### Session Y3: Multi-valley Electron Systems in the Quantum Limit

 Friday, March 20, 2009 8:00AM - 8:36AM Y3.00001: Phase Transitions of Dirac Electrons in Bismuth Invited Speaker: Lu Li The Fermi Surface (FS) in elemental bismuth consists of 3 electron ellipsoids and one hole ellipsoid [1]. The accidental coincidence of the hole and electron caliper areas when the field $\bf H$ is aligned with the trigonal axis $\bf Z$ has long stymied analyses of the quantum oscillations. Because of current strong interest in how electrons with Dirac dispersion behave in intense fields, we have renewed attack on this problem [2] using high-resolution torque magnetometry in fields up to 31 T and at temperatures $T$ down to 300 mK. When $\bf H$ is tilted with respect to $\bf Z$ by a slight angle $\theta$, the torque $\vec{\tau}$ on the sample derived from the 3 electron ellipsoids dominates the torque from the hole FS, allowing the Landau Level crossings of the Dirac electron to be resolved. By measuring the curves of $\vec{\tau}$ vs. $H$ at 19 values of $\theta$ straddling the trigonal axes, we completely resolve the Landau Levels of the Dirac electrons. A new result is the detection of jumps in the transverse magnetization when $H$ exceeds the quantum limit of the electron pockets. By tracing the jumps in the plane of $H$ vs. $\theta$, we uncover a region in which the Dirac electrons enter a new ground state. Within this cone-shaped region, Landau Level anomalies are severely suppressed. We interpret the state as one in which the 3-fold valley degeneracy of the Dirac gas is lifted to form a many-body state. The unusual nature of the magnetization within this region will be described. \\[4pt] [1] M. H. Cohen and E. I. Blount, Phil. Mag. {\bf 5}, 115 (1960). \\[0pt] [2] Lu Li {\it et al.}, Science {\bf 321}, 547 (2008). Friday, March 20, 2009 8:36AM - 9:12AM Y3.00002: Hall Plateaus at magic angles in ultraquantum Bismuth Invited Speaker: Fauqu\'e Beno\^It The behaviour of a three-dimensional electron gas in the presence of a magnetic field strong enough to put all carriers in the first Landau level (i.e. beyond the quantum limit) is a longstanding question of theoretical condensed matter physics [1]. This issue has been recently explored by two high-field experiments on elemental semi-metal Bismuth. In a first study of transport coefficients (which are dominated by hole-like carriers), the Nernst coefficient presented three unexpected maxima that are concomitant with quasi-plateaux in the Hall coefficient [2]. In a second series of experiments, torque magnetometry (which mainly probes the three Dirac valley electron pockets) detected a field-induced phase transition [3]. The full understanding of the electron and hole behaviours above the quantum limit of pure Bi is therefore still under debate. In this talk, we will present our measurement of the Hall resistivity and torque magnetometry with magnetic field up to 31 T and rotating in the trigonal-bisectrix plane [4]. The Hall response is dominated by the hole pockets according to its sign as well as the period and the angular dependence of its quantum oscillations. In the vicinity of the quantum limit, it presents additional anomalies which are the fingerprints of the electron pockets. We found that for particular orientations of the magnetic field (namely magic angles''), the Hall response becomes field-independent within the experimental resolution around 20T. This drastic dependence of the plateaux on the field orientation provides strong constraints for theoretical scenarios. \\[4pt] [1] Bertrand I. Halperin, \emph{Japanese Journal of Applied Physics}, {\bf 26}, Supplement 26-3 (1987).\\[0pt] [2] Kamran Behnia, Luis Balicas, Yakov Kopelevich, \emph{Science}, {\bf 317}, 1729 (2008).\\[0pt] [3] Lu Li, J. G. Checkelsky, Y. S. Hor, C. Uher, A. F. Hebard, R. J. Cava, and N. P. Ong , \emph{Science}, {\bf 321}, 5888 (2008).