Quantum Interference in Graphene Nanoconstrictions

Magnetotransport spectroscopy of electroburnt graphene nanojunctions

We have reported the precise methodology for fabricating graphene quantum dots through electroburning and performed measurements on the Coulomb blockade and oscillation phenomena. The diameters of graphene quantum dots can be estimated to range from several to tens of nanometers, utilizing the disk capacitance model and the two-dimensional quantum well model. By subjecting the quantum dots to a vertical magnetic field, an obvious alteration in conductance can be detected at the point of resonance tunneling. This observed phenomenon can be attributed to the modification in the density of states of Landau levels within the graphene leads. Moreover, by manipulating the gate voltage, it is possible to regulate the Fermi level of the lead, resulting in distinct magnetoresistance of different electron states. The presence of this lead effect may potentially disrupt the magnetic response analysis of graphene-based single-molecule transistors, necessitating a comprehensive theoretical examination to mitigate such interference.

Regulation of quantum spin conversions in a single molecular radical

Free radicals, generally formed through the cleavage of covalent electron-pair bonds, play an important role in diverse fields ranging from synthetic chemistry to spintronics and nonlinear optics. However, the characterization and regulation of the radical state at a single-molecule level face formidable challenges. Here we present the detection and sophisticated tuning of the open-shell character of individual diradicals with a donor-acceptor structure via a sensitive single-molecule electrical approach. The radical is sandwiched between nanogapped graphene electrodes via covalent amide bonds to construct stable graphene-molecule-graphene single-molecule junctions. We measure the electrical conductance as a function of temperature and track the evolution of the closed-shell and open-shell electronic structures in real time, the open-shell triplet state being stabilized with increasing temperature. Furthermore, we tune the spin states by external stimuli, such as electrical and magnetic fields, and extract thermodynamic and kinetic parameters of the transition between closed-shell and open-shell states. Our findings provide insights into the evolution of single-molecule radicals under external stimuli, which may proof instrumental for the development of functional quantum spin-based molecular devices.

Subthreshold slope below 60 mV/decade in graphene transistors induced by channel geometry at the wafer-scale

In this paper, we demonstrate experimentally that field-effect transistors with nanoconstricted graphene monolayer channels have a subthreshold swing (SS) below 60 mV/dec, which is slightly dependent on temperature. Two shapes of nanoconstricted graphene monolayers are considered: (i) a bow-tie shape, representative for a symmetric channel, and (ii) a trapezoidal shape, which illustrates an asymmetric channel. While both types of nonuniform channels are opening a bandgap in graphene, thus showing an on/off ratio of 105, the SS in the graphene bow-tie channel is below 60 mV/dec in the temperature range 25 °C-44 °C.

Thermoelectric Response Enhanced by Surface/Edge States in Physical Nanogaps

Current solid-state thermoelectric converters have poor performance, which typically renders them useless for practical applications. This problem is evidenced by the small figures of merit of typical thermoelectric materials, which tend to be much smaller than 1. Increasing this parameter is then key for the development of functional devices in technologically viable applications that can work optimally. We propose here a feasible and effective design of new thermoelectric systems based on physical gaps in nanoscale junctions. We show that, depending on the type of features, i.e., the character of surface/edge states, on both sides of the gap, it is possible to achieve high figures of merit. In particular, we show that, for configurations that have localized states at the surfaces/edges, which translate into sharp resonances in the transmission, it is possible to achieve large Seebeck coefficients and figures of merit by carefully tuning their energy and their coupling to other states. We calculate the thermoelectric coefficients as a function of different parameters and find non-obvious behaviors, such as the existence of a certain coupling between the localized and bulk states for which these quantities have a maximum. The highest Seebeck coefficients and figures of merit are achieved for symmetric junctions, which have the same coupling between the localized state and the bulk states on both sides of the gap. The features and trends of the thermoelectric properties and their changes with various parameters that we find here can be applied not only to systems with nanogaps but also to many other nanoscale junctions, such as those that have surface states or states localized near the contacts between the nanoscale object and the electrodes. The model presented here can, therefore, be used to characterize and predict the thermoelectric properties of many different nanoscale junctions and can also serve as a guide for studying other systems. These results pave the way for the design and fabrication of stable next-generation thermoelectric devices with robust features and improved performance.

