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Advances and challenges in single-molecule electron transport

Abstract

Electronic transport properties of single-molecule junctions have been widely measured by several techniques, including mechanically controllable break junctions, electromigration break junctions, and by means of scanning tunneling microscopes. In parallel, many theoretical tools have been developed and refined for describing such transport properties and for obtaining numerical predictions. Most prominent among these theoretical tools are those based upon density functional theory. In this review, theory and experiment are critically compared, and this confrontation leads to several important conclusions. The theoretically predicted trends nowadays reproduce the experimental findings well for series of molecules with a single well-defined control parameter, such as the length of the molecules. The quantitative agreement between theory and experiment usually is less convincing, however. Two main sources for the quantitative discrepancies can be identified. Experimentally, the atomic structure of the junction typically realized in the measurement is not well known, so simulations rely on plausible scenarios. In theory, correlation effects can be included only in approximations that are difficult to control for experimentally relevant situations. Therefore, one typically expects qualitative agreement with present modeling tools; in exceptional cases a quantitative agreement has already been achieved. For further progress, benchmark systems are required that are sufficiently well defined by experiment to allow quantitative testing of the approximation schemes underlying the theoretical modeling. Several key experiments can be identified suggesting that the present description may even be qualitatively incomplete in some cases. Such key experimental observations and their current models are also discussed here, leading to several suggestions for extensions of the models toward including dynamic image charges, electron correlations, and polaron formation.

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DOI:https://doi.org/10.1103/RevModPhys.92.035001

1. Research Areas
1. Physical Systems
Condensed Matter & Materials Physics

Authors & Affiliations

Ferdinand Evers

• Institut für Theoretische Physik, Universität Regensburg, D-93053 Regensburg, Germany

• Department of Condensed Matter Physics, Faculty of Mathematics and Physics, Charles University, Ke Karlovu 5, 121 16 Praha 2, Czech Republic

• Huygens-Kamerlingh Onnes Laboratory, Leiden University, Niels Bohrweg 2, 2333 CA Leiden, Netherlands and Department of Materials, University of Oxford, OX1 3PH Oxford, United Kingdom

• Huygens-Kamerlingh Onnes Laboratory, Leiden University, Niels Bohrweg 2, 2333 CA Leiden, Netherlands

• *ruitenbeek@physics.leidenuniv.nl

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References

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Vol. 92, Iss. 3 — July - September 2020

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Images

• Figure 1

Relaxed geometries representing three typical arrangements of an alkanedithiol molecule bridged between Au electrodes and calculated length dependence of the conductance for these arrangements. The results illustrate the spreading of conductances that can occur due to structural modifications. From [252].

• Figure 2

Overview of experimental techniques aimed at measuring single-molecule transport. (a) Notched-wire mechanically controllable break junction. (b) Lithographically fabricated mechanically controllable break junction. (c) Electromigration break junctions. (d) STM break junction by repeated indentation. (e) $I\left(t\right)$ or $I\left(s\right)$ operation of STM. (f) Low-temperature UHV STM manipulation of individual molecules.

• Figure 3

Experiment probing the conductance of a single molecule by repeated indentation of a Au STM tip into the Au metal surface, in solution of 4,4-bipyridine. The breaking of the metal-metal contact is observed as steps in the conductance (a) near multiples of ${G}_{0}$, giving rise to (b) peaks in the conductance histogram. (c) Enlarging to lower conductance, additional steps are resolved, and (d) for many repeats of breaking this produces a new series of peaks in the conductance histograms at small fractions of ${G}_{0}$. (e),(f) Tests with pure solvent show only tunneling characteristics. From [520].

• Figure 4

Illustration of partitioning in model calculations: molecule, extended molecule (shown in red), semi-infinite leads. Adapted from [74].

• Figure 5

Transmission function of 4,4’-vinylenedipyridine junction with Au and Ag electrodes (yellow and gray lines, respectively) calculated using STAIT. The vertical dashed line indicates the Fermi energy ${\epsilon }_{F}$. Dashed curves are Lorentzian fits. (Inse) The relaxed geometry with gold leads. From [1].

• Figure 6

Computational results for Au-benzenedithiol-Au junctions under high applied bias ${V}_{b}$. Atomic structure is indicated together with the electronic orbitals (density clouds) nearest ${\epsilon }_{F}$. At ${V}_{b}=1.1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{V}$ (right panels) the orbitals shift in energy, but they are also heavily distorted relative to 0 V (left panels). From [12].

