Session W24: Integer Quantum Hall Effect

A quantitative examination of the collapse of spin splitting in the quantum Hall regime

There is a great deal of current interest in understanding electron spin physics in semiconductors for potential quantum computation applications. The quantum Hall effect in the two-dimensional electron system (2DES) has proved to be a unique system in this avenue due to a tunability in the difference of spin population and thus the strength of exchange interaction provided by the formation of Landau levels. In this talk, we want to present our experimental results to \textit{quantitatively} examine the theoretical model of spin splitting collapse in the quantum Hall regime [by Fogler and Shklovskii, Phys. Rev. B 52, 17366 (1995)] at fixed magnetic fields as a function of electron density in a high quality heterojunction insulated-gate field effect transistor. In the density range between $n$ = 2$\times $10$^{10}$ and 2$\times $10$^{11}$ cm$^{-2}$, the Landau level number $N$ follows a power-law dependence on the critical electron density $n_{c}$, where the spin splitting collapses, and $N$=11.47$\times n_{c}^{0.64\pm 0.01}$. This power law dependence is in good agreement with the theoretical prediction in the low density regime.   

Single electron wave packets probed by Hanbury-Brown and Twiss interferometry

The ballistic propagation of electronic waves in the quantum Hall edge channels of a 2DEG bears strong analogies with photon optics which inspired a whole set of experiments, including the realization of electronic Mach-Zehnder [1] and Hanbury-Brown and Twiss [2] interferometers. So far, these experiments have been performed with continuous sources, but the recent realization of on-demand single electron emitters [3] has risen the hope to reach, in these experiments, the single charge control. We report here on the first realization of a Hanbury-Brown and Twiss experiment on a single electron beam generated by the single electron emitter recently developed by our group [3]. Using the chiral edge channels of the quantum Hall effect, single electron emitted by the source are directed towards an electronic beam-splitter. From the low frequency current correlations at the output of the beam splitter, we are able to count and characterize the elementary excitations produced by the source. By analyzing their antibunching with thermal excitations, we show that we are able to shape single particle states in a tuneable way. [1] Ji et al., Nature 422, 415 (2003) [2] Henny et al., Science 284 296 (1999) [3] F\`{e}ve et al., Science 316, 1169 (2007)   

Hall Viscosity I: Linear Response Theory for Viscosity

In two dimensional systems with broken time-reversal symmetry, there can exist a non-dissipative viscosity coefficient [1,2,3]. This Hall viscosity is similar in nature to the non-dissipative Hall conductivity. In order to investigate this phenomenon further, we develop a linear response formalism for viscosity. We derive a Kubo formula for the frequency dependent viscosity tensor in the long wavelength limit. We compute the viscosity tensor for the free electron gas, integer quantum Hall systems, and two-dimensional paired superfluids. In the zero frequency limit, we show how the known results [3,4] for the Hall viscosity are recovered.\\[4pt] [1] J. Avron, R. Seiler, and P. Zograf, Phys. Rev. Lett. {\bf 75}, 697 (1995).\\[0pt] [2] P. Levay, J. Math. Phys. {\bf 36}, 2792 (1995).\\[0pt] [3] N. Read, Phys. Rev. B {\bf 79}, 045308 (2009).\\[0pt] [4] N. Read and E. Rezayi, Phys. Rev. B {\bf 84}, 085316 (2011).   

Hall Viscosity II: Extracting Viscosity from Conductivity

When time reversal symmetry is broken, the viscosity tensor of a fluid can have non-dissipative components, similarly to the non-dissipative off-diagonal Hall conductivity. This ``Hall viscosity'' was recently shown to be half the particle density times the orbital angular momentum per particle. Its observation can thus help elucidate the nature of the more exotic quantum Hall states and related systems (e.g., p+ip superconductors). However, no concrete measurement scheme has hitherto been proposed. Motivated by this question we use linear response theory to derive a general relation between the viscosity tensor and the wave-vector dependent conductivity tensor for a Galilean-invariant quantum fluid. This relation enables one to extract the Hall viscosity, as well as other viscosity coefficients (shear and bulk) when relevant, from electromagnetic response measurements. We also discuss the connection between this result and a similar one recently derived by C. Hoyos and D. T. Son [arXiv:1109.2651].   

