Session R6: Focus Session: Graphene - Defects, Edges, Theory

Imaging defects on epitaxial graphene/SiC(0001) using non-contact AFM

Graphene exhibits linear dispersion at the Dirac point, which leads to novel properties that can be further tailored by the introduction of defects into the honeycomb lattice. In this work, we created defects on epitaxial graphene/SiC(0001) using N and Ar plasma, and studied the atomic structure of the defects using an integrated approach with non-contact atomic force microscopy (AFM) with Q-plus sensor and density functional theory (DFT) calculations. With atomic resolution AFM imaging, straightforward identifications of single- and di-vacancy defects, as well as other more convoluted vacancy complexes can be made. In addition, local contact potentials of these defects are also obtained by frequency shift-bias spectroscopy. These results and comparisons with DFT calculations will be discussed at the meeting.   

Graphene nanoflakes with defective edge terminations

Systematic tight-binding investigations of the electronic spectra (as a function of the magnetic field) are presented\footnote{I. Romanovsky, C. Yannouleas, and U. Landman, Phys. Rev. B {\bf 86}, 165440 (2012)} for trigonal graphene nanoflakes with reconstructed zigzag edges, where a succession of pentagons and heptagons, that is 5-7 defects, replaces the hexagons at the zigzag edge. For nanoflakes with such reczag defective edges, emphasis is placed on topological aspects and connections underlying the patterns dominating these spectra. In addition to features that are well known to appear for graphene dots with zigzag edge termination, the electronic spectra of trigonal graphene nanoflakes with reczag edge terminations exhibit unique features. These unique features appear within a stripe of negative energies $E_b < E < 0$ and along a separate regime forming a constant-energy line outside this stripe. The lower bound $(E_b)$ specifying the energy stripe is independent of size. A main finding concerns the limited applicability of the continuous Dirac-Weyl equation, since the latter does not reproduce the special reczag features.   

Magnetic-field effects in graphene nanorings: armchair versus zigzag edge terminations

Dirac quasiparticles in narrow graphene nanorings exhibit characteristic differences in their behavior depending on the shape (e.g., trigonal vs. hexagonal) and the type of edge terminations (armchair vs. zigzag). The differences are manifested in the tight-binding single-particle spectra as a function of the magnetic field $B$ and in the patterns of the Aharonov-Bohm oscillations. The symmetry of shape leads to the appearance of three-member (triangles) or six-member (hexagons) braid bands.\footnote{% I. Romanovsky, C. Yannouleas, and U. Landman, Phys. Rev. B {\bf 85}, 165434 (2012)} With the exception of the formation of the braid bands, the characteristic differences maintain in the energy spectra of the continuous Dirac-Weyl equation for a circular ring of finite width. These differences will be further analyzed with the help of a relativistic superlattice model.   

Electron transport in graphene monolayers

We demonstrate electronic transmission of a monolayer can be reduced when covered by a nanoribbon. The transmission reduction occurs at different energies determined by the width of the nanoribbon. We explain the transmission reduction by using of interference between the wavefunctions in the monolayer and the nanoribbon. Furthermore, we show the transmission reduction of a monolayer is combinable and propose a concept of ``combination of control'' for nano-application design.   

Unraveling the interlayer-related phonon self-energy renormalization in bilayer graphene

In this work, we present a step towards further understanding of the bilayer graphene (2LG) interlayer (IL)-related phonon combination modes and overtones as well as their phonon self-energy renormalizations by using both gate-modulated and laser-energy dependent inelastic scattering spectroscopy. We show that although the IL interactions are weak, their respective phonon renormalization response is significant. Particularly special, the IL interactions are mediated by Van der Waals forces and are fundamental for understanding low-energy phenomena such as transport and infrared optics. Our approach opens up a new route to understanding fundamental properties of IL interactions which can be extended to any graphene-like material, such as MoS$_{2}$, WSe$_{2}$, oxides and hydroxides. Furthermore, we report a previously elusive crossing between IL-related phonon combination modes in 2LG, which might have important technological applications.   

