Session A21: Focus Session: Graphene: Quantum Interference and Transport

Imaging coherent transport in mesoscopic graphene

To understand the coherent flow of electrons through a graphene device, we must employ nanoscale probes that can access the relevant length scales. At low temperatures and small size scales, diffusive trajectories of electrons interfere with each other, resulting in a coherent correction to the conductivity known as universal conductance fluctuations (UCF). Here, we use a liquid-He-cooled scanning probe microscope (SPM) tip to induce an additional, movable scatterer in a graphene device. By scanning the tip over a device, we map the conductance fluctuations vs. scatterer position. We find that the conductance is highly sensitive to the position of this scatterer, producing $\delta G \sim e^2/h$ fluctuations when the scatterer is displaced by a distance comparable to the electron wavelength. These measurements, in combination with numerical simulations, demonstrate the value of this cooled SPM technique to probe coherent transport in graphene.   

Simulating a movable scatterer in coherent graphene devices

We study coherent transport of electrons through a graphene device. At sufficiently low temperatures in mesoscopic samples, this diffusive transport becomes coherent. Thus, small changes in the device, such as modifying the chemical potential or changing the location of impurities, induces electrical conductance fluctuations. The magnitude of these fluctuations has a universal value (of order $e^2/h$) independent of the quality of the sample and at low temperatures of its size. A recent experiment by Berezovsky et al, employed a liquid-He-cooled scanning probe microscope (SPM) tip to induce a movable scatterer in a graphene device. The experiment found that a small change to the disorder configuration can drastically change the electrical conductance. We present numerical calculations of transport in realistic grapehene devices studied with the SPM technique. We find good agreement with experimental results and report the effects of varying the details of the disorder.   

Quantum interference controlled graphene nanoribbon/molecule junctions: First-principles modeling of multiterminal nanoelectronic devices

The recent fabrication of graphene nanoribbons (GNRs) has opened unforeseen avenues for carbon nanoelectronics by providing new type of semiconducting channel for field-effect transistors (FETs) or interconnect electrodes for molecular devices. GNRs can resolve one of the key challenges for molecular electronics---a well-defined molecule-electrode contact with good transparency and reproducibility--through a unified $\pi$-bonded network across GNR and conjugated molecule. However, very little is known about such devices, which is partly due to the lack of first-principles tools that can handle atomistic and electronic structure of the device while taking into account more than two electrodes at finite bias voltage. Here we propose a {\em three-terminal} device composed of [18]annulene ring-like organic molecule attached to two GNRs with zigzag edges in a configuration that ensures destructive quantum interference of electron paths around the ring and minuscule transmission at the Fermi level as the off-state. The third electrode is then coupled to the device to switch it into the on-state. Using our recently developed nonequilibrium Green function formalism combined with the density functional theory for multiterminal devices, we demonstrate FET-like operation of this heterojunction.   

Spin-Resolved Quantum Interference in Graphene

Graphene's spin transport properties have attracted a great deal of attention, due in part to the potential for long spin lifetimes, and to the unusual spin structures that are predicted to exist at the edge of a graphene flake. The ability to measure spin polarized electrical currents is an important step toward testing these predictions, and toward achieving coherent spin control in graphene. Here, we resolve spin transport directly from conductance features that are caused by quantum interference. These features split visibly in an in-plane magnetic field, similar to Zeeman splitting in atomic and quantum dot systems. Graphene's g-factor and density of states can be determined to high precision from the magnitude of the splitting. These spin-polarized conductance features may, in the future, lead to the development of graphene devices incorporating interference-based spin filters. It has been shown by numerous imaging techniques that exfoliated graphene flakes are modulated by nanometer-scale ripples. The in-plane magnetic field used to spin-split conductance fluctuations also generates a random vector potential due to the rippled topography. This random vector potential leads to anisotropic momentum scattering and an effective dephasing rate, both of which can be clearly observed in transport. Based on these measurements, the ripple geometry can be extracted from graphene's in-plane magnetoresistance.   

Masses, topological phase transitions and fractionalized particles in graphene

Interaction effects in graphene between electrons or between electrons and phonons, when sufficiently strong, for example when a magnetic field is switched on, can give a mass to the Dirac particles. This phenomenon is the graphene realization of the Higgs mechanism that gives fundamental particles their masses. We classify all possible patterns of symmetry breaking in graphene (i.e., all possible masses). Some of these masses are dual to each other in the sense that they each support defects carrying complementary topological charges. For example, a topological defect in the Kekule pattern binds a unit electric charge at its core, while a superconducting vortex in graphene traps a unit valley-pseudo spin. The topological defects also carry just a fraction of Fermi statistics, when time-reversal symmetry is broken by the anomalous Hall effect. We calculate the charge and statistical angle of topological defects by integrating out the massive fermions and constructing the effective field theory for the system. This duality allows for direct continuous phase transitions between two unrelated symmetry-broken phases through the deconfinement transition of defects, making graphene an ideal testbed for the concept of deconfined criticality.   