Four-point characterization using capacitive and ohmic contacts

A four-point characterization method is developed for semiconductor samples that have either capacitive or ohmic contacts. When capacitive contacts are used, capacitive current- and voltage-dividers result in a capacitive scaling factor which is not present in four-point measurements with only ohmic contacts. Both lock-in amplifier and pre-amplifier are used to measure low-noise response over a wide frequency range from 1 Hz -- 100 kHz. From a circuit equivalent of the complete measurement system after carefully being modeled, both the measurement frequency band and capacitive scaling factor can be determined for various four-point characterization configurations. This technique is first demonstrated with a discrete element four-point test device and then with a capacitively and ohmically contacted Hall bar sample using lock-in measurement techniques. In all cases, data fit well to a circuit simulation of the entire measurement system over the whole frequency range of interest, and best results are achieved with large area capacitive contacts and a high input-impedance preamplifier stage. Results of samples (substrates grown by Max Bichler Dieter Schuh, and Frank Fischer of the WSI) measured in the QHE regime in magnetic fields up to 15 T at temperatures down to 1.5 K will also be shown.   

Growth and magnetotransport measurements of triple-valley high mobility, miscut (111) AlAs quantum wells

We optimize growth of AlAs on (111)B GaAs substrates and perform magnetotransport measurements on vicinal (111)AlAs quantum wells (QWs). Previous literature reports that MBE growth on exactly oriented GaAs (111)B substrates is difficult, and the grown epi-layers are obscured by pyramid-like surface faceting and twin defect formation; slight substrate miscut results in stable step-flow growth. We perform a combined structural analysis with AFM, TEM and XRD to correlate MBE growth conditions with defect density scaling. We find that a high growth temperature of 690$^{\circ}$ C and low As beam fluxes reduce micro-twin formation for exactly oriented substrates and eliminate them for miscut substrates. A slight miscut of 2$^{\circ}$, at which slip-step growth is known to occur, lead to AlAs QWs with record electron mobility $\mu$ = $13000 ~$cm$^2$/Vs at a sheet density $n_{\mathrm{2D}}$ = $2.17\times 10^{11}$ cm$^{-2}$. Numerical calculations reveal that valley splitting is about 1 meV per degree of miscut, which compares to typical Fermi energies of 2DEGs in AlAs QWs. Signatures in the transport data indicate that not only miscut but also exchange splitting between valleys can play an important role. Magnetotransport data at 15 mK in magnetic fields up to 15 T will also be presented.   

Chiral heat transport in driven quantum Hall and spin Hall edge states

We consider a model for an edge state of electronic systems in the quantum Hall regime with filling $\nu=1$ as well as in the quantum spin Hall regime. In both cases the system is in contact with two reservoirs by tunneling at point contacts. Both systems are locally driven by applying an ac voltage in one of the contacts. By weakly coupling them to a third reservoir, the transport of the generated heat is studied in two different ways: i) when the third reservoir acts as a thermometer the local temperature is sensed, and ii) when the third reservoir acts as a voltage probe the time-dependent local voltage is sensed. Our results indicate a chiral propagation of the heat along the edge in the quantum Hall case and in the quantum spin Hall case (if the injected electrons are spin polarized). The temperature profile shows that electrons along the edge thermalize with the closest upstream reservoir.   

Wigner path integral solution for the integer quantum Hall effect

The real time propagator of the Wigner distribution function can be constructed from the Wigner-Liouville equation as a phase space path integral. By analogy with the Feynman path integral one can define a new effective Lagrangian of the system in the Wigner-Weyl representation. The effects of gauge transformations and geometric constraints on the action are discussed. In particular we discuss the dynamics of a non-interacting 2DEG on a Hall strip.   

Microwave reflection study of GaAs/AlGaAs devices in the regime of the radiation-induced magnetoresistance oscillations

The microwave-induced magnetoresistance oscillations are revealed in the GaAs/AlGaAs two dimensional electron system (2DES) under microwave and terahertz photo-excitation at liquid helium temperatures. Such oscillations are understood in terms of the displacement and inelastic models for photo-excited transport in this system. In order to identify the relative physical contributions, we have concurrently examined magnetotransport and microwave reflection from the 2DES. For the reflection measurements, a sensitive microwave detector was assimilated into the standard experimental setup. Here, we report on the observed magnetic field induced changes in the microwave reflection, and correlate the observations with concurrent transport response of the photo-excited 2DES.   