Space dependent Fermi velocity in strained graphene

We investigate some apparent discrepancies between two different models for curved graphene: the one based on tight binding and elasticity theory, and the covariant approach based on quantum field theory in curved space. We demonstrate that strained or corrugated samples will have a space dependent Fermi velocity in either approach that can affect the interpretation of local probes experiments in graphene. We also generalize the tight binding approach to general inhomogeneous strain and find a vector field proportional to the derivative of the strain tensor that has the same form as the spin connection obtained in the covariant approach.   

Moir\'{e} minibands in graphene heterojunctions with hexagonal 2D crystals

The transformation of the linear Dirac spectrum of electrons in monolayer graphene and parabolic spectrum in bilayer graphene due to the influence of a tightly bound insulating or semiconducting layer is studied. We present a symmetry-based classification and quantitative analysis of generic miniband structures for electrons in graphene heterojunction with a 2D crystal with the hexagonal Bravais symmetry, such as boron nitride. In particular, we identify conditions at which the first moire miniband is separated from the rest of the spectrum by either one or a group of three isolated mini Dirac points and is not obscured by dispersion surfaces coming from other minibands. In such cases the Hall coefficient exhibits two distinct alternations of its sign as a function of charge carrier density. Then, we study the Hofstadter spectrum of electrons in a magnetic field.   

Electric-Field Dependence of the Effective Dieletric Constant in Graphene Materials

The dielectric constant of a material is one of the fundamental features used to characterize its electrostatic properties such as capacitance, charge screening, and energy storage capability. Here we address the issue of the effective dielectric constant ($\varepsilon_{G}$) in a few-layer graphene materials (e.g. graphene, MoS$_2$, WS$_2$, etc.) subjected to an external electric field. In particular for graphene, the value of $\varepsilon_{G}$ has attracted interest due to contradictory reports from theoretical and experimental studies. Through extensive first-principles electronic structure calculations, including van der Waals interactions, we show that the graphene dielectric constant depends on the value of the external field ($E_{\rm ext}$): it is nearly constant at $\varepsilon_{G} \sim 3$ for low fields ($E_{\rm ext}<0.1$ V/\AA) but increases at higher fields, reaching $\varepsilon_{G}$=4.5 at $E_{\rm ext} =1.7$ V/\AA. Further increase of $E_{\rm ext}$ drives the system to an unstable state where the layers are decoupled and can be easily separated. Calculations performed for other layered materials follow the same trend. Our results point to a promising way of understanding and controlling the screening properties of few-layer graphene materials via electrical means.   

RKKY interaction in monolayer and bilayer graphene: Exact results in terms of the Meijer-G functions

We present the results for the RKKY interaction in monolayer and bilayer graphene in terms of Meijer-G functions for the undoped, doped and biased cases. The results are obtained from the linear-response expression for susceptibility written in terms of the integral over Green's functions and using Dirac bands. The salient features of the large-distance behavior in each case will be discussed. For instance, for doped monolayer graphene, the interaction falls off as $R^{-2}$ and oscillates as the product of two terms, one being a $\{1+ \cos[(K-K').R]\}$-like interference term from both Dirac cones and the second term scaled by Fermi momentum $k_F$. For doped and unbiased bilayer graphene, the interaction decays as $R^{-2}\cos(k_FR)[e^{-k_FR}+\sin(k_FR)]\Phi_{K,K'}$ where $\Phi_{K,K'}$ is a similar Dirac-cone factor. For the gated bilayer graphene, $k_F$ must be replaced by another scaled momentum $k_U$, which depends on Fermi energy, gate voltage and the interlayer hopping energy, allowing the possibility of tuning of the interaction by gate voltage. References: M. Sherafati and S. Satpathy, PRB, 83, 165425 (2011); PRB, 84, 125416 (2011); AIP Conf. Proc. 1461, 24 (2012)   

Graphene multilayers in the crossed in-plane magnetic and out-of-plane electric fields

