Session A5: Fractional Quantum Hall Effect I

Properties of in-situ back-gated two-dimensional electron gases in GaAs/AlGaAs for the study of electron correlations in the 2$^{nd}$ Landau level

We report on growth and processing optimization of in-situ back-gated two-dimensional electron gases in GaAs/AlGaAs quantum wells. We find that gate leakage currents as small as 4 pA can cause noticeable heating of the electrons if the lattice is not properly thermally anchored to the cryostat. However, we also show that when the devices are properly optimized gate voltages as large as 4V can be applied before leakage turns on, allowing the density to be tuned over a large range from near depletion to over 4 x 10$^{11}$ cm$^{-2}$. In these optimized devices heating effects at dilution refrigerator temperatures are negligible and the gap at $\nu \quad =$ 5/2 can be tuned continuously with density to a maximum \textgreater 400 mK. Such devices should prove useful for the study of electron transport in nanostructures in the 2$^{nd}$ Landau level.   

Analyzing the Disorder Broadening of the Even Denominator Fractional Quantum Hall States in the Presence of Alloy Disorder

The unique character and potential application of the even denominator v$=$5/2 fractional quantum hall state has elicited significant interest. Yet, the most basic properties of this ground state remain unexplained. One poorly understood effect is that of the various types of disorder. We report energy gaps at the filling factor v$=$7/2 in a series of samples into which we intentionally added aluminum impurities during the MBE growth. These data, together with the availability of energy gaps at v$=$5/2 in the same samples, allows us to quantify the disorder broadening and the intrinsic gap of the even denominator fractional quantum Hall states. This work was supported by DOE DE-SC000671.   

Experimental constraints and a possible quantum Hall state at $\nu$=5/2

Several topological orders have been proposed to explain the quantum Hall plateau at $\nu$=5/2. The observation of an upstream neutral mode on the sample edge [Bid {\it et al.}, Nature (London) {\bf 466}, 585 (2010)] supports the non-Abelian anti-Pfaffian state. On the other hand, the tunneling experiments [Radu {\it et al.}, Science {\bf 320}, 899 (2008); Lin {\it et al.}, Phys. Rev. B {\bf 85}, 165321 (2012); Baer {\it et al.}, Phys. Rev. B {\bf 90}, 075403 (2014)] favor the 331 state which exhibits no upstream modes. We find a topological order, compatible with the results of both types of experiments. That order allows both finite and zero spin polarizations. It is Abelian but its signatures in Aharonov-Bohm interferometry can be similar to those of the Pfaffian and anti-Pfaffian states.   

Entanglement Entropy of Quantum Hall Systems with Short Range Disorder

The critical value of the mobility for which the filling 5/2 quantum Hall effect is destroyed by short range disorder is determined from an earlier calculation of the entanglement entropy. The value agrees well with experiment; this agreement is particularly significant in that there are no adjustable parameters. Entanglement entropy vs. disorder strength for filling 1/2, filling 9/2 and filling 7/3 is calculated. For filling 1/2 there is no evidence for a transition for the disorder strengths considered; for filling 9/2 there appears to be a stripe-liquid transition. For filling 7/3 there again appears to be a transition at similar value of the disorder strength as the 5/2 transition but there are stronger finite size effects.   

Hall viscosity of hierarchical quantum Hall states

We construct model wave functions on a torus for all chiral states in the abelian quantum Hall hierarchy. These functions have no variational parameters, and they transform under the modular group in the same way as the multicomponent generalizations of the Laughlin wave functions. Assuming the absence of Berry phases upon adiabatic variations of the modular parameter $\tau$, we calculate the quantum Hall viscosity and find it to be in agreement with the formula, given by Read, which relates the viscosity to the average orbital spin of the electrons. For the filling factor $\nu=2/5$ Jain state, which is at the second level in the hierarchy, we compare our model wave function with the numerically obtained ground state of the Coulomb interaction Hamiltonian in the lowest Landau level, and find very good agreement in a large region of the complex $\tau$-plane. For the same example, we also numerically compute the Hall viscosity and find good agreement with the analytical result for both the model wave function and the numerically obtained Coulomb wave function. We argue that this supports the notion of a generalized plasma analogy that would ensure that wave functions obtained using conformal field theory methods do not acquire Berry phases upon adiabatic evolution.   

