Session G46: QHE, Including Graphene

Edge States in Graphene

We investigate edge states in graphene subject to both realistically described spin-orbit coupling and an external magnetic field. Localized edge states are responsible for the charge and spin transport when the electron chemical potential is in a gap of the bulk spectrum. We derive expressions for the edge state energy dispersions and wave functions both analytically and numerically and discuss specific features that distinguish them from conventional edge states due to magnetic field in electron gas. Finally we discuss the relationship between edge state characteristics and both spin and charge response functions.   

Quantum Hall effect and edge states in graphene.

The experimentally observed quantum Hall effect (QHE) in graphene has renewed the interest in the study of multivalley semiconductors in high magnetic fields. In this work we study some properties of graphene in presence of a high magnetic field. \par \noindent 1) We discuss the form of the quantization of the Hall conductivity. We argue that the Hall conductance can be understood in the general framework of the theory of the QHE in two-dimensional systems.\par \noindent 2) We study the properties of undoped graphene in the QHE regime. We find that the Zeeman coupling combined with the electron-electron interaction favors a spin-polarized ground state against a valley-polarized state. This ground state support low energy collective excitations that are combinations of spin and valley density waves. We discuss the possibility that spin texture excitations, Skyrmions, become the low energy charged excitations in the spin-polarized ground state.\par \noindent 3) Finally we analyze the properties of the edge states in the QHE regime. Due to the valley degeneracy occurring in graphene, electron-like and hole-like Landau levels with different spin and valley orientation cross at the edge of the sample. In the undoped samples Coulomb interaction produces repulsion between the states and forces the states to anti-cross, creating a valley and spin coherent stripe at the edge of the sample. We analyze the excitations occurring in this stripe and discuss their possible relevance in tunnelling experiments.   

Quantum Hall Effect in Graphene

We study the integer and fractional quantum Hall effect on a honeycomb lattice at half-filling (graphene) in the presence of disorder and electron-electron interactions. We show that the interactions between the delocalized chiral edge states (generated by the magnetic field) and Anderson-localized surface states (created by the presence of zig-zag edges) lead to edge reconstruction. As a consequence, the point contact tunneling on a graphene edge has a non-universal tunneling exponent, and the Hall conductivity is not perfectly quantized in units of $e^2/h$. We argue that the magneto-transport properties of graphene depend strongly on the strength of electron-electron interactions, the amount of disorder, and the details of the edges.   

Band Collapse as Route to Odd-Integer Quantum Hall Effect In Graphene

The recent Quantum Hall experiments in graphene have confirmed the theoretically well-understood picture of the quantum Hall (QH) conductance in fermion systems with continuum Dirac spectrum. In this paper we take into account the lattice, and perform an exact diagonalization of the Landau problem on the hexagonal lattice. At very large magnetic fields the Dirac argument fails completely and the Hall conductance, given by the number of edgestates present in the gaps of our spectrum, is dominated by lattice effects. As the field is lowered, the experimentally observed situation is recovered through a phenomenon which we call band collapse. As a corollary, for low magnetic field, graphene will exhibit two qualitatively different QHE's: at low filling, the QHE will be dominated by the ``relativistic'' Dirac spectrum and the Hall conductance will be odd-integer; above a certain filling, the QHE will be dominated by a non-relativistic spectrum, and the Hall conductance will span all integers, even and odd.   

Quantum Hall Ferromagnetism in Graphene

In a magnetic field non-interacting electrons with the graphene band structure have four nearly degenerate Landau levels close to the intrinsic Fermi level. The four levels are associated with different spin states and with the two different sites in the honeycomb lattice unit cell. In a single-particle theory weak spin-orbit coupling and Zeeman fields split the four levels by an amount that is small compared to the electronic self-energy. We discuss the quantum Hall effect in graphene using at integer filling factors $\nu=-1$, $\nu=0$, and $\nu=1$ in a picture where the ground state is a broken symmetry SU(4) ferromagnet and these weak single-particle terms act as symmetry breaking external fields. We also discuss the consequences of the difference in interaction strength beteween electrons on the same and different sublattice and sample quality criteria necessary to realize quantum Hall ferromagnetism in graphene.   

Local defects and ferromagnetic interactions in 2D graphite

Recent experiments have shown unexpected properties in carbon-based materials. Ferromagnetic behavior enhanced by proton bombardment has been reported in graphite and, more recently, quantum Hall effect has been experimentally observed in graphene sheets. These results suggest that electronic correlations play an important role in these materials. The anomalous electronic and transport properties reported by the experiments agree with theoretical predictions of 2D models considering only ${\pi}$-electrons. The vanishing of the density of states at the Fermi energy and the absence of a true Fermi surface have important consequences in the electronic behavior of 2D graphite. Although at present there is not a microscopic explanation of the graphite ferromagnetic properties, they seem to be related to topological defects, as pointed out by AFM measurements. In this work we show that lattice defects and vacancies in the graphene structure give rise to localized states at the Fermi energy. Repulsive electron-electron interactions lead to the formation of local moments by polarization of local states. Due to the lack of a Fermi surface the RKKY-like interaction does not have oscillations instead it decays as $r^{-3}$, where $r$ is the distance between defects. The interaction is then ferromagnetic.   