• Figure 7

Impact of self-consistency achieved under bias in transport computations for the Au-BDT-Au junction of Fig. 6. (Top panel) $I\text{−}V$ curve (right axis) and differential conductance $dI/dV$ (left axis). (Bottom panel) Comparison of the transmission at zero bias and at ${V}_{b}=0.82\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{V}$. The vertical dashed lines are placed at ${\epsilon }_{F}-{V}_{b},{\epsilon }_{F},{\epsilon }_{F}+{V}_{b}$. The three peaks visible at zero bias correspond to LUMO, the pair HOMO and HOMO-1, and HOMO-2. The central peak is suppressed at the finite bias; see the text for an explanation. From [12].

• Figure 8

Schematic illustration of the similarity between two models at two extreme limits. (a) When the single-particle level separation $\mathrm{\Delta }={ϵ}_{1}-{ϵ}_{0}$ and the level broadening $\mathrm{\Gamma }$ are much larger than the on-site Coulomb repulsion $U$, we may neglect the latter. The transmission has two spin degenerate resonances, and the $I\text{−}V$ characteristics (b) are expected to show a generic $\mathsf{S}$ shape determined predominantly by the nearest level, in this example the HOMO. (c) In the other limit, when $U\gg \mathrm{\Gamma }$ we again obtain two resonances, but here they are single-spin resonances separated by $U$. However, when measuring an $I\text{−}V$ curve the shape (d) will be indistinguishable from the noninteracting two-level model, although the amplitudes may differ. Here we assume that ${T}_{K}\ll T$.

• Figure 9

The differential conductance of a TIAM near the singlet-triplet transition displayed in the plane of gate or bias voltage (units, ${U}_{1}={U}_{2}$). Effectively, the gate voltage fine-tunes the difference between singlet and triplet energies with ${V}_{G}^{*}=0.25$ at criticality. At ${V}_{G}>{V}_{G}^{*}$, a Kondo peak and triplet-singlet side peaks emerge, while ${V}_{G}<{V}_{G}^{*}$ there is a singlet gap. Color coding is such that bright yellow implies unitary conductance. From [109].

• Figure 10

Schematic illustration of a molecule with two-point contacts and the sublattice structure of the honeycomb lattice.

• Figure 11

Double-dot molecule used in the experiment ([22]). Owing to the presence of the side groups, the center benzene rings are tilted against each other, leading to a partial decoupling of the left and right $\pi$ systems.

• Figure 12

Two examples of edge structures that are always associated with supernumerary zero modes. (Left panel) The two singly connecting sites share the same binding partner. Therefore, only one of them can form a double bond; the other becomes a radical. (Right panel) A similar pattern. From [503].

• Figure 13

Structures of (left panel) the molecules anthraquinone and (right panel) the molecule LC2. From [348].

• Figure 14

Azulene with different positions of the linker group (denoted by “Ar-”) and corresponding ab initio transmissions. The molecule is not bipartite, so the ab initio transmissions exhibit nodes shifted from the gap center, in agreement with the rules discussed in the text. From [512].

• Figure 15

Conductance measured for a series of diaminoacenes as a function of the number $N$ of benzo rings (red dots, left axis). When measured in the transverse direction (upper data points) the conductance increases with $N$, while when measured longitudinally the conductance decreases. The log-linear plot in the inset shows that the latter is not a simple exponential decrease. The two trends are reproduced by calculations for the square of the tunnel coupling (blue crosses, right scale). From [365].

• Figure 16

Transmission resonances of oligoacenes directly bound to Ag contacts for geometries as exemplified in the inset for anthracene. This provides an example of the evolution of the transmission resonances that exhibit growth of the transmission with increasing molecular length. The numbers above the curves indicate the numbers of carbon rings: benzene (1), naphthalene (2), anthracene (3), tetracene (4), pentacene (5), and hexacene (6). From [528].

• Figure 17

Conductance as a function of the twist angle between two phenyl rings. (Left panel) By design, steric interactions constrain the twist angles between the two phenyl rings of a series of molecules. The smaller the angle $\theta$ the larger the overlap of the wave functions (the conjugation) between the two ring sections. (Right panel) The measured conductance follows the expected ${\mathrm{cos}}^{2}\theta$ dependence. Adapted from [491].

• Figure 18

Transmission functions of Au-alkanedithiol-Au series, where the number of carbons is denoted by $n$. The length dependence of the conductance is determined by the spatially delocalized ${\mathrm{HOMO}}^{*}$ resonance and a broad resonance around $E=-1.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{eV}$, which corresponds to the gateway state (HOMO, “$S$ level”). (Inset) The wave function isosurface of the HOMO and the ball-and-stick model with carbons (black), hydrogens (white), and sulphurs at the ends (yellow). From [252].

