Session Y22: Graphene Theory

Many-body Signature of Coulomb Implosion in Graphene

We develop an asymptotic analysis for the scattering of a single electron off a Coulomb impurity in the Dirac equation description of (undoped) graphene, to demonstrate that the penetration of the centrifugal barrier that occurs in this problem may be assessed in a momentum representation. The method is directly generalizable to particle-hole scattering, which supports a similar phenomenon. We derive a Bethe-Salpeter equation for the 3-leg vertex for the sublattice antisymmetric response in the ladder approximation. We solve the integral equations for the lowest $(m=0)$ angular components numerically for both $q=0$ and $q\ll1$ but nonzero, where $q$ is the momentum transfer. In the $q=0$ case there is a clear power law behavior in the solution of the vertex function and the exponents become complex when the coupling constant is above a threshold. We also find that the response can have poles above a critical coupling constant, which we associate with a transition to an exciton condensate state. We can reproduce the poles by solving the integral equations with a model kernel analytically. However, we find that a small momentum cut-off is necessary for the existence of these poles in our analytical solution. In the case of nonzero but small $q$, we find that the correction to the antisymmetric response has a power law behavior in the coupling constant.   

Localized plasmons around impurities in graphene

We explore the collective plasmonic excitations in graphene, in particular localized modes around impurities. We show that the impurity can be used to tune the dielectric response of graphene, thus making it a natural plasmonic material. We focus on the dependence of spatial modulation and frequency of localized plasmons on the electronic filling and impurity strength and show that these parameters can be tuned to control some targeted features of the plasmonic modes. It is possible to verify our predictions through scanning tunneling microscope experiments.   

Unstable plasmons in extrinsic graphene with injected carriers

We theoretically study the plasmons ({\em i.e.}, the charge density collective modes) of a graphene system into which is injected carriers with a sharply peaked distribution function in momentum-space. We find that when this stream of carriers is injected into intrinsic graphene, the collective modes are stable. However, when the stream of carriers is injected into extrinsic ({\em i.e.}, doped) graphene, under certain circumstances the plasmons of the system become unstable and (initially) grow exponentially. This is analogous to the two-stream instablity in plasma physics, which occurs when, for example, a stream of electrons is injected into a stationary plasma. In the graphene system, the dopant carriers in extrinsic graphene and the injected stream of carriers play the role of the stationary plasma and the injected electrons in the plasma case, respectively.   

A theoretical study of the structure of suspended graphene

Only one atom thick, graphene represents the ultimate crystalline membrane. I will use a self consistent extension of second order expansion, to show that this implies a flat phase characterized by scale invariant perturbations. Graphene, however, shows a preferred lengthscale, reveiling itself in the form of ripples on the surface of size 100-300 Angstrom. I trace this unusual feature to the free electrons on the surface. I show that ripples spontaneously appear, correlated to charge inhomogeneities, in agreement with the experimental data. In addition, a unique structural response is predicted in the case of doped graphene, with an inherent asymmetry between electron and hole doping.   

Low energy electronic properties of misoriented sheets of graphene

We study the electronic properties of graphene multilayers in a tight binding framework. We focus on the effect of the relative orientation of the layers on the electronic properties in the neighborhood of the Dirac points. We present numerical calculations, as well as an approximate analytical result that describes the low energy inter-layer coupling as a function of their orientation. The theory is also applied to study the low energy electronic properties of two crossed metallic tubes of general chirality and crossing angle.   

The gapped state of a carbon mono-layer in periodic magnetic and electric fields

When smooth, zero-on-average, periodic magnetic and electric fields are applied to a carbon mono-layer (graphene), a gap between the filled valence and empty conduction band is introduced. However, this gapped state does not correspond to a band insulator: a constant electric field induces a quantized Hall current even though the magnetic flux through the sample is zero. The phenomenon is of the same type as in Haldane's model for a quantum Hall effect without Landau levels, although there is the following important difference between the two models: Haldane's model requires control over external magnetic fields on length scales less than an angstrom. For the model studied here, control over external fields on length scales that are larger by several orders of magnitude is sufficient. The Hall effect is explained in terms of the topological theory of Thouless, Kohmoto, Nightingale and den Nijs. A complementary explanation in terms of simple physical principles is also presented.   

