals to the conductance of molecular electronic devices

We present a theoretical approach which allows one to extract the orbital contribution to the conductance of molecular electronic devices. This is achieved by calculating the scattering wave functions after the Hamiltonian matrix of the extended molecule is obtained from a self-consistent calculation that combines the nonequilibrium Green's function formalism with density functional theory employing a finite basis of local atomic orbitals. As an example, the contribution of molecular orbitals to the conductance of a model system consisting of a 4,4-bipyridine molecule connected to two semi-infinite gold monatomic chains is explored, illustrating the capability of our approach.

Dithiocarbamate Anchoring in Molecular Wire Junctions: A First Principles Study

Recent experimental realization [J. Am. Chem. Soc., 127 (2005) 7328] of various dithiocarbamate self-assembly on gold surface opens the possibility for use of dithiocarbamate linkers to anchor molecular wires to gold electrodes. In this paper, we explore this hypothesis computationally. We computed the electron transport properties of 4,4'-bipyridine (BP), 4,4'-bipyridinium-1,1'-bis(carbodithioate) (BPBC), 4-(4'-pyridyl)-peridium-1-carbodithioate (BPC) molecule junctions based on the density functional theory and nonequilibrium Green's functions. We demonstrated that the stronger molecule-electrode coupling associated with the conjugated dithiocarbamate linker broadens transmission resonances near the Fermi energy. The broadening effect along with the extension of the pi conjugation from the molecule to the gold electrodes lead to enhanced electrical conductance for BPBC molecule. The conductance enhancement factor is as large as 25 at applied voltage bias 1.0 V. Rectification behavior is predicted for BPC molecular wire junction, which has the asymmetric anchoring groups.

Quantum Transport Effects in Copper(II) Phthalocyanine Sandwiched between Gold Nanoelectrodes

The electrical transmission of copper(II) phthalocyanine (CuPc) sandwiched between gold nanoelectrodes is studied on the basis of the Green function formalism coupled with the Gaussian-broadening technique. In the Au-CuPc-Au junction, broadened density of states (DOS) of the Au chains is defined as continuous DOS of electrodes to calculate the Green function of the electrodes. Two peaks of the transmission function found in the vicinity of the Fermi level are analyzed in terms of molecular orbitals (MOs). A convenient procedure to analyze MO contribution to a transmission peak is proposed. It is found that (I) symmetry-matched interactions between CuPc and the gold nanoelectrodes are important to the enhancement of the transmission function and (II) the nanoelectrodes have almost no effect on the electronic states of CuPc.