• Figure 19

One-path and two-path molecules used for demonstrating constructive quantum interference in the experiment by [490].

• Figure 20

Conducting-tip AFM measurements of (left panel) the conductance of linearly (anthracene-based) and (right panel) cross-conjugated (anthraquinone-based) molecules. Two-dimensional conductance histograms are shown, constructed by logarithmic binning of $dI/dV$ in units ${\mathrm{\Omega }}^{-1}$ versus bias voltage. The color scale indicates the number of counts, ranging from black (0 counts) to white ($>40$ counts). Destructive QI suppresses transport in cross-conjugated molecules (right panel). Adapted from [146].

• Figure 21

Transmission curves from ab initio for anthracene (AC) and anthraquinone (AQ) molecules shown in Fig. 20; AQ-MT (AQ-monothiol) contains only one thiol end group. Vertical bars at the bottom indicate resonance centers of HOMO-1, HOMO, and LUMO for AC-DT (top), AQ-DT (middle), and AQ-MT (bottom). The AC molecules show a pronounced dip between the HOMO and LUMO peaks, which is consistent with rules for bipartite lattices discussed in Sec. 4b4. Adapted from [146].

• Figure 22

(Left panel) Molecules investigated by [290]. Similar species with mixed end groups were investigated by [22]. (Right panel) Molecules investigated by [14].

• Figure 23

Molecules with anthanthrene core studied by [133].

• Figure 24

Schematics of molecular levels leading to asymmetric and nonmonotonous $I\text{−}V$ characteristics. In these diagrams the vertical scale represents energy, while the horizontal axis is a space coordinate. Filled states in leads are indicated by the blue shaded range. The lines at the top show the local electrostatic potential (vacuum level). The gray bars show effective energy barriers. The asymmetry can be due to differences in coupling to the two leads, represented by different barrier widths (a). When the molecule has two sites separated by an internal barrier (b), the current becomes for coherent transport high when the two levels are shifted into resonance by the applied potential. Level alignment was not considered in the original proposal by Aviram and Ratner. In that proposal, the high current was taken to result from a cascade between the levels, involving relaxation by vibron excitation.

• Figure 25

Temperature dependence of the linear conductance of ${\mathrm{C}}_{60}$ in a gold junction for different electrode separations . Solid lines are fits to $G\left(T\right)={G}_{0}{\left[1+{\left(T/{T}_{K}\right)}^{2}\left({2}^{1/s}-1\right)\right]}^{-s}+{G}_{b}$, where $s=0.22$ and ${G}_{b},{G}_{0},{T}_{K}$ are parameters. The widely used formula is a fit to the theoretical $G\left(T\right)$ from a numerical renormalization-group calculation ([62]; [139]). The extracted ${T}_{K}$ are (from the top trace to the bottom) $60.3±2.4$, $55.5±0.9$, $45.6±1.9$, and $38.1±1.2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{K}$. (Inset) $dI/dV$ traces at ${x}_{0}+0.7\text{\hspace{0.17em}}\text{\hspace{0.17em}}Å$. From [337].

• Figure 26

Illustration of the Franck-Condon blockade in a molecular junction assuming a parabolic potential surface with eigenenergies $n\hslash {\omega }_{0}$ and vibronic (harmonic oscillator) wave functions. (a) At low excitation levels transport is suppressed because the oscillator overlap between $|N,0⟩$ and $|N+1,1⟩$ is small. (b) At higher bias, higher vibrational levels $|N+1,n⟩$, the overlap with $|N,0⟩$ is enhanced and current starts to flow. Adapted from [246].

• Figure 27

(a) Current stability diagram of a functionalized ${\mathrm{C}}_{60}$ bound to graphene nanoelectrodes at 20 mK. (b) Simulated current stability diagram with $\lambda =3$, $\hslash {\omega }_{0}=1.7\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{meV}$. The current is suppressed for low bias voltage $|{V}_{b}|\lesssim 6\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mV}$ because of the Franck-Condon blockade. The blockade cannot be lifted by changing the gate voltage ${V}_{g}$. For higher bias, the current is restored; its detailed ${V}_{g,b}$ dependence bears a strong imprint of vibronic excitations. From [246].

• Figure 28

Proposed $\mathrm{Pt}\text{−}{\mathrm{H}}_{2}\text{−}\mathrm{Pt}$ molecular configuration. The dark spheres represent Pt atoms of the metallic leads, for which the detailed arrangement is not known. The white spheres represent the H atoms that are aligned with the axis of the junction.