High-mobility hydrogen-terminated Si(111) transistors for measurement of six-fold valley degenerate two-dimensional electron systems in fractional quantum Hall regime

The quality of hydrogen-terminated Si(111) (H-Si(111)) transistors has improved significantly. Peak electron mobility of 325,000 cm$^{\mathrm{2}}$/Vs was achieved at 90 mK, and the fractional quantum Hall effect (FQHE) at $1<\nu <2$was studied extensively [1]. We have further improved the device by solving gate leakage and contact problems with an updated design, in which a Si piece with thermal oxide acts as a gate through a vacuum cavity, and PN junctions are used to define a hexagonal two-dimensional (2D) region on a H-Si(111) piece. The device operates as an ambipolar transistor, in which a 2D electron system (2DES) and a 2D hole system can be induced at the same H-Si(111) surface. Peak electron mobility of more than 200,000 cm$^{\mathrm{2}}$/Vs is routinely achieved at 300 mK. The Si(111) surface has a six-fold valley degeneracy. The hexagonal device is designed to investigate the symmetry of the 2DES. Preliminary data show that the transport anisotropy at $\nu <6$ can be explained by the valley occupancy. The details of the valley occupancy can be caused by several mechanisms, such as miscut, magnetic field, pseudospin quantum Hall ferromagnetism (QHFM), and nematic valley polarization phases [2]. The FQHE is investigated in magnetic fields up to 35T, and the properties of composite fermions will be discussed. [1] T.M. Kott, B.H. Hu, S.H. Brown, B.E. Kane, Phys. Rev. B 89, 041107(R) (2014) [2] D. A. Abanin, S. A. Parameswaran, S. A. Kivelson, and S. L. Sondhi. Phys. Rev. B, 82, 035428 (2010)   

Quantum Hall Systems on Toroidal Geometries

We present results of recent numerical calculations of second Landau level (LL) states on toroidal geometries. Calculations on the torus generally allow for smaller particle numbers than those on the sphere, due to less powerful symmetries. However, on the torus, different candidate states for particular quantum Hall plateaus appear at equal flux, in contrast to the situation on the sphere or plane. This means that working on the torus allows for more direct comparisons of trial states and reduces the problem of aliasing. Moreover, the torus brings interesting geometry, described by a modular parameter $\tau$. This potentially allows for a larger variety of phases as well as some interesting limits which can be treated analytically. It also allows for the calculation of the Hall viscosity, a quantity which corresponds to the shift on the sphere. Among fillings considered are $\nu=\frac{12}{5}$ and $\nu=\frac{5}{2}$ where states hosting non-Abelian anyons have been conjectured.   

Disc configuration exact diagonalization studies of the phase diagram and edge states of the $\nu=5/2$ fractional quantum Hall state with Landau level mixing and finite well thickness

The $\nu=5/2$ fractional quantum Hall effect is of experimental and theoretical interest as a possible manifestation of non-Abelian statistics. The nature of this state has yet to be fully determined. The leading candidates are the Moore-Read Pfaffian state and its particle-hole conjugate, the anti-Pfaffian. When effects which break particle-hole symmetry are not included, these states are degenerate. We carry out an exact diagonalization calculation in a disk of neutralizing charge configuration, which breaks this degeneracy, and include Landau level mixing interactions arising from a diagrammatic expansion of the Coulomb potential and the effects of finite thickness. The Pfaffian sector is shown to favor strong interactions with the neutralizing background and strong Landau level(LL) mixing, while the anti-Pfaffian state occurs at weak LL mixing and background interactions. We find that there is a phase transition from the anti-Pfaffian to the Pfaffian state through a series of compressible stripe states as LL mixing is turned on. Furthermore, LL mixing interactions lead to an increased quasihole size and can overcome the effects of edge reconstruction. When the effects of the finite thickness of the confining quantum well are included, we observe enhancement these properties.   

Non-Abelian statistics of Luttinger holes in quantum wells

Non-Abelian quasiparticle excitations represent a key element of topologically protected quantum computing. Such exotic states appear in fractional quantum Hall (FQH) effect as eigenstates of $N$-body interaction potential. These potentials can be obtained by renormalization of electron-electron interactions in the presence of Landau level (LL) mixing. The properties of valence band holes makes them fundamentally different from electrons. In the presence of magnetic field, low-lying states do not exhibit fan-like diagram and several of the levels cross. Variation of magnetic field in the vicinity of level crossings serves as a knob that tunes LL mixing and enhances the 3-body interaction. $1/2$ filling factor FQH is a state that was not observed in electron liquid, but has been observed for holes. The properties of the two dimensional charged quantum hole liquid in the presence of magnetic field are studied using the spherical geometry. The properties of the novel $1/2$ state are discussed.   

Competing quantum Hall phases in the second Landau level in low density limit

We present here the results from two high quality, low density GaAs quantum wells. In sample A of electron density n $=$ 5.0 x 10$^{\mathrm{10}}$ cm$^{\mathrm{-2}}$, anisotropic electronic transport behavior was observed at $\nu =$7/2 in the second Landau level. We believe that the anisotropy is due to the large Landau level mixing effect in this sample. In sample B of density 4.1 x 10$^{\mathrm{10}}$ cm$^{\mathrm{-2}}$, strong 8/3, 5/2, and 7/3 fractional quantum Hall states were observed. Furthermore, our energy gap data obtained in various samples of different densities suggest that the 5/2 state may be spin unpolarized in the low density limit. The results from both samples show that the strong electron-electron interactions and a large Landau level mixing effect play an import role in the competing ground states in the second landau level. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.   

