Session X9: QHE: High Filling Factors and Weak Fields

Preferential orientation of stripes in high Landau levels

Near half-filled Landau Levels at $n>2$, two-dimensional systems are unstable with respect to the formation of a stripe phase, with charge carrier guiding centers forming a one-dimensional periodic pattern. The stripe phase is consistent with experimentally observed anisotropic magnetoresistance. The puzzling experimental feature has been the preferential orientation of the stripes along $[110]$ crystallographic direction in GaAs heterostructures with spatial confinement along $[001]$. Here we present the Hartree-Fock analysis demonstrating that shear strain $\epsilon_{xy}$ defines a preferential direction of instability with respect to the formation of the stripe phase. The analysis is valid for both hole and electron gases, but the effect is stronger in hole systems. In the absence of external strain, the preferential orientation originates from the built-in electric fields in heterostructures that induce internal strain due piezoelectric effect in GaAs. The internal strain leads to the orientation of stripes along $[110]$. Application of external strain can compensate the internal strains, and change the preferential orientaton to $[1\overline{1}0]$, as observed experimentally. Our analysis also accounts for the experimentally observed effects of the in-plane magnetic field on the orientation of stripes in electronic samples.   

Strain control of the orientation of stripe phases in quantum Hall regime

Ground state of a two dimensional electron gas in partially occupied Landau levels (LLs) is unstable against the formation of nematic phases. Near half-filled LLs, charge density wave (CDW) is expected to form a unidirectional stripe phase. The existence of striped phase has received experimental support from the observation of large anisotropy of magnetoresistance in high mobility 2D electron and hole gases. The puzzling experimental feature has been the preferential orientation of the stripes along [110] crystallographic direction in a variety of GaAs heterostructures. Here we show that the orientation of stripes can be manipulated by uniaxial strain. We study hole samples fabricated in the Van der Pauw geometry from Carbon-doped GaAs/AlGaAs heterostructure grown on (001) substrate. We apply uniaxial strain along the two orthogonal directions $[110]$ and $[1\overline{1}0]$ and study the transport properties in a perpendicular field at $10$mK. Strain, apart from switching the hard and easy transport axes, also induces stripe phases at filling fractions of $\nu=5/2$ and $9/2$, which are isotropic in unstrained samples. From our experimental and theoretical investigation we conclude that piezoelectricity is at the origin of preferential orientation of stripes in two dimensional systems.   

Effects of Landau-level mixing on charge density wave formation in quantum Hall systems

Anisotropic transport in half-filled Landau levels has been explained in terms of interaction induced charge-density wave (CDW) formation. We use the Hartree-Fock approximation to study the influence of electron-electron interactions and Landau level mixing on the formation of CDWs in two-dimensional electron and hole systems. For the situation of two nearly degenerate levels near the Fermi energy, we construct a Landau theory appropriate for competing order parameters that allows for both striped and triangular CDW formation and use this to construct a phase diagram as a function of filling, Landau level mixing parameter and temperature. We find the possibility of coexisting CDW ordering in the two states, along with possible hysteretic behaviour. We discuss how an external tuning parameter such as Rashba spin-orbit coupling might be used to explore this phase diagram experimentally.   

Observation of a Spin-polarization Driven Insulating Phase between the Quantum Hall States in a Dilute Metallic 2D Hole System

The origin of Metal-Insulator Transition (MIT) in strongly correlated 2D electron systems has long been of great interest. Here we present the transport properties of 10nm wide GaAs quantum wells with hole densities around the critical point of the 2D MIT (critical density $p_{c}\approx $ 0.92 $\times $ 10$^{10}$cm$^{-2})$. For hole density above $p_{c}$ but less than 1.2$\times $ 10$^{10}$cm$^{-2}$, an insulating phase is observed between the $\nu $=1 and $\nu $=2 quantum Hall states which both show metallic temperature dependence (d$R_{xx}$/d$T>$0). Through studying the evolution of this insulating phase versus 2D hole density, we show that it is connected to the zero field insulating phase. The interplay between the spin-polarization induced 2D MIT physics and the formation of quantum Hall states is suggested to explain the insulating phase we report. We will also present detailed data on the Hall resistivity $\rho _{xy}$ of our dilute 2D hole system near $p_{c}$.   