Effect of Impurity Adsorption on the Electronic and Transport Properties of Graphene Nanogaps

Graphene stands out as a versatile material with several uses in fields that range from electronics to biology. In particular, graphene has been proposed as an electrode in molecular electronics devices that are expected to be more stable and reproducible than typical ones based on metallic electrodes. In this work, we study by means of first principles, simulations and a tight-binding model the electronic and transport properties of graphene nanogaps with straight edges and different passivating atoms: Hydrogen or elements of the second row of the periodic table (boron, carbon, nitrogen, oxygen, and fluoride). We use the tight-binding model to reproduce the main ab-initio results and elucidate the physics behind the transport properties. We observe clear patterns that emerge in the conductance and the current as one moves from boron to fluoride. In particular, we find that the conductance decreases and the tunneling decaying factor increases from the former to the latter. We explain these trends in terms of the size of the atom and its onsite energy. We also find a similar pattern for the current, which is ohmic and smooth in general. However, when the size of the simulation cell is the smallest one along the direction perpendicular to the transport direction, we obtain highly non-linear behavior with negative differential resistance. This interesting and surprising behavior can be explained by taking into account the presence of Fano resonances and other interference effects, which emerge due to couplings to side atoms at the edges and other couplings across the gap. Such features enter the bias window as the bias increases and strongly affect the current, giving rise to the non-linear evolution. As a whole, these results can be used as a template to understand the transport properties of straight graphene nanogaps and similar systems and distinguish the presence of different elements in the junction.

Quantum Circuit Rules for Molecular Electronic Systems: Where Are We Headed Based on the Current Understanding of Quantum Interference, Thermoelectric, and Molecular Spintronics Phenomena?

In this minireview, we discuss important aspects of the various quantum phenomena (such as quantum interference, spin-dependent charge transport, and thermoelectric effects) relevant in single-molecule charge transport and list some of the basic circuit rules devised for different molecular systems. These quantum phenomena, in conjunction with the existing empirical circuit rules, can help in predicting some of the structure-property relationships in molecular circuits. However, a universal circuit law that predicts the charge transport properties of a molecular circuit has not been derived yet. Having such law(s) will help to design and build a complex molecular device leading to exciting unique applications that are not possible with the traditional silicon-based technologies. Based on the existing knowledge in the literature, here we open the discussion on the possible future research directions for deriving unified circuit law(s) to predict the charge transport in complex single-molecule circuits.

Electrode effects on the observability of destructive quantum interference in single-molecule junctions

Destructive quantum interference (QI) has been a source of interest as a new paradigm for molecular electronics as the electronic conductance is widely dependent on the occurrence or absence of destructive QI effects. In order to interpret experimentally observed transmission features, it is necessary to understand the effects of all components of the junction on electron transport. We perform non-equilibrium Green's function calculations within the framework of density functional theory to assess the structure-function relationship of transport through pyrene molecular junctions with distinct QI properties. The chemical nature of the anchor groups and the electrodes controls the Fermi level alignment, which determines the observability of destructive QI. A thorough analysis allows to disentangle the transmission features arising from the molecule and the electrodes. Interestingly, graphene electrodes introduce features in the low-bias regime, which can either mask or be misinterpreted as QI effects, while instead originating from the topological properties of the edges. Thus, this first principles analysis provides clear indications to guide the interpretation of experimental studies, which cannot be obtained from simple Hückel model calculations.

Graphene nanogaps for the directed assembly of single-nanoparticle devices

Significant advances in the synthesis of low-dimensional materials with unique and tuneable electrical, optical and magnetic properties has led to an explosion of possibilities for realising hybrid nanomaterial devices with unconventional and desirable characteristics. However, the lack of ability to precisely integrate individual nanoparticles into devices at scale limits their technological application. Here, we report on a graphene nanogap based platform which employs the large electric fields generated around the point-like, atomically sharp nanogap electrodes to capture single nanoparticles from solution at predefined locations. We demonstrate how gold nanoparticles can be trapped and contacted to form single-electron transistors with a large coupling to a buried electrostatic gate. This platform offers a route to the creation of novel low-dimensional devices, nano- and optoelectronic applications, and the study of fundamental transport phenomena.