Observation of linear-polarization-sensitivity in the microwave-radiation-induced magneto-resistance oscillations

In the quasi two-dimensional GaAs/AlGaAs system, we investigate the effect of rotating \textit{in-situ} the electric field of linearly polarized microwaves relative to the current, on the microwave-radiation-induced magneto-resistance oscillations. We find that the frequency and the phase of the photo-excited magneto-resistance oscillations are insensitive to the polarization. On the other hand, the amplitude of the resistance oscillations are remarkably responsive to the relative orientation between the microwave antenna and the current-axis in the specimen. The results suggest a striking linear polarization sensitivity in the radiation-induced magnetoresistance oscillations.   

Non-equilibrium bosonization and its applications to quantum Hall systems

Bosonization is a powerful theoretical method allowing one to treat interactions in 1D systems non-perturbatively. This technique in its original formulation applies to equilibrium states. In order to describe recent experiments on 1D systems far from equilibrium, we introduce a new bosonization method: A non-equilibrium state is described by imposing non-trivial boundary conditions for collective boson fields. This method allows to reduce the problem of finding correlation functions in an interacting 1D system to the calculation of full counting statistics of a process, which creates a non-equilibrium state. The full counting statistics has been extensively studied, and it is well known in several important situations. We apply the non-equilibrium bosonization technique to explain recent experiments on noise-induced dephasing in quantum Hall interferometers and to the energy equilibration at quantum Hall edge states.   

Single-electron quantum tomography in quantum Hall edge channels

The recent demonstration of an on demand single electron source [1,2] has opened the way to a new generation of``electron quantum optics'' experiments aimed at preparing, manipulating and measuring coherent single electron excitations propagating in ballistic conductors such as the edge channels of a 2DEG in the integer quantum Hall regime. In this talk, I will describe a proposal [3] for measuring single electron coherence using an Hanbury Brown and Twiss interferometer. This quantum tomography protocol could be used to characterize single electron sources and to perform quantitative studies of decoherence.\\[4pt] [1] Science 316, 1169 (2007)\\[0pt] [2] Phys. Rev. B 82, 201309 (2010)\\[0pt] [3] New Journal of Physics 13, 093007 (2011)   

Linear polarization rotation study of the radiation-induced magnetoresistance oscillations

The polarization sensitivity of microwave-radiation-induced magneto-resistance oscillations is investigated by rotating, by an angle \textit{$\theta $}, the polarization of linearly polarized microwaves with respect to the long-axis of GaAs/AlGaAs Hall-bar electron devices. At low microwave power, $P$, experiments show a strong sinusoidal variation in the diagonal resistance $R_{xx}$ vs. \textit{$\theta $} at the oscillatory extrema, indicating linear polarization sensitivity in the microwave radiation-induced magneto-resistance oscillations. Surprisingly, the phase shift \textit{$\theta $}$_{0}$ for maximal oscillatory $R_{xx}$ response under photo-excitation appears dependent upon the radiation-frequency $f$, the extremum in question, and the magnetic field orientation or \textit{sgn}(B).   

Wave function multifractality and dephasing at quantum Hall transitions: A numerical investigation

To understand the effect of the Coulomb interaction is one of the most challenging problems in the context of Anderson localization and the quantum Hall effect. In our study we address this question by following a perturbation theory in the interaction near the non-interacting fixed point. In each order diagrams appear which contain correlation functions characterizing the fluctuation properties of wavefunctions at the (non-interacting) critical fixed point. It turns out that the correlators relevant for dephasing combine in a way such that the {\em leading} multifractal powerlaws cancel; the subleading terms govern the interaction corrections. We present a numerical study based on the Chalker-Coddington network, in which we determine quantitatively the subleading multifractal exponents of the salient wavefunction correlators.   

Corrections to scaling near the quantum Hall transition

Corrections to scaling near critical points are important to understand, because they superimpose and often obscure the true asymptotics of critical scaling laws. This is true, in particular, for studies near the quantum Hall transition where recent numerical work by Slevin and Ohtsuki (Phys. Rev. B {\bf 80}, 041304 (2009)) reports a very small value for the leading irrelevant scaling index $|y|\approx 0.17$. We here report a numerical study of two-point conductances and two-terminal conductances at the integer quantum Hall transition within the Chalker-Coddington network. The scaling of these observables will be analyzed in the two-dimensional and the quasi-onedimensional geometries. We confirm the relation between the conductance exponents $X_q$ and the anomalous dimensions $\Delta_q$ known from the multifractal wavefunction analysis: $X_q=2\Delta_q$. For a consistent picure it is essential to carefully account for corrections to scaling due to subleading power laws and irrelevant scaling operators.