Weak-field quantum Hall transition: microscopic verification

Levitation scenario: the higher is the Fermi level the lower is the magnetic field at which transition into $\sigma_{xy}=1$ quantum Hall phase takes place, was put forward by Khmelnitskii more than 25 years ago. It was based on field-theoretical arguments. While zero-field complete localization of 2D electron states even at high energies was confirmed by numerical treatment of the Anderson Hamiltonian, no microscopic description of low-field quantum Hall transition existed so far. We constructed a {\it weakly-chiral} network model [Phys. Rev. Lett. 103, 066801 (2009)] which, depending on node parameters, captures both the Anderson insulator ($\sigma_{xy}=0 $) phase and the quantum Hall ($\sigma_{xy}=1$) phase. Numerical analysis of this model, as well as analytical treatment of its classical limit, are in full agreement with each other; they both reveal delocalization transition in non- quantizing magnetic field, where electron trajectories are only slightly curved. At low-field transition, electron states can be viewed as two weakly coupled by disorder Chalker-Coddington networks, with {\it opposite} chiralities.   

Scanning Gate Microscopy on a Quantum Hall Interferometer

We perform scanning gate microscopy (SGM) experiments [1] at very low temperature (down to 100 mK) in the Quantum Hall regime on a mesoscopic quantum ring (QR) patterned in an InGaAs/InAlAs heterostructure. Close to integer filling factors $\nu$=6, 8 and 10,the magnetoresistance of the QR is decorated with fast periodic oscillations, with a magnetic field period close to AB/$\nu$, where AB is the Aharonov-Bohm period. We analyze the data in terms of electron tunneling between edge states trapped inside the QR and those transmitted through the QR openings [2]. SGM images reveal that the tip-induced perturbation of the electron confining potential gives rise to a rich pattern of narrow and wide concentric conductance fringes in the vicinity of the QR. [1] F. Martins et al. Phys. Rev. Lett. 99 136807 (2007); B. Hackens et al. Nat. Phys. 2 826 (2006). [2] B. Rosenow and B. I. Halperin, Phys. Rev. Lett. 98, 106801 (2007).   

Gated Magnetotransport in a Very-High Mobility GaAs/AlGaAs Quantum Well

The ability to control electron density and potential profile in a very-high mobility, modulation-doped GaAs/AlGaAs 2D electron system by potential gates is a key ingredient in realizing confinement of quasiparticles in the fractional quantum Hall effect. In this study, top-gated Hall bar samples were constructed on a 20-nm GaAs/AlGaAs 2D electron gas quantum well (QW) using a Si$_{3}$N$_{4}$ dielectric layer. Before the gate was processed, the wafer had an electron density $n=6.3\times 10^{11}$ cm$^{-2}$ and a mobility $\mu =1\times 10^7$cm$^{2}$/Vs at 300 mK. By magnetotransport measurements in the quantum Hall region we found that electrons can be depleted uniformly (from 5.64 to $0.6\times 10^{11}$cm$^{-2})$, and the $n$ vs. gate-potential shows simple capacitive characteristics in this range. The \textit{$\mu $} vs. $n$ can be described by a power law, $\mu =A\cdot n^{1.9}$. We discuss the applications of the gate technique for quantum transport studies in very-high mobility QWs at ultralow temperature region produced by nuclear demagnetization refrigerator. Ref. R. L. Willett, et al, Appl. Phys. Lett. 89, 242107 (2006)   

Density dependence of mobility in a high quality Carbon-doped GaAs two-dimensional hole system

