Session P29: Focus Session: Carbon Nanotubes and Related Materials IX: Graphene Electronic Structure

Veselago lens and p-n junctions in graphene.

Both monolayer and bilayer graphene are gapless semiconductors. Their electrostatic modulation can be used to generate single and multiple p-n junctions. We have shown [V. Cheianov and V.I. Fal'ko - Phys. Rev. Lett. 97, 226801 (2006)] that p-n junctions in monolayer graphene are transparent for incident electrons. In particular, those electrons approaching the n-p interface in an almost perpendicular direction can cross it without reflection. Moreover, in graphene the transmission of charge through the n-p interface is quite similar to the refraction of electromagnetic waves at the interface where the refractive index inverts sign [J. Pendry - Nature 423, 22 (2003); J. Pendry - Phys. Rev. Lett. 85, 3966 (2000)]. This is because the electron dispersion in the conduction and valence bands in graphene is such that so that, after an electron crosses the n-p interface, from the n- to p-side, its wave vector becomes directed opposite to its velocity. As a result, n-p junctions in graphene possess intriguing and very promising transport properties: a single straight p-n interface can focus electrons [V. Cheianov, V.I. Fal'ko, B.L. Altshuler - Science 315, 1252 (2007)]. This situation is realised in the n-p junction with equal densities of carriers in the n- and p-regions. Also, we have shown that by varying the carrier density in, e.g., p-side of the junction the focus can be smeared into a pair of caustics meeting each other in a cusp, and calculated the characteristic interference pattern of electron waves in the vicinity of the cusp. Using the idea of fine-tuned focusing of electron flow by the p-n interface, we propose to use n-p-n junction In a bipolar graphene-based transistor to create Veselago lens and focused beam splitters for electrons.   

Intraband Landau level transitions in monolayer graphene

We study the cyclotron resonance of electrons and holes in monolayer graphene, via infrared transmission measurements in a magnetic field, $B$, up to 18 T. We find that, instead of having a single resonance energy as in a traditional two-dimensional system, a wide range of transitions between different sets of Landau levels (LLs) can be uniquely distinguished in monolayer graphene. We have observed intraband transitions between neighboring LLs up to $n=7$, where $n$ is the LL index. As expected from the unusual linear dispersion of the low-energy electronic band of monolayer graphene, we show that the corresponding energies of all observed LL transitions are proportional to $\sqrt{B}$. In addition, beyond such a simple linear dispersion, we find that the measured band velocity near the charge-neutral Dirac point ($E=0$) is $\sim$$12\%$ larger than that at higher energies. The LL transitions in the electron and hole bands of monolayer graphene show a considerable asymmetric behavior.   

High Field Magnetoresistance of Graphene at the Dirac Point

The longitudinal and Hall resistance of graphene near the charge neutral point have been studied down to low temperature (20 mK) in high magnetic field (20 T). At issue is the nature of the ground state in the vicinity of the Dirac point in high magnetic fields. In samples in which the offset voltage is small, we observe a highly unusual approach to an insulating state as the field increases. In samples with $\mu > 0.5$ T$^{-1}$ and $V_{0} < 3$ V, the resistance at the Dirac point R0 increases divergently to M$\Omega$ in fields of 14-20 T at temperatures T $<$ 2 K. This divergent behavior is suppressed in samples with large $V_{0}$. Surprisingly, this rise shows little temperature dependence below 2 K. The acute dependence on magnetic field and accompanying lack of activated behavior with temperature provides evidence for an unusual cross-over or transition to the insulating state. Implications for theoretical models including gapless edge modes and Quantum Hall Ferromagnetism will be discussed in the context of these results.   

Effect of disorder on transport in a graphene p-n junction

We evaluate the resistance of a gate-tunable graphene ${p}$-${n}$ junction, in which the gradient of the carrier density is controlled by the gate voltage. Depending on this gradient and on the density of charged impurities, the junction resistance is dominated by either diffusive or ballistic contribution. We find the conditions for observing ballistic transport and show that in recent experiments they were satisfied at best marginally. We make suggestions how the disorder effects can be reduced.   

Nonlinear screening and ballistic transport in a graphene p-n junction

Our theoretical work is devoted to a new class of graphene devices: lateral p-n junctions. Such structures have been recently realized experimentally by modulating the electron density in graphene samples with external gates. We study the charge density distribution, the electric field profile, and the resistance of such $p$-$n$ junctions. We show that the proper treatment of the electrostatic screening, including nonlinear effects, is crucial for obtaining the correct results for all these quantities. In particular, we show that the total electric field at the interface of the electron and hole regions is strongly enhanced due to limited screening capacity of Dirac quasiparticles. Accordingly, the junction resistance is significantly lower than estimated in previous theoretical literature. At the same time, our new theory enables us to achieve a closer agreement with the recent experiments.   

Quantum Critical Scaling of Graphene

We show that the emergent relativistic symmetry of electrons in graphene near its quantum critical point (QCP) implies a crucial importance of the Coulomb interaction. We derive scaling laws, valid near the QCP, that dictate the nontrivial magnetic and charge response of interacting graphene. Our analysis yields numerous predictions for how the Coulomb interaction will be manifested in experimental observables such as the diamagnetic response and electronic compressibility.   

