Session D7: Quantum Hall Effect in Mono-, Bi- and Trilayer Graphene

Correlated random hopping disorder in graphene at high magnetic fields: Landau level broadening and localization properties

Disorder is key to understand the electronic transport properties in graphene, particularly in the quantum Hall regime. There is still some debate on the most relevant disorder mechanisms for transport in graphene. Among those, ripple disorder is believed to play an important role. Static ripples give rise to random correlated hopping disorder, which is the disorder mechanism analyzed in this work. We study the density of states and localization properties of the lowest Landau levels of graphene at high magnetic fields, focusing on the effects caused by correlated long-range hopping disorder. We find that the broadening of the lowest Landau level shrinks exponentially with increasing disorder correlation length. The broadening also grows linearly with magnetic field and with disorder amplitudes. More importantly, we observe that the ratio between the n=1 and n=0 Landau level widths depends only on the correlation length and is rather insensitive to the disorder strength and to the magnitude of the magnetic field. This allows a closer contact of our results with experiments. In addition, the lowest Landau level peak shows a robust splitting (inferred from the analysis of the participation ratio), whose origin we identify as the breaking of the sublattice (valley) degeneracy.   

Electrically tunable quantum anomalous Hall effect in graphene decorated by $5d$ transition-metal adatoms

Based on first-principles calculations, we predict that $5d$ transition-metals on graphene present a unique class of hybrid systems exhibiting topological transport effects that can be manipulated effectively by external electric fields [1]. The origin of this phenomenon lies in the exceptional magnetic properties and the large spin-orbit interaction of the $5d$ metals leading to significant magnetic moments accompanied with colossal magnetocrystalline anisotropy energies. A strong magneto-electric response is predicted that offers the possibility to switch the spontaneous magnetization direction by moderate electric fields, enabling an electrically tunable quantum anomalous Hall effect.\\[4pt] [1] preprint: http://arxiv.org/abs/1108.5915   

Scanning gate microscopy on graphene in the quantum Hall regime

Scanning Gate Microscopy was performed on monolayer graphene devices in the quantum hall regime. The devices studied consisted of exfoliated graphene deposited on SiO2, etched in a Hall Bar configuration, and electrically contacted by standard lithographic techniques. Using a custom built scanning probe microscope (SPM), operating at liquid Helium temperatures and below, in magnetic fields up to 16 T, we spatially mapped the position dependent effects of a movable gate, i.e. the charged tip of the SPM, on the conductivity of the graphene device. Using a global backgate to modulate the carrier density, we can visually observe the transitions between filling factors. Striking features are observed in the resistance versus position maps, offering insights into the microscopic properties of graphene in the quantum Hall regime.   

Confining potential and Landau level edge states in graphene

Two-dimensional electron systems in the Quantum-Hall (QH) regime host gapless one-dimensional chiral edge states which are responsible for the quantization of the Hall conductivity. In the regime of the fractional QH effect the edge states form a chiral Luttinger liquid which presents unusual quantum properties such as fractionally charged excitations and interference patterns that could serve as building blocks for QH qubits. Observing and exploiting these properties requires precise control of the edges, but in semiconductor-based 2DES were edge states were studied thus far, achieving the necessary control was difficult. This is because the smooth confinement potential in these systems imposes a length scale which is much larger than the magnetic length leading to incompressible strips and to non-universal behavior. We will show that this limitation is not present in graphene. Using scanning-tunneling-microscopy and spectroscopy to follow the spatial evolution of Landau levels toward an edge, we demonstrated that in graphene it is possible to control the edge states by varying the distance to the screening plane and by controlling the confinement geometry.   

Unconventional Sequence of Fractional Quantum Hall States in Suspended Graphene

Graphene provides a unique platform to study many-body correlations due to the relativistic nature of its charge carriers and their fourfold degeneracy. We report local electronic compressibility measurements of a suspended graphene flake performed using a scanning single-electron transistor. Between filling factors $v$ = 0 and 1, our measurements reveal incompressible fractional quantum Hall states at $v$ = 1/3, 2/3, 2/5, 3/5, 3/7, 4/7 and 4/9, which clearly follow the standard composite fermion sequence. In contrast, between $v$ = 1 and 2, incompressible states occur only at $v$ = 4/3, 8/5, 10/7 and 14/9. These fractions correspond to a subset of the composite fermion sequence involving only even numerators, suggesting a robust underlying symmetry. We extract the energy gaps of each fractional quantum Hall state as a function of magnetic field and find that $v$ = 1/3, 2/3, 4/3, and 8/5 are strongest at low field, persisting below 1.5 T. Our results provide insight into the interplay between electronic correlations and SU(4) symmetry in graphene.   

Quantum Hall effect on centimeter scale chemical vapor deposited graphene films

We report observations of well developed half integer quantum Hall effect on mono layer graphene films of 7 mm by 7 mm in size. The graphene films are grown by chemical vapor deposition on copper, then transferred to SiO$_{2}$/Si substrates, with typical carrier mobilities $\approx $4000 cm$^{2}$/Vs. The large size graphene with excellent quality and electronic homogeneity demonstrated in this work is promising for graphene-based quantum Hall resistance standards, and can also facilitate a wide range of experiments on quantum Hall physics of graphene and practical applications exploiting the exceptional properties of graphene.   