\\[0pt] [4] Beno\^it Fauqu\'e, Luis Balicas, Ilya Sheikin, Jean Paul Issi and Kamran Behnia, to be published Friday, March 20, 2009 9:12AM - 9:48AM Y3.00003: Bismuth and graphite in the ultraquantum limit: signatures of fractional quantum Hall effect Invited Speaker: Yakov Kopelevich Bismuth and graphite are semimetals that possess both conventional massive and Dirac-like quasiparticle spectra. High quality graphite is a multi-layer system with nearly decoupled two-dimensional (2D) graphene planes, in which the integer quantum Hall effect has already been found [1]. On the other hand, the fractional quantum Hall effect (FQHE) has been observed for 3D bismuth in the ultraquantum limit (UCL), i. e. above the field that pulls all carriers into the lowest Landau level [2]. Recent measurements performed on quasi-2D graphite in magnetic field up to B = 57 T revealed well defined plateaus in the Hall resistance for B $>$ 10 T, suggesting also the FQHE occurrence in graphite in the UCL [3]. A striking similarity of the obtained results with the FQHE measured for 2D electron system in a GaAs/AlGaAs quantum well [4] is found. Our present results indicate the interplay between FQHE and charge density wave states in graphite. We discuss the FQHE occurrence in bismuth and graphite within the framework of available theoretical models. \\[4pt] [1] Y. Kopelevich, J. H. S. Torres, R. R. da Silva et al., Phys. Rev. Lett. 90,156402 (2003). \\[0pt] [2] [K. Behnia, L. Balicas, and Y. Kopelevich, Science 317, 1729 (2007). \\[0pt] [3] Y. Kopelevich, B. Raquet, M. Goiran et al. (unpublished). \\[0pt] [4] W. Pan, H. L. Stormer, D. C. Tsui et al., Phys. Rev. Lett. 88, 176802 (2002). Friday, March 20, 2009 9:48AM - 10:24AM Y3.00004: AlAs 2D Electrons at High Magnetic Field: The Role of Spin and Valley Degree of Freedom Invited Speaker: Mansour Shayegan Two-dimensional (2D) electrons in AlAs quantum wells occupy multiple conduction-band minima (or valleys) at the X point of the Brillouin zone. These valleys have large effective mass (m*) and g-factor compared to the standard GaAs electrons, and are also highly anisotropic. The system is rather unique in that, with proper choice of well width and by applying in situ symmetry-breaking strain in the plane, one can control the occupation of different valleys, thus rendering a system with tuneable m*, g-factor, Fermi contour anisotropy, and with single, double, or triple valley degeneracy. By adding a magnetic field, we obtain a system which allows us to explore very rich physics ensuing from the valley and spin degrees of freedom in a strongly interacting system. In this presentation, I will highlight some of our latest results on 2D electrons confined to wide AlAs quantum wells where the electrons reside in two in-plane valleys whose occupation can be controlled via the application of strain. I will present the results of our m* measurements, via analyzing the temperature dependence of the Shubnikov-de Haas oscillations. The measured m* shows a strong dependence on the occupation of valley and spin subbands, reflecting the electron- electron interaction in this system. Most remarkably, m* is suppressed with respect the band value when the 2D electrons are fully spin- and valley-polarized. I will also discuss the relation of m* suppression to the 2D metal-insulator transition problem. Our studies also include measurements of the valley susceptibility (dependence of valley population on applied strain) and the valley polarization of the fractional quantum Hall effect composite fermions. While part of our observations can be explained well by a simple Landau level fan diagram for composite fermions with a valley degree of freedom, there are some surprises. Friday, March 20, 2009 10:24AM - 11:00AM Y3.00005: High Mobility Sixfold Valley Degenerate Electrons on Silicon [111] Surfaces Invited Speaker: Robert N. McFarland The 111 surface of silicon is predicted to retain the sixfold valley degeneracy of the ideal bulk crystal. We have developed a method for fabricating field effect transistors using vacuum as a dielectric in order to study electron transport on the bare hydrogen-terminated surface, free from the complications created by intrinsic disorder at Si-Si$O_2$ interfaces. The resulting devices display very high mobilities (up to 110,000 $cm^2/Vs$ at 70mK, more than twice as large as the best silicon MOSFETs), enabling us to probe valley-dependent transport dynamics to a much greater degree than previously possible. Measurements made on a recent device over a density range of $n_s=0.7-7\times10^{11}/cm^2$ reveal considerable information about the nature of this degeneracy and its role in 2D transport. In particular, we find (at $n_s$=6.7) that 1) low field Shubnikov-de Haas oscillations reveal a clearly sixfold degenerate system and allow us to establish an upper bound on the valley splitting of 0.2K 2