Graphene zigzag ribbons, square lattice models and quantum spin chains

In our previous work, based on the exact solutions of tight-binding model of zigzag graphene ribbons (ZGRs), we suggested a new description of ZGRs in terms of coupled quantum chains. Treating ZGRs as coupled quantum chains reveals a close connection between the low energy properties of the ZGR model and a continuous family of square lattice model Hamiltonians with similar width-dependent properties, that includes the $\pi$-flux and the trivial square lattice models. We use this mapping between ZGRs Hamiltonian and these models to study the effect of higher order hopping terms of the tight-binding model and spin- orbit interactions in graphene ribbons. [1] \\[4pt] [1] M. Zarea and N. Sandler, New Journal of Physics 11 (2009) 095014   

Effective Time Reversal Symmetry Breaking and Energy Spectra of Graphene Armchair Rings

Under certain circumstances, the low-energy electronic states of graphene are well-described by a Dirac equation with a valley-dependent gauge field. This can lead to phenomena reflecting ``effective broken time reversal symmetry'' (EBTRS). We study the energy spectra and wavefunctions of graphene rings formed from metallic armchair ribbons, near zero energy, to search for such properties. Appropriately chosen corner junctions [1] are shown to impose phase shifts in the wavefunctions that at low energies have the same effect as flux tubes passing near the ribbon surface. Closing the ribbon into a ring captures this flux and yields properties that may be understood as signatures of EBTRS [2]. These include a gap in the spectrum around zero energy, which can be removed by the application of real magnetic flux through the ring. The stability of the spectra to various perturbations is examined. \hfil\break [1] A. Iyengar, T. Luo, H.A. Fertig, L. Brey, Phys. Rev. B {\bf 78}, 235411 (2008).\hfil\break [2] T. Luo, A.P. Iyengar, H.A. Fertig, and L. Brey, Phys. Rev. B {\bf 80}, 165310 (2009).   

Quantum lattice Boltzmann scheme for the Dirac equation with second-order accuracy

We present a second-order accurate quantum lattice Boltzmann formulation for the Dirac equation with a scalar potential. Following earlier work of Succi and co-workers, the scheme is derived by integrating the Majorana form of the Dirac equation along light-cone characteristics using the trapezoidal rule. However, our scheme differs by consistently using the same characteristic for the streaming and algebraic terms, improving the accuracy from first to second order. This improvement comes at the price of implicitness, but the implicit coupled equations are rendered fully explicit by a unitary change of variables analogous to that used in the derivation of lattice Boltzmann hydrodynamics. We thus obtain a unitary, second-order accurate, and readily parallelisable numerical scheme. Some computations of the time-dependent Klein paradox for the tunnelling of wave-packets through a strong potential barrier will be presented in relation to recent experimental studies of the Klein paradox in graphene.   

Hexagonal lattice Green functions applied to graphene

Horiguchi showed how to relate adjacency Green functions (FG) on the honeycomb to those of the triangle lattice, and so find them in closed form. We extend these results to include 2$^{nd}$, 3$^{rd}$, and 4$^{th}$ neighbors: \[ \tilde {H}=a\,H^{(1)}+b\,H^{(2)}+c\,\left( {H^{(3)}+\frac{1}{2}H^{(4)}} \right) \] The GFs are applied to find LDOS or spectral density for confined regions and for lattice modifications that are periodic or of finite support, such as punctures, edges and tears. Application to SMT and ARPES on graphene are indicated.   

Conductivity switching of two benzene rings under electric field

We study the electron transport and switching of S-C$_6$H$_3$F(CH3)-C$_6$H$_3$(CH3)F-S molecule sandwiched between two Au(111) electrodes using plane wave quantum transport calculation method described in [1-2]. A nonlocal pseudopotential method is used to describe the system, and scattering states are calculated. We found that under zero external electric field, the two rings have a perpendicular configuration, while under strong external electric field they change into a planar configuration. As a result, the quantum conductivity of the system will increase for more than ten times. We propose to use this as a molecular switch, serving the function of a transistor.\\[4pt] [1] Lin-Wang Wang, PRB \textbf{72}, 045417 (2005)\\[0pt] [2] A. Garc\'{\i}a-Lekue and Lin-Wang Wang, PRB \textbf{74}, 245404 (2006)   

High controllability of ferromagnetism in grapheme

We address the issue of high controllability of ferromagnetism in graphene-based samples. To study magnetic correlations in graphene, we systematically carry out quantum Monte Carlo simulations of the Hubbard model on a honeycomb lattice. In the filling region below the Van Hove singularity, the system shows a short-range ferromagnetic correlation, which is slightly strengthened by the on-site Coulomb interaction and markedly by the next-nearest-neighbor hoping integral. The ferromagnetic properties depend on the electron filling strongly, which may be manipulated at ease by the electric gate. Due to its resultant high controllability of ferromagnetism, graphene-based samples may facilitate the new development of many applications.