Graphene Superconducting Quantum Interference Device

Graphene can support Cooper pair transport when contacted with two superconducting electrodes, resulting in the well-known Josephson effect. By depositing aluminum/palladium electrodes in the geometry of a loop onto a single graphene sheet, we fabricate a two junction dc superconducting quantum interference device (SQUID). ~Not only an the supercurrent in this device be increased by moving the electrostatic gate away from the Dirac point, but it can also be modulated periodically by an applied magnetic field---a potentially powerful probe of electronic transport in graphene. ~We analyze the magnetic field modulation of the critical current with the asymmetric/inductive SQUID model of Fulton and Dynes and discuss the variation of the fitting parameters with gate voltage.   

Klein tunneling in disordered graphene p-n-p junctions

The unavoidable presence of disorder makes the interpretation of the experimental results on ``Klein tunneling'' effects in graphene systems not trivial. Charged impurities profoundly alter the ideal potential profile created electrostatically in graphene p-n-p junctions to observe the Klein tunneling. We calculate the screened potential profile in graphene p-n-p junctions taking into account the presence of charged impurities and many-body effects. We then solve the full quantum mechanical transport problem for the massless Dirac fermions in presence of the calculated screened potential profile. Using our theoretical model we are able to quantify the effects of disorder on the signatures of the ``Klein tunneling'' and identify the necessary experimental conditions to unambiguously observe ``Klein tunneling'' phenomena in graphene.   

Quantum transport at the Dirac point and Klein tunneling in disordered graphene

We describe a robust method to obtain transport properties at the Dirac point in disordered graphene that uniquely combines three crucial features: 1) fully quantum mechanical calculation of transport properties using the transfer matrix approach, 2) a microscopic treatment of the effects of charged disorder with the self-consistent Thomas-Fermi-Dirac density functional method, and 3) the ability to treat experimentally relevant system sizes. As an application we discuss the effects of disorder on Klein tunneling in p-n-p junctions.   

Wave packet scattering in graphene and the Klein paradox

Using the tight binding Hamiltonian for a honeycomb lattice, we develop a computational technique for the construction and time evolution of Gaussian wave packets in graphene. Employing this approach, we study the scattering across barriers and compute reflection and transmission coefficients. Given the nature of energy dispersion in graphene, it is anticipated that electrons with momenta close to the ``Dirac points'' would suffer perfect transmission for very high barriers. We analyze the effects of wave packet scattering close to the Dirac points in search of the Klein paradox.   

Quantum pumping in graphene

We show that graphene-based quantum pumps can tap into evanescent modes, which penetrate deeply into the device as a consequence of Klein tunneling. The evanescent modes dominate pumping at the Dirac point, and give rise to a universal response under weak driving for short and wide pumps, in close analogy to their role for the minimal conductivity in ballistic transport. In contrast, evanescent modes contribute negligibly to normal pumps. Our findings add a new incentive for the exploration of graphene-based nanoelectronic devices.   

Tunneling conductance between misaligned graphene layers

In graphene multi-layer systems, adjacent layers are often slightly misaligned compared to the simple hexagonal or Bernal stacking arrangements. One important example of this behavior is the structure of multi-layers grown on the carbon face of SiC. We report on an estimate of the specific conductance between two misaligned layers. In our theory the lifetime of Bloch states within the semi-isolated layers plays a key role in determining the inter-layer conductance. We discuss how our theoretical picture can be tested by examining the in-plane field dependence of the conductance and comment on current- paths in multi-layer transport experiments with top layer contacts.   

Hyperfine-induced valley mixing and the spin-valley blockade in carbon-based quantum dots

Hyperfine interaction (HFI) in carbon nanotube and graphene quantum dots is due to the presence of $^{13}$C atoms. We theoretically show [1] that in these structures the short-range nature of the HFI gives rise to a coupling between the valley degree of freedom of the electron and the nuclear spin, in addition to the usual electron spin-nuclear spin coupling. We predict that this property of the HFI affects the Pauli blockade transport in carbon-based double quantum dots. In particular, we show that transport is blocked only if both the spin and the valley degeneracies of the quantum dot levels are lifted, e.g., by an appropriately oriented magnetic field. The blockade is caused by four ``supertriplet'' states in the (1,1) charge configuration. \\[4pt] [1] A. Palyi and G. Burkard, http://arxiv.org/abs/0908.1054 (accepted to PRB)   

Spin-dependent conductance of small graphene flakes

Using {\it ab initio} density-functional theory and quantum transport calculations based on nonequilibrium Green's function formalism we study structural, electronic, and transport properties of small graphene flakes. Rectangular and triangular graphene flakes are stable, having magnetically ordered edge states. We show that a spin-polarized current can be produced in pure, hydrogenated, rectangular graphene flakes by exploiting the spatially separated edge states of the flake using asymmetric, nonmagnetic contacts (1). Sharp discontinuities in the transmission spectra which arise from Fano resonances of localized states in the flake are also predicted. Functionalization of the graphene flake with magnetic adatoms such as vanadium also leads to spin-polarized currents even with symmetric contacts. Ground state of triangular flakes have non-zero magnetic moments and their conductance are spin polarized. \\ (1) H. Sahin and R. T. Senger, Phys. Rev. B \textbf{78}, 205423 (2008).