Disorder and the integer quantum Hall plateau-to-plateau transition

We have studied the temperature scaling of integer quantum Hall plateau-to-plateau transition in various disordered systems down to 10mK. The samples are Al$_{x}$Ga$_{1-x}$As-Al$_{0.33}$Ga$_{0.67}$As heterostructures with different Al concentration $x$. Previous work has shown a perfect power law (dR$_{xy}$/dB)$\vert _{Bc}\propto $T$^{-\kappa }$ with $\kappa $=0.42 over two decades of temperatures in the sample with $x$=0.85{\%}. In the sample of $x$=0, $\kappa $=0.58 was observed at high temperatures, while we have now found $\kappa $=0.42 restored below a crossover temperature T$_{c}$=120mK. T$_{c}$ increases to 250mK for $x$=0.21{\%}, and is not observable in the experimental temperature range in the sample of $x$=0.85{\%}. We propose that the quantum phase coherence length exceeds the disorder correlation length below T$_{c}$, shifting the nature of the disorder in the system from long-ranged to short-ranged. On the other end of the sample series with $x$=4.1{\%}, where alloy clustering is believed to be significant, $\kappa $=0.58 is found to persist down to the lowest temperature of the measurement limit, suggesting a different universal class of the transition in long-range disordered systems.   

Resistively-Detected NMR in a 2DES near $\nu = 1$: Clues to the Origin of the Dispersive Lineshape

Resistively-detected NMR (RDNMR) measurements on 2D electron systems near the $\nu = 1$ quantum Hall state are reported. In agreement with previous RDNMR studies, we observe a dispersive lineshape, where $R_{XX}$ vs. frequency shows both negative and positive excursions from the equilibrium $R_{XX}$ value. However, in contrast to recent results of Gervais, et al. [Phys. Rev. Lett. 94, 196803 (2005)], this dispersive lineshape is found at all RF powers, and a conventional, Korringa temperature dependence ($T_1 T = const$) of the nuclear spin- lattice relaxation rate is observed. The shape of the unexplained dispersive lineshape is found to invert when then temperature derivative of the longitudinal resistance changes sign. This suggests that both Zeeman and thermal effects are important to RDNMR in this regime.   

Anisotropic Electronic Transport in a Two-dimensional Hole System under a Tilted Magnetic Field

We study the electrical transport properties of a high mobility two-dimensional hole system (2DHS) confined in a GaAs/AlGaAs quantum well grown on the (100) surface of GaAs. We observed a remarkable magnetotransport anisotropy in the $N=1$ Landau level, different from the two-dimensional electron gas (2DEG), in which the anisotropy shows exclusively in the higher Landau levels ($N\geq2$). Under a tilted magnetic field, we find that the anisotropy can be either reduced or enhanced by the in- plane magnetic field, depending on the direction of the field with respect to the lattice orientation of the sample. This behavior is again different from the previous reported results in 2DEG, where we expect the anisotropy will collapse precipitously as applying a much smaller in-plane magnetic field than what we have applied on the 2DHS. Particularly, at high tilting angles, we observed a ``spike'' like feature developed in the magnetoresistance ($R_{xx}$) at the filling factor $2<\nu\leq5/2$. However, unlike the resistance spike features associated with the first order magnetic transition between quantum Hall ferromagnets, no correlated spike has been observed in the Hall resistance ($R_{xy}$) of our sample.   

Evidence for reentrant striped phases in a two-dimensional hole system

Anisotropic transport is observed at low temperatures in the N=1 and N=2 Landau levels of a very clean two-dimensional (2D) hole system. At $\nu$ = 7/2 and $\nu$ =11/2, the longitudinal magnetoresistance develops strong anisotropies which depend on the direction of current flow and temperature. Interestingly, the transport at $\nu$=9/2 is {\it isotropic} for all temperatures. Our results for a two-dimensional hole system contrast sharply with 2D electron transport where no anisotropy has been observed in the N=1 Landau level, the strongest anisotropy occurs at $\nu$=9/2, and no reentrant behavior is evident.   

Spin Transition in a 2DES at $\nu = 1/2$

The transition from partial to complete electron spin polarization as a function of density in a 2DES at $\nu = 1/2$ has been probed using a resistively-detected NMR (RDNMR) technique. Both the nuclear spin lattice relaxation time $T_1$ of $^{71}$Ga and the response in resistance to a change in the nuclear spin polarization appear to reflect this transition. At low densities, where the electron spin polarization is partial, the $T_1$ time is relatively short, due to the presence of both electron spin states at the Fermi level. In this regime $T_1$ is density independent and has a Korringa-like temperature dependence. Above a critical density $T_1$ increases and the RDNMR signal eventually vanishes, consistent with a transition to complete electron spin polarization. In the transition region we observe a non- Korringa $T_1$ temperature dependence and an unexpected enhancement of the RDNMR signal.   