A Mechanically Tunable Quantum Dot in a Graphene Break Junction

Graphene quantum dots (QDs) are intensively studied as platforms for the next generation of quantum electronic devices. Fine tuning of the transport properties in monolayer graphene QDs, in particular with respect to the independent modulation of the tunnel barrier transparencies, remains challenging and is typically addressed using electrostatic gating. We investigate charge transport in back-gated graphene mechanical break junctions and reveal Coulomb blockade physics characteristic of a single, high-quality QD when a nanogap is opened in a graphene constriction. By mechanically controlling the distance across the newly formed graphene nanogap, we achieve reversible tunability of the tunnel coupling to the drain electrode by 5 orders of magnitude, while keeping the source-QD tunnel coupling constant. The break junction device can therefore become a powerful platform to study the physical parameters that are crucial to the development of future graphene-based devices, including energy converters and quantum calorimeters.

Sub-10 nm nanogap fabrication on suspended glassy carbon nanofibers

Glassy carbon nanofibers (GCNFs) are considered promising candidates for the fabrication of nanosensors for biosensing applications. Importantly, in part due to their great stability, carbon electrodes with sub-10 nm nanogaps represent an attractive platform for probing the electrical characteristics of molecules. The fabrication of sub-10 nm nanogap electrodes in these GCNFs, which is achieved by electrically stimulating the fibers until they break, was previously found to require fibers shorter than 2 µm; however, this process is generally hampered by the limitations inherent to photolithographic methods. In this work, to obtain nanogaps on the order of 10 nm without the need for sub-2 µm GCNFs, we employed a fabrication strategy in which the fibers were gradually thinned down by continuously monitoring the changes in the electrical resistance of the fiber and adjusting the applied voltage accordingly. To further reduce the nanogap size, we studied the mechanism behind the thinning and eventual breakdown of the suspended GCNFs by controlling the environmental conditions and pressure during the experiment. Following this approach, which includes performing the experiments in a high-vacuum chamber after a series of carbon dioxide (CO₂) purging cycles, nanogaps on the order of 10 nm were produced in suspended GCNFs 52 µm in length, much longer than the ~2 µm GCNFs needed to produce such small gaps without the procedure employed in this work. Furthermore, the electrodes showed no apparent change in their shape or nanogap width after being stored at room temperature for approximately 6 months.

Robust graphene-based molecular devices

One of the main challenges to upscale the fabrication of molecular devices is to achieve a mechanically stable device with reproducible and controllable electronic features that operates at room temperature^1,2. This is crucial because structural and electronic fluctuations can lead to significant changes in the transport characteristics at the electrode-molecule interface^3,4. In this study, we report on the realization of a mechanically and electronically robust graphene-based molecular junction. Robustness was achieved by separating the requirements for mechanical and electronic stability at the molecular level. Mechanical stability was obtained by anchoring molecules directly to the substrate, rather than to graphene electrodes, using a silanization reaction. Electronic stability was achieved by adjusting the π-π orbitals overlap of the conjugated head groups between neighbouring molecules. The molecular devices exhibited stable current-voltage (I-V) characteristics up to bias voltages of 2.0 V with reproducible transport features in the temperature range from 20 to 300 K.

Charge-state assignment of nanoscale single-electron transistors from their current-voltage characteristics

The electronic and magnetic properties of single-molecule transistors depend critically on the molecular charge state. Charge transport in single-molecule transistors is characterized by Coulomb-blocked regions in which the charge state of the molecule is fixed and current is suppressed, separated by high-conductance, sequential-tunneling regions. It is often difficult to assign the charge state of the molecular species in each Coulomb-blocked region due to variability in the work-function of the electrodes. In this work, we provide a simple and fast method to assign the charge state of the molecular species in the Coulomb-blocked regions based on signatures of electron-phonon coupling together with the Pauli-exclusion principle, simply by observing the asymmetry in the current in high-conductance regions of the stability diagram. We demonstrate that charge-state assignments determined in this way are consistent with those obtained from measurements of Zeeman splittings. Our method is applicable at 77 K, in contrast to magnetic-field-dependent measurements, which generally require low temperatures (below 4 K). Due to the ubiquity of electron-phonon coupling in molecular junctions, we expect this method to be widely applicable to single-electron transistors based on single molecules and graphene quantum dots. The correct assignment of charge states allows researchers to better understand the fundamental charge-transport properties of single-molecule transistors.