Spin-skyrmion in graphene

In the presence of a magnetic field, the band structure of the two-dimensional electron gas in graphene consists in a series of Landau levels with energy $E_{n}=\pm \sqrt{2} v_{F}\sqrt{\left\vert n\right\vert }/\ell .$ Each Landau level is 4-fold degenerate including valley and spin degrees of freedom. Working in the Hartree-Fock approximation and a finite Zeeman coupling, we compute the energy required to excite a spin-skyrmion (or antiskyrmion) in Landau levels $n=1,2$ at filling factors $\nu =4,8.$ We show that a skyrmion-antiskyrmion pair has lower energy than an electron-hole pair at these two filling factors. We compare our results with a field-theoretical calculation [1] and with a recent experimental measurement of the transport activation gap in this system. \\[4pt] [1] Kun Yang, S. Das Sarma and A. H. MacDonald, Phys. Rev. B \textbf{74}, 075423 (2006).   

Coulomb impurity under magnetic field in graphene: a semiclassical approach

We address the problem of a Coulomb impurity in graphene in the presence of a perpendicular uniform magnetic field. We show that the problem can be solved below the supercritical impurity magnitude within the WKB approximation. Without impurity the semiclassical energy correctly reproduces the Landau level spectrum. For values below the supercritical impurity magnitude the energy spectrum still evolves as square root B with a renormalized fine structure constant. For a given Landau level the WKB energy depends on the absolute value of angular momentum in a way which is consistent with the exact diagonalization result. Below the supercritical impurity magnitude, the WKB solution can be expanded as a convergent series in powers of the effective fine structure constant. Relevance of our results to validity of the widely used Landau level projection approximation is discussed.   

Chern-Simons theory of an anomalous metallic state in half-filled monolayer graphene

The ground state of half-filled monolayer graphene undergoes a novel metal-insulator transition with increasing strength of applied magnetic field. In a weak magnetic field the ground state at half-filling corresponds to a critical metallic state, that governs the $\nu=-2$ to $\nu=2$ quantum Hall plateau transition. In the strong magnetic field regime this critical state gives way to an interaction driven quantum Hall insulator state. Currently there is no satisfactory theoretical explanation of the insulating phase and the phase transition. Motivated by this issue, we investigate the nature of the ground state in clean half-filled monolayer graphene, using a lattice Chern-Simons theory. In contrast to the results obtained previously by mean-field calculations in the Landau level basis, our analysis in the unpolarized regime shows the existence of an anomalous semimetallic state up to a critical strength of magnetic field, and the critical strength is determined by non-universal details of interaction strength. In the polarized regime the dynamics of the relevant metallic state changes dramatically, and new insulating phases do emerge.   

Bilayer graphene as a helical quantum Hall ferromagnet

The two-dimensional electron gas (2DEG) in a graphene bilayer supports a variety of uniform broken-symmetry ground states in Landau level $N=0$ and at integer filling factors $\nu \in \left[ -3,4\right] .$ When a bias is applied between the layers at filling factors $\nu =1,3$, the ground state evolves from an interlayer coherent state at small bias to a state with orbital coherence at higher bias where \textit{electric} dipoles associated with the orbital pseudospins order spontaneously in the plane of the layers. We show that by further increasing the bias the 2DEG goes first through an electron crystal with an orbital pseudospin texture at each site and then into a helical state where the pseudospins rotate in space. The pseudospin textures in the crystal and the helical states are due to the presence of a Dzyaloshinsky-Moriya interaction in the effective pseudospin Hamiltonian when orbital coherence is present in the ground state. We study in detail the electronic structure these nonuniform states as well as their collective excitations and compute their electromagnetic absorption.   

Theory of integer quantum Hall effect in bi-layer graphene

Bi-layer graphene in a quantizing external magnetic field exhibits plateaus of Hall conductivity at various integer fillings. Moreover, electron-electron interactions in suspended doubly gated bi-layer graphene appear sufficiently strong (and short-ranged) to result in a finite gap persisting down to zero magnetic field. In this talk we will demonstrate the competition of various orders within the zeroth Landau level and how their interplay is influenced by the filled Landau levels lying below the Fermi energy. Scaling behavior of the gap at the neutrality point will be discussed. Besides the splitting of the zeroth Landau level, degeneracy lifting of the rest of the Landau levels will be discussed.   

Electronic transport in ABA trilayer graphene

We present measurements of the electronic transport in ABA trilayer graphene field-effect devices fabricated with both a back and top gate, at zero, low and high magnetic fields. While the zero field resistivity exhibits a saddle point as a function of the two gate voltages that is similar to bilayer graphene, the low-field Hall data are consistent with a two-band system having a band overlap characteristic of semimetals. At high magnetic fields the quantum Hall effect is clearly observed, and displays features which can be traced to the underlying band overlap, as well as a lifting of the lattice mirror symmetry. Overall the transport in ABA trilayers is that of a semimetal in which the band structure can be strongly modified via the electric field effect. This work is supported by the DOE under grant No. DE-FG03-99ER45766 and the Gordon and Betty Moore Foundation.   

Quantum Hall liquid to Charge density wave phase transitions in ABC-trilayer graphene

We study interaction driven states within ABC-stacked trilayer graphene's 12-fold degenerate Landau level which appear near the neutral system Fermi level. The 12-fold degeneracy of the zero-energy LL is due to spin and valley degeneracy along with a degenerate set of triplet ($n=0,1,2$) LL orbitals. We predict that at filling factors $\nu = - 5,-2, 1,4$ a quantum phase transition from a quantum Hall liquid state to a triangular charge density wave occurs as a function of the single-particle induced LL orbital splitting. This transition is preceded by a softening of the magneto-roton minima of the quantum Hall liquid which appears at $ql_{B} \sim 2.4$. The charge density wave is a manifestation of the LL orbital pseudospin textures with nonzero winding numbers. The phase diagrams at other filling factors along with the experimental consequences of our theoretical predictions will also be addressed.