Spin polarization at fractional filling factors

We measure the spin polarization of a two-dimensional electron system (2DES) in the Quantum Hall regime by photocurrent interband absorption spectroscopy. The sample is a single GaAs quantum well that is placed in a dilution fridge with optical windows at low temperatures down to 70 mK. The 2DES density is tuned by a back-gate. We illuminate with circularly polarized light. This allows us to discriminate optical transitions into each electronic spin level. We show that the spectra consist of many-body electron-hole complexes. The lowest in energy is a singlet trion-like transition \emph{T} (J.G. Groshaus \emph{et.al.}, \emph{Phys. Rev. Lett.} \textbf{93}, 96802, 2004). In the \emph{T}-absorption process, the photocreated electron- hole pair binds to an electron from the 2DES with spin that is opposite to that of the photocreated one. We model this process taking into account correlations and phase space considerations. Using this model and measurements of the \emph{T}-absorption, we obtain the spin polarization of the 2DES. We find that at $\nu =1/3$ the 2DES is fully polarized. As $\nu$ is increased there is loss of spin polarization. Around $\nu =2/3$ the 2DES is half polarized. At this $\nu$, the 2DES remains half polarized for the relatively wide range of magnetic fields of 2 Tesla. This points to the existence of spatial magnetic domains, as proposed by G. Murthy (\emph{Phys. Rev. Lett.} \textbf{84}, 350, 2000).   

Small length-scale probes of 2D electron conduction and correlations.

An important approach to revealing correlation effects in high mobility 2D electron systems is through conduction measurements, and this is particularly true when a conduction measurement can be applied over small length-scales. Relevant but attainable significant length-scales include composite fermion mean-free-paths for the various composite particle constructions throughout the magnetic field spectrum. These length-scales are dependent upon the sample quality such that higher mobilities will typically allow a larger conductance probe to examine the correlation effects. The specific correlations will present different length-scales: composite fermion mean-free-path at filling factor 1/2 is substantially larger than the mean-free-path of the 5/2 state precursor particle at high temperatures. We will present here various probes recently used to try to access these small length scale effects. The various techniques will be reviewed and the complicating factors for these probes will be discussed, in addition to prospects for improved examination of relative small correlation energy states.   

High Frequency Magneto Oscillations in low Electron Density GaAs/AlGaAs Quantum Wells

We have observed exceptional magneto oscillations in modulation doped GaAs/AlGaAs quantum well structures with low ($10\%$) Al barriers. These reproducible, rapid oscillations appear in our low density samples ($n\sim6.5\times 10^{10}/cm^{2}$) after standard illumination and are robust against thermal cycling. They are periodic in \textit{inverse} magnetic field (1/B) and the amplitude increases with temperature up to $\sim600mK$. With increasing temperature, these oscillations shift from the $\nu\sim1/2$ range towards the $\nu\sim 1/3$ range of the underlying, low density 2DES. Some aspects are reminiscent of recent work by G. A. Cs\'{a}thy et. al. (PRL, \textbf{92}, 256804(2004)) Under tilt, these oscillations follow the standard $\cos\theta$ behavior of a 2D system. The period, if naively translates into density, yields $1.4\times10^{13}/cm^{2} $, which significantly outnumbers all countable carriers in the device. A backside gate affects the QHE in the 2DES as well as the rapid oscillations, although the density shifts for the latter are highly incompatible with dimensions and screening considerations.   

Linear magnetoresistance in a high quality two-dimensional electron system

In a high quality two-dimensional electron system of density $n \sim 1\times10^{11}$ cm$^{-2}$ and mobility $\mu \sim 10 \times10^6$ cm$^2$/Vs, at the temperature ($T$) of 1.2K, the diagonal magnetoresistance, $R_{xx}$, shows a strictly linear magnetic ($B$) field dependence, except for sharp spikes at $B$- fields where the integer quantum Hall effect develops. As $T$ is lowered to $\sim$ 35 mK, the main feature of $R_{xx}$ is now dominated by multiple minima and peaks, due to the formation of integer and fractional quantum Hall states. However, when plotting $R_{xx}$ at the even-denominator fillings ($\nu=1/4$, 1/2, 3/4, and 3/2) as a function of $B$ field, the same linear $B$ field dependence is recovered. Interestingly, this linear magnetoresistance cannot be understood under the composite fermion model. Rather, it can be explained in terms of a slight, unintentional electron density gradient in our sample: Practically all $R_{xx}$ features can be reproduced quantitatively through $R_{xy}$. We will discuss the implications of this finding.