Driving interference control by side carbon chains in molecular and two-dimensional nano-constrictions

Interference pattern modulation by side carbon chains is a general phenomenon, which is demonstrated in a benzene molecular device, a zigzag graphene nanoribbon device and a SiC nanoribbon device.

Mechanically controlled quantum interference in graphene break junctions

The ability to detect and distinguish quantum interference signatures is important for both fundamental research and for the realization of devices such as electron resonators¹, interferometers² and interference-based spin filters³. Consistent with the principles of subwavelength optics, the wave nature of electrons can give rise to various types of interference effects⁴, such as Fabry-Pérot resonances⁵, Fano resonances⁶ and the Aharonov-Bohm effect⁷. Quantum interference conductance oscillations⁸ have, indeed, been predicted for multiwall carbon nanotube shuttles and telescopes, and arise from atomic-scale displacements between the inner and outer tubes^9,10. Previous theoretical work on graphene bilayers indicates that these systems may display similar interference features as a function of the relative position of the two sheets^11,12. Experimental verification is, however, still lacking. Graphene nanoconstrictions represent an ideal model system to study quantum transport phenomena^13-15 due to the electronic coherence¹⁶ and the transverse confinement of the carriers¹⁷. Here, we demonstrate the fabrication of bowtie-shaped nanoconstrictions with mechanically controlled break junctions made from a single layer of graphene. Their electrical conductance displays pronounced oscillations at room temperature, with amplitudes that modulate over an order of magnitude as a function of subnanometre displacements. Surprisingly, the oscillations exhibit a period larger than the graphene lattice constant. Charge-transport calculations show that the periodicity originates from a combination of the quantum interference and lattice commensuration effects of two graphene layers that slide across each other. Our results provide direct experimental observation of a Fabry-Pérot-like interference of electron waves that are partially reflected and/or transmitted at the edges of the graphene bilayer overlap region.

Stable-radicals increase the conductance and Seebeck coefficient of graphene nanoconstrictions

Nanoscale thermoelectricity is an attractive target technology, because it can convert ambient heat into electricity for powering embedded devices in the internet of things. We demonstrate that the thermoelectric performance of graphene nanoconstrictions can be significantly enhanced by the presence of stable radical adsorbates, because radical molecules adsorbed on the graphene nanoconstrictions create singly-occupied orbitals in the vicinity of Fermi energy. This in turn leads to sharp features in their transmission functions close to Fermi energy, which increases the electrical conductance and Seebeck coefficient of the nanoconstrictions. This is a generic feature of radical adsorbates and can be employed in the design of new thermoelectric devices and materials.

Spin signatures in the electrical response of graphene nanogaps

We analyse the electrical response of narrow graphene nanogaps in search for transport signatures stemming from spin-polarized edge states. We find that the electrical transport across graphene nanogaps having perfectly defined zigzag edges does not carry any spin-related signature. We also analyse the magnetic and electrical properties of nanogaps whose electrodes have wedges that possibly occur in the currently fabricated nanogaps. These wedges can host spin polarized wedge low-energy states due to the bipartite nature of the graphene lattice. We find that these spin-polarized low-energy modes give rise to low-voltage signatures in the differential conductance and to distinctive features in the stability diagrams. These are caused by fully spin-polarized currents.