Heterostructure design and growth conditions necessary for electron mobility exceeding 30x10$^{6}$ cm$^{2}$/Vs in GaAs quantum wells

Ultra-high purity GaAs/AlGaAs heterostructures remain the preeminent semiconductor platform for the study of strong correlations in low dimensions. In particular, the study of fragile fractional quantum Hall states such as $\nu =$5/2 and $\nu =$12/5 in the 2$^{\mathrm{nd}}$ Landau level requires low disorder samples. While low temperature mobility is often specified as a parameter quantifying sample quality, it does not encode all information necessary to quantify disorder relevant to the fractional quantum Hall effect. Here we describe the heterostructure design considerations and molecular-beam-epitaxy growth conditions needed to achieve an electron mobility \textgreater 30x10$^{6}$cm$^{2}$/Vs. In particular, we report on the impact of several modulation doping schemes on mobility and the quality of transport in the 2$^{\mathrm{nd}}$ Landau level. We also detail constraints on starting source material purity for the achievement of high mobility. In our work high mobility has been achieved primarily through improvements in starting source materials and heterostructure design rather than improvements in vacuum quality.   

Phase Diagram of Fractional Quantum Hall Effect of Composite Fermions in Multi-Component Systems

The fractional quantum Hall effect (FQHE) of composite fermions (CFs) produces delicate states arising from a weak residual interaction between CFs. We study the spin phase diagram of these states, motivated by the recent experimental observation by Liu et al. [1] of several spin-polarization transitions at 4/5, 5/7, 6/5, 9/7, 7/9, 8/11 and 10/13 in GaAs systems. We show [2] that the FQHE of CFs is much more prevalent in multicomponent systems, and consider the feasibility of such states for systems with N components for an SU(N) symmetric interaction. Our results apply to GaAs quantum wells, wherein electrons have two components, to AlAs quantum wells and graphene, wherein electrons have four components (two spins and two valleys), and to an H-terminated Si(111) surface, which can have six components. We provide a fairly comprehensive list of possible incompressible FQH states of CFs, their SU(N) spin content, their energies, and their phase diagram as a function of the generalized ``Zeeman" energy. The results are in good agreement with available experiments.\\ $[1]$ Yang Liu, D. Kamburov, S. Hasdemir, M. Shayegan, L.N. Pfeifer, K.W. West, K.W. Baldwin, arXiv:1407.7846\\ $[2]$ Ajit C. Balram, Csaba T\H oke, A. W\'ojs, J. K. Jain, arXiv:1410.7447   

The Phase Diagram of the $\nu=5/2$ Fractional Quantum Hall Effect: Effects of Landau Level Mixing and Non-Zero Width

We study the phase diagram of the $\nu=5/2$ state by exactly diagonalizing an effective Hamiltonian describing the fractional quantum Hall effect of electrons under realistic conditions in GaAs semiconductors. The effective Hamiltonian takes Landau level mixing into account to lowest-order perturbatively in $\kappa$, the ratio of the Coulomb energy scale to the cyclotron gap and we incorporate non-zero width $w$ of the quantum well and sub-band mixing. Using the torus and sphere, we analyze the non-trivial competition between candidate ground states with 4 criteria: overlaps with trial wave functions; the size of energy gaps; the sign of an order parameter for particle-hole symmetry breaking; and entanglement spectrum. We find the ground state is in the universality class of the Moore-Read Pfaffian state, rather than anti-Pfaffian, for $\kappa<\kappa_c(w)$, where $0.6< \kappa_c (w) <1$. Landau level mixing and non-zero width suppress the excitation gap with Landau level mixing having a larger effect. Our findings have important implications for the identification of non-Abelian fractional quantum Hall states.   

Fractional Quantum Hall Effect and Wigner Crystal of Interacting Composite Fermions

In two-dimensional electron systems confined to GaAs quantum wells, as a function of either tilting the sample in magnetic field or increasing density, we observe multiple transitions of the fractional quantum Hall states (FQHSs) near filling factors $\nu$ = 3/4 and 5/4. The data reveal that these are spin-polarization transitions of interacting two-flux composite Fermions, which form their own FQHSs at these fillings. The fact that the reentrant integer quantum Hall effect near $\nu$ = 4/5 always develops following the transition to full spin polarization of the $\nu$ = 4/5 FQHS strongly links the reentrant phase to a pinned ferromagnetic Wigner crystal of two-flux composite Fermions.