• Figure 29

Single-molecule measurements on a thiol-coupled molecule (shown in the inset in the top panel) having side groups that make it asymmetric. The current (right axis) and differential conductance (left axis) reflect the asymmetry of the molecule. The sign of the asymmetry is random and changes between different measured junctions. The gap in the conductance observed (a) at room temperature become much more pronounced (b) at 30 K. For each curve, several consecutive sweeps are shown. Adapted from [380].

• Figure 30

A selection of conductance histogram data for Au-BDT-Au junctions illustrating the wide variation and sensitivity to experimental conditions. The histograms are rescaled for ease of comparison, and have been adapted from [517]; [286]; [42]; [222]; and [537]. The differences in appearance of the data are due to the choice of the methods of constructing histograms; see also Sec. 2d. Three methods have been applied: (1) First binning data in linear conductance and representing this in a histogram in linear conductance. (2) First binning data in linear conductance and representing this in a histogram in the logarithm of conductance. (3) Binning the logarithm of conductance and representing this in a histogram. The third method is now the method of choice, but earlier data used the other two. The data by Xiao, Xu, and Tao (top panel) uses method (1) and is mapped onto this logarithmic scale. The data by Kim et al. uses method (2). This enhances the data at low conductance, and the peaks are observed against a decaying background. The other three examples employ method (3).

• Figure 31

Overview of measured conductance values for Au-ADT-Au junctions plotted on a semilogarithmic scale as a function of the length of the backbone in terms of the number of ${\mathrm{CH}}_{2}$ groups. Symbols are used for distinguishing low-conductance peaks (green stars), medium-conductance peaks (red diamonds), and high-conductance peaks (blue triangles), following the categories introduced by [252]. Histograms that show only a single peak and for which the identification in terms of these three categories is not discussed are shown as gray bullets. The data include published work from [68], [520], [521], [156], [155][52], [140], [167], [194], [253], [252], [412], [475], [507], [180], [314], [335], [171], [182], [218], [513], [424], [425][284], [13], [221], [223], [372], [441], and [136]. The spread in the data is large, but the systematic trends in the separate works are shown by connecting the data points obtained from three works, namely, [475], the H peak from [252], and the H peak from [155]. This indicates that the decay constant $\beta$ is a more robust and reproducible property, as illustrated for the complete dataset by the histogram in the inset. More complete details on the data collected here are given in the Supplemental Material ([542]).

• Figure 32

Comparison of conductance histograms for OPE3 obtained with different experimental methods, under different conditions, and by different experimental groups ([181]; [511]; [208]; [118]). In all cases, the logarithm of the conductance was used for binning the data into a semilog plot of counts as a function of the logarithm of the conductance. The position of the peak in the conductance reproduces to within a factor of 2 in conductance, and the width of the peak is also fairly reproducible.

• Figure 33

STM experiment on the tripod molecule propynyltrioxatriangulenium (P-TOTA) and a comparison with computations. (a) Structure of the molecule. (b) Height profile of a single P-TOTA molecule as observed in a STM image recorded at 4.5 K for a 100 mV bias and at a 30 pA current set point. (c) Height profile along the trajectory indicated in (b) and compared with computations. (d) Conductance-distance dependence for approach of the tip to the center of the molecule from above. The experimental curves are shown for 40 forward and backward traces. The configurations of the molecule in the computations for the positions of the tip at the numbered stages are shown in the lower panels. From [196].

• Figure 34

Observations of anomalous behavior for an OPE3 molecule coupled by thiol bonds to Au in MCBJ experiments at 10 K. (a) Traces of conductance recorded while breaking the molecular junction, shown for low bias [obtained from the slope of $I\left(V\right)$ around $V=0$] and high bias voltage (obtained from the ratio $I/V$ at $V=0.9\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{V}$). While the two curves nearly coincide up to the point marked as ${d}_{0}$, at this point there is a sudden break in the curve for low bias voltage, while the high-bias curve for the same junction continues without interruption. (b) Current-voltage curves shown for the four different electrode separations marked by the colored arrows in (a). A vertical offset is applied for clarity. Following a smooth evolution of $I\left(V\right)$ at lower electrode separation, a sharp transition is seen to a gapped state. From [118].

• Figure 35

A thiolated arylethynylene molecule with a 9,10-dihydroanthracene core (the three rings in the center). The core divides the molecule into two conjugated parts.

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