Quantum Interference Assisted Spin Filtering in Graphene Nanoflakes

We demonstrate that hexagonal graphene nanoflakes with zigzag edges display quantum interference (QI) patterns analogous to benzene molecular junctions. In contrast with graphene sheets, these nanoflakes also host magnetism. The cooperative effect of QI and magnetism enables spin-dependent quantum interference effects that result in a nearly complete spin polarization of the current and holds a huge potential for spintronic applications. We understand the origin of QI in terms of symmetry arguments, which show the robustness and generality of the effect. This also allows us to devise a concrete protocol for the electrostatic control of the spin polarization of the current by breaking the sublattice symmetry of graphene, by deposition on hexagonal boron nitride, paving the way to switchable spin filters. Such a system benefits from all of the extraordinary conduction properties of graphene, and at the same time, it does not require any external magnetic field to select the spin polarization, as magnetism emerges spontaneously at the edges of the nanoflake.

Building nanogapped graphene electrode arrays by electroburning

Carbon nanoelectrodes with nanogap are reliable platforms for achieving ultra-small electronic devices. One of the main challenges in fabricating nanogapped carbon electrodes is precise control of the gap size. Herein, we put forward an electroburning approach for controllable fabrication of graphene nanoelectrodes from preprocessed nanoconstriction arrays. The electroburning behavior was investigated in detail, which revealed a dependence on the size of nanoconstriction units. The electroburnt nanoscale electrodes showed the capacity to build molecular devices. The methodology and mechanism presented in this study provide significant guidance for the fabrication of proper graphene and other carbon nanoelectrodes.

An approach for the efficient fabrication of graphene nanoelectrodes through the combination of dash-line lithography and electroburning is demonstrated in detail.

From electroburning to sublimation: substrate and environmental effects in the electrical breakdown process of monolayer graphene

We report on the characterization of the electrical breakdown (EB) process for the formation of tunneling nanogaps in single-layer graphene. In particular, we investigated the role of oxygen in the breakdown process by varying the environmental conditions (vacuum and ambient conditions). We show that the density of oxygen molecules in the chamber is a crucial parameter that defines the physical breakdown process: at low density, the graphene lattice is sublimating, whereas at high density, the process involved is oxidation, independent of the substrate material. To estimate the activation energies of the two processes, we use a scheme which consists of applying voltage pulses across the junction during the breakdown. By systematically varying the voltage pulse length, and estimating the junction temperature from a 1D thermal model, we extract activation energies which are consistent with the sublimation of graphene under high vacuum and the electroburning process under air. Our study demonstrates that, in our system, a better control of the gap formation is achieved in the sublimation regime.

Field-Effect Control of Graphene-Fullerene Thermoelectric Nanodevices

Although it was demonstrated that discrete molecular levels determine the sign and magnitude of the thermoelectric effect in single-molecule junctions, full electrostatic control of these levels has not been achieved to date. Here, we show that graphene nanogaps combined with gold microheaters serve as a testbed for studying single-molecule thermoelectricity. Reduced screening of the gate electric field compared to conventional metal electrodes allows control of the position of the dominant transport orbital by hundreds of meV. We find that the power factor of graphene-fullerene junctions can be tuned over several orders of magnitude to a value close to the theoretical limit of an isolated Breit-Wigner resonance. Furthermore, our data suggest that the power factor of an isolated level is only given by the tunnel coupling to the leads and temperature. These results open up new avenues for exploring thermoelectricity and charge transport in individual molecules and highlight the importance of level alignment and coupling to the electrodes for optimum energy conversion in organic thermoelectric materials.

Single Electron Transistor with Single Aromatic Ring Molecule Covalently Connected to Graphene Nanogaps

We report a robust approach to fabricate single-molecule transistors with covalent electrode-molecule-electrode chemical bonds, ultrashort (∼1 nm) molecular channels, and high coupling yield. We obtain nanometer-scale gaps from feedback-controlled electroburning of graphene constrictions and bridge these gaps with molecules using reaction chemistry on the oxidized graphene edges. Using these nanogaps, we are able to optimize the coupling chemistry to achieve high reconnection yield with ultrashort covalent single-molecule bridges. The length of the molecule is found to influence the fraction of covalently reconnected nanogaps. Finally, we discuss the tunneling nature of the covalent contacts using gate-dependent transport measurements, where we observe single electron transport via large energy Coulomb blockade even at room temperature. This study charts a clear path toward the assembling of ultraminiaturized electronics, sensors, and switches.