Two-dimensional hole systems (2DHSs) in GaAs/AlGaAs heterostructures offer a unique platform for the study of the concomitance of strong Coulomb interactions and spin-orbit coupling. Recent advances in the growth by molecular beam epitaxy of Carbon-doped 2DHS in (001) oriented GaAs heterostructures have resulted in a substantial increase in the achievable low temperature mobility. Here we report on a systematic study of mobility in a series of high quality Carbon-doped (001) GaAs 2DHS samples. Several samples consisting of 16{\%} AlGaAs barriers and 20nm GaAs quantum wells where grown with varying doping profiles to investigate the density dependence of mobility. Using an 80nm setback, a low temperature (T$\sim $50mK) mobility of 2.6x10$^{6}$cm$^{2}$/Vs at a density p=6.2x10$^{10}$cm$^{-2}$ was observed. Surprisingly, samples with similar structural design but with a higher as-grown 2D density of 1x10$^{11}$cm$^{-2}$ displayed a reduced mobility of 2.0x10$^{6}$cm$^{2}$/Vs. This behavior is distinct from that observed in high mobility two-dimensional electrons. In order to better understand the mechanisms influencing mobility in our 2DHSs we have studied the density dependence of mobility in a single sample using a backgate to modulate the density from 2x10$^{10}$cm$^{-2}$ to 1x10$^{11}$cm$^{-2}$.   

Experimental methods for fabricating ultra-shallow undoped 2DEGs in GaAs-AlGaAs heterostructures

The 2-dimensional electron gas (2DEG) at a GaAs-AlGaAs heterointerface forms the starting point for several experiments with 1-dimensional channels, quantum dots and mesoscopic rings. Usually for these experiments, the despth of the 2DEG from the wafer surface is about 100 nm. In structures where modulation doping or delta doping is used this distance is dictated by the need to have a large enough spacer layer to reduce scattering and switching noise from remote ionsied dopants. Very shallow high mobility 2DEGs would be useful in defining smaller lithographic features that conveniently approach the single electron limit. These would also enable several experiments with laterally coupled 1-dimensional channels and rings. In this talk we present an experimental method of making fully undoped and shallow ($<$ 30nm) 2DEGs. We have developed a method of making very low resistance (less than 50 $\Omega$) contacts to these structures and systematically studied the evloution of the mobility as a function of the depth of the 2DEG (from 300nm to 30nm). We also demonstrate a way of extracting quantitative information about the surface states from the data.   

Evidence of fully spin polarized $\nu$ = 3 in single valley (110)-AlAs quantum wells

We observe a spike in the longitudinal resistance of a single valley (110)-AlAs quantum well between $\nu$ = 3 and 4 which is interpreted as evidence of a quantum Hall ferromagnetic transition. This feature occurs at a magnetic field $B$ = 2.85 T in a sample with densities $n = 1.5 - 2.17\times 10^{11}$ cm$^{-2}$ in a perpendicular field with no external strain. The spike disappears on further lowering the density of the sample or on raising its temperature above 600 mK. The spike also shows magnetic hysteresis. Since AlAs is a heavy mass system, the exchange enhanced Zeeman energies become comparable to the cyclotron energies and can lead to Landau level crossings even in a purely perpendicular magnetic field. Being a single valley system, there are fewer quantum numbers for the Landau levels than in standard (001) double-valley AlAs wells, and exchange interactions can reorder the levels differently. The spike feature may suggest that up to three completely spin polarized levels exist before the occupation of minority spin levels lowers the exchange interaction energy causing the ground state transition. The (110)-AlAs quantum wells may thus be particularly suited to study exchange enhancement effects.   