Atomic Collapse and Quasi-Rydberg States in Graphene

We demonstrate that graphene opens a way to investigate in the laboratory a fundamental quantum relativistic phenomenon, that is, atomic collapse in a strong Coulomb electric field, long-sought for, but still inaccessible in high-energy experiments. We consider charged impurities in graphene and show that an impurity can host an infinite family of Rydberg-like resonance states of massless Dirac particles. Strong coupling of these states to the Dirac continuum via Klein tunneling leads to striking resonance effects with direct signatures in transport and local properties.   

Delocalization of disordered massless Dirac fermions in graphene

Graphene is a two-dimensional carbon material with a honeycomb lattice and Dirac-like low-energy excitations. Motivated by recent graphene transport experiments, we have undertaken a numerical study of the conductivity of disordered two-dimensional massless Dirac fermions. The beta function of the Dirac hamiltonian subject to a random scalar potential is computed numerically. Although it belongs to, from a symmetry standpoint, the two-dimensional symplectic class, the beta function monotonically increases with decreasing the dimensionless conductance. We also provide an argument based on the spectral flows under twisting boundary conditions, which shows that none of states of the massless Dirac Hamiltonian can be localized. K. Nomura, M. Koshino, S. Ryu, Phys. Rev. Lett. 99, 146806 (2007).   

Strain induced phonon softening in graphene

We have developed a process to transfer single layer graphene from silicon dioxide to polydimethylsiloxane (PDMS). This allows the straightforward application of uniaxial strain by bending of the PDMS. Using this technique, we examine the Raman scattering spectra of graphene sheets under uniaxial strain. The spectra display significant strain-induced downshifts, as predicted by theory. The frequency shows up to 14 and 30 cm$^{-1}$ downshifts for the G mode and D$^{\ast }$ mode, respectively; the softening rate ($\Delta \omega $/$\omega )$ is about 1{\%} for both. This behavior is explained by the weakening of carbon-carbon bonds due to elongation, and is consistent with previous studies on carbon nanotubes.   

Structural and Electronic Properties of Bilayer Epitaxial Graphene

Scanning tunneling microscopy (STM) and spectroscopy (STS) are used to study the structural and electronic properties of epitaxial graphene on SiC(0001) [1]. We address in this talk the surface morphology and stacking sequence of bilayer graphene. STM topographic images show that in the initial stages of growth, the surface morphology of graphene conforms to an underlying SiC interface reconstruction [1]. In bilayer epitaxial graphene, the top graphene layer forms a continuous sheet across steps separating adjoining terraces. A change in the apparent height between the two graphene basis atoms is observed as a function of tunneling bias. We model the relative heights based on a simple form for the local density of states in AB layer stacking (Bernal, as typical for bulk graphite) [2], and predict a smooth transition from imaging a single sublattice to imaging both sublattices. The experimentally-observed transition is consistent with Bernal stacking of the epitaxial bilayer, and an interlayer hopping energy of 0.4 eV. This work was supported in part by NSF grant ECS-0404084 and Dept. of Commerce/NIST grant 60NANB7D6166. [1] G. M. Rutter et al., Science \textbf{317,} 219 (2007); arXiv:0711.2523. [2] Z. F. Wang et al., Phys. Rev. B \textbf{75,} 085424 (2007).   

Aharonov-Bohm effect and broken valley-degeneracy in graphene rings

We analyze theoretically the electronic properties of Aharonov-Bohm rings made of graphene. We show that the combined effect of the ring confinement and applied magnetic flux offers a controllable way to lift the orbital degeneracy originating from the two valleys, even in the absence of intervalley scattering. The phenomenon has observable consequences on the persistent current circulating around the closed graphene ring, as well as on the ring conductance. We explicitly confirm this prediction analytically for a circular ring with a smooth boundary modelled by a space-dependent mass term in the Dirac equation. This model describes rings with zero or weak intervalley scattering so that the valley isospin is a good quantum number. The tunable breaking of the valley degeneracy by the flux allows for the controlled manipulation of valley isospins. We compare our analytical model to another type of ring with strong intervalley scattering. For the latter case, we study a ring of hexagonal form with lattice-terminated zigzag edges numerically. We find for the hexagonal ring that the orbital degeneracy can still be controlled via the flux, similar to the ring with the mass confinement.   

Topological confinement in bilayer graphene

We study a new type of one-dimensional chiral states that can be created in bilayer graphene (BLG) by electrostatic lateral confinement. These states appear on the domain walls separating insulating regions experiencing the opposite gating polarity. While the states are similar to conventional solitonic zero-modes, their properties are defined by the unusual chiral BLG quasiparticles, from which they derive. The number of zero-mode branches is fixed by the topological vacuum charge of the insulating BLG state. We discuss how these chiral states can manifest experimentally, and emphasize their relevance for valleytronics.