Robust Molecular Anchoring to Graphene Electrodes

Recent advances in the engineering of picoscale gaps between electroburnt graphene electrodes provide new opportunities for studying electron transport through electrostatically gated single molecules. But first we need to understand and develop strategies for anchoring single molecules to such electrodes. Here, for the first time we present a systematic theoretical study of transport properties using four different modes of anchoring zinc-porphyrin monomer, dimer, and trimer molecular wires to graphene electrodes. These involve either amine anchor groups, covalent C-C bonds to the edges of the graphene, or coupling via π-π stacking of planar polyaromatic hydrocarbons formed from pyrene or tetrabenzofluorene (TBF). π-π stacked pyrene anchors are particularly stable, which may be advantageous for forming robust single-molecule transistors. Despite their planar, multiatom coupling to the electrodes, pyrene anchors can exhibit both destructive interference and different degrees of constructive interference, depending on their connectivity to the porphyrin wire, which makes them attractive also for thermoelectricity. TBF anchors are more weakly coupled to both the graphene and the porphyrin wires and induce negative differential conductance at finite source-drain voltages. Furthermore, although direct C-C covalent bonding to the edges of graphene electrodes yields the highest electrical conductance, electron transport is significantly affected by the shape and size of the graphene electrodes because the local density of states at the carbon atoms connecting the electrode edges to the molecule is sensitive to the electrode surface shape. This sensitivity suggests that direct C-C bonding may be the most desirable for sensing applications. The ordering of the low-bias electrical conductances with different anchors is as follows: direct C-C coupling > π-π stacking with the pyrene anchors > direct coupling via amine anchors > π-π stacking with TBF anchors. Despite this dependency of conductances on the mode of anchoring, the decay of conductance with the length of the zinc-porphyrin wires is relatively insensitive with the associated attenuation factor β lying between 0.9 and 0.11 Å^-1.

Distinguishing Lead and Molecule States in Graphene-Based Single-Electron Transistors

Graphene provides a two-dimensional platform for contacting individual molecules, which enables transport spectroscopy of molecular orbital, spin, and vibrational states. Here we report single-electron tunneling through a molecule that has been anchored to two graphene leads. Quantum interference within the graphene leads gives rise to an energy-dependent transmission and fluctuations in the sequential tunnel-rates. The lead states are electrostatically tuned by a global back-gate, resulting in a distinct pattern of varying intensity in the measured conductance maps. This pattern could potentially obscure transport features that are intrinsic to the molecule under investigation. Using ensemble averaged magneto-conductance measurements, lead and molecule states are disentangled, enabling spectroscopic investigation of the single molecule.

Atomistic Analysis of Room Temperature Quantum Coherence in Two-Dimensional CdSe Nanostructures

Recent experiments on CdSe nanoplatelets synthesized with precisely controlled thickness that eliminates ensemble disorder have allowed accurate measurement of quantum coherence at room temperature. Matching exactly the CdSe cores of the experimentally studied particles and considering several defects, we establish the atomistic origins of the loss of coherence between heavy and light hole excitations in two-dimensional CdSe and CdSe/CdZnS core/shell structures. The coherence times obtained using molecular dynamics based on tight-binding density functional theory are in excellent agreement with the measured values. We show that a long coherence time is a consequence of both small fluctuations in the energy gap between the excited state pair, which is much less than thermal energy, and a slow decay of correlation between the energies of the two states. Anionic defects at the core/shell interface have little effect on the coherence lifetime, while cationic defects strongly perturb the electronic structure, destroying the experimentally observed coherence. By coupling to the same phonon modes, the heavy and light holes synchronize their energy fluctuations, facilitating long-lived coherence. We further demonstrate that the electronic excitations are localized close to the surface of these narrow nanoscale systems, and therefore, they couple most strongly to surface acoustic phonons. The established features of electron-phonon coupling and the influence of defects, surfaces, and core/shell interfaces provide important insights into quantum coherence in nanoscale materials in general.

Connectivity dependence of Fano resonances in single molecules

Using a first principles approach combined with analysis of heuristic tight-binding models, we examine the connectivity dependence of two forms of quantum interference in single molecules.