Correlation between Hall plateau overshoot and Landau level coincidences in Si/ Si$_{1-x}$Ge heterostructures

We observe anamolous overshoots in the Hall resistance $R_{\mathrm {xy}}$ at even quantized plateaus and compare their positions to the Landau level crossings in the longitudinal resistance $R_{\mathrm {xx}}$ in a double valley degenerate Si/ Si$_{1-x}$Ge heterostructure in a tilted magnetic field $B$. Though such $R_{\mathrm {xy}}$ overshoots have been previously observed, their underlying cause is still in dispute. The magneto-transport data is measured on a Hall bar 20 $\mu$m wide with a length to width ratio of 20, etched on a 2DEG with density $n$ = $3.4\times 10^{11}$ cm$^{-2}$ and mobility $\mu$ = 70,250 cm$^{2}$/Vs. The sample shows overshoot at 0$^{\circ}$ tilt angle. Overshoots also appear at the even filling factors $\nu$ = 6, 10, 14 {\em etc.} at an angle of 76.3$^{\circ}$ and at $\nu$ = 8, 12, 16 {\em etc.} at 79.41$^{\circ}$ while disappearing at the previous factors. The angles at which the Landau level's coincide are identified from the longitudinal resistance $R_{\mathrm {xx}}$ minima and differ from the angles of maximum overshoot. These results will be discussed in the light of the recent theory of remnant incompressible strips by Siddiki et al.[1] \newline\newline[1] A. Siddiki, J. Horas, J. Moser, W. Wegscheider, and S. Ludwig. Euro. Phys. Lett., 88(1), OCT 2009.   

Probing low-lying excitations of the $\nu=2$ incompressible fluid with a quantum antidot

In the integer quantum Hall (IQH) regime, electrons can be manipulated coherently in edge states. We have begun to control spins in such systems, in order to utilise both long spin-coherence timescales and controllable electron transport along edge states. Using quantum point contacts to selectively inject and detect non-equilibrium edge-state populations, we perform spin-resolved spectroscopy of a quantum antidot (AD) in the IQH regime. At filling factor two, we find that AD transmission resonances are not spin-selective, contrary to the prediction of the conventional non-interacting picture of lowest-Landau-level (LLL) quantum states, and implying a small spin-excitation energy relative to the thermal energy. In apparent contradiction, we also observe a much larger orbital excitation energy scale in non-linear transport spectroscopy measurements. By treating the AD as a `dot of holes' in the LLL, we find that our observations are consistent with the predicted spin-charge separation that occurs at the edge of an interacting maximum-density droplet of the IQH incompressible fluid. Thus, we believe these experiments offer a direct probe of the physics of a large IQH droplet ($\sim$150 particles), with an edge which is well-described by a Luttinger-liquid model in this regime, with different scales for spin and charge excitations.   

Coherence length in the quantum Hall effect

The Hall effect resistivity plateau has only one e and hence only one electron and h/e comes from the flux quantization. Hence the number of electrons at the plateau is one or a small number. As the field moves away from the plateau, there is a phase transition from single electron state to a cluster state. As the field further moves, the cluster breaks and single/small particle state is formed again. The coherence length which depends on the temperature, determines the variation of resistivity as a function of magnetic field. The exponent of temperature which determines $\partial $R/$\partial $B is the inverse of the exponent of the coherence length. At the plateau the electrons are localized according to the Anderson model except that the plateau location depends on spin. [1] K. N. Shrivastava, AIP Conf. Proc.1017,47-56(2008); 1017, 422-428(2008); 1136, 469-473(2009); 1150, 59-67(2009); 1169, 48-54(2009).   

Anisotropy of electronic quantum Hall phases at filling factor 9/2

Strongly correlated two-dimensional electronic systems in a perpendicular magnetic field develop very pronounced magneto-transport anisotropies at particular filling factors of high Landau levels. The most robust anisotropic phases generally occur around half filling of the upper Landau levels. In this study we focus on the nature of anisotropic quantum Hall phases observed around filling factor 9/2. Despite the efforts, there are many questions that still remain about the microscopic origin of anisotropy and the physical mechanism of stabilization of anisotropic phases. In this study we view the appearance of anisotropy as signature of a phase transition from an isotropic phase to an anisotropic liquid crystalline phase. We study the stabilization of anisotropic phases in this regime by means of a microscopic wave function with no rotational symmetry intrinsically containing nematic order. Monte Carlo results using an effectively projected interaction potential indicate the stabilization of such an anisotropic liquid crystalline states at this filling factor.