We report an experimental study of the out-of-plane differential conductivity $dI/dV$ in graphite mesas as a function of applied out-of-plane voltage $V$ in the in-plane magnetic fields $B_y$ up to 55 T. The spectrum $dI/dV$ vs $V$ has a pronounced peak at the critical voltage $V_0$, which grows linearly with the magnetic field $V_0\propto B_y$. The experimental results are consistent with a theoretical model. The electronic energy spectrum on each graphene layer is given by the two-dimensional (2D) Dirac cone $\varepsilon = v |p|$, where $v$ and $p = (p_x,p_y)$ are the velocity and 2D momentum. As a result of magnetic field $B_y$, the Dirac cones of the consecutive layers are shifted in the momentum space by $\Delta p_x = eB_yd$, where $d$ is a distance between the layers. Whereas electric field $E_z$ shifts the energy by $\Delta \varepsilon = E_z d$. For generic $E_z$ and $B_y$, the wave functions are localized on a finite number of layers in the $z$ direction. However, when the resonant condition $\Delta \varepsilon = v\Delta p_x$ is achieved, i.e. when $E_z = vB_y$, the Dirac cones align, and wave functions become delocalized in the $z$ direction. We believe that the resonant delocalization of the wave functions corresponds to the peak in differential conductance.   

Interactions between adsorbates on graphene

Interactions between adsorbates on a surface of graphene play an important role in many applications. We offer a detailed analysis of interactions between two adsorbed atoms and molecules. We compare the first principles DFT, numerical tight-binding, and analytical functional integral calculations to identify the microscopic nature of the adsorbate-adsorbate interaction and the role of different contributions. The interaction has two distinct regimes: a weak coupling regime, which is akin to RKKY (Ruderman-Kittel-Kasuya-Yosida) magnetic interaction, and a strong coupling regime which is dominated by interaction via a many-body electronic dressing ``cloud'' around each adsorbate. We show that the interplay between these two regimes provides an opportunity to manipulate the magnitude and the structure of the adsorbate-adsorbate interaction (up to complete reversal of sign) via a variety of easily accessible properties, such chemical potential via back-gating, type of an adsorbed atom, electronic configuration of an adsorbed molecule, and strain.   

\emph{Ab Initio} Many-body Study of Cobalt Adatoms Adsorbed on Graphene

Research interest in the adsorption of transition metal adatoms on graphene has grown rapidly because of their promising use in spintronics. Single Co atoms on graphene have been extensively studied recently, and possible Kondo effects have been considered. However, these calculations show significantly varying results on the bonding nature of Co/graphene system. We use auxiliary-field quantum Monte Carlo (AFQMC) and a size-correction embedding scheme to accurately calculate the binding energy of Co/graphene.\footnote{Y. Virgus, W. Purwanto, H. Krakauer, and S. Zhang, arXiv:1210.6973.} We find that as a function of the distance $h$ between the Co atom and the six-fold hollow site, there are two states that provide binding and exhibit a double-well feature with nearly equal binding energy of $0.4$~eV at $h = 1.51$ and $h = 1.65$~\AA, corresponding to low-spin $^2$Co ($3d^{9}4s^{0}$) and high-spin $^4$Co ($3d^{8}4s^{1}$), respectively. Binding of Co on bilayer graphene is also investigated.   

Polarization Waves around Coulomb Impurities in Strained Graphene

We study the distribution of polarization charge around external Coulomb centers in graphene. We consider uniaxially strained Graphene so that the Dirac cones are anisotropic and there is a preferred direction on the lattice. Under these conditions we find that the polarization charge exhibits oscillations around the impurity with predominant d-wave symmetry for small anisotropy (strain) and also admixture of g-wave as well as higher waves with increasing anisotropy. The total polarization in the ground state is zero. This rich variety of behavior is in stark contrast to the situation in undeformed graphene, where the polarization charge away from the impurity is identically zero. Thus our results could be used for detection of Coulomb impurity physics even in the case of relatively small charge ions as long as there is sufficient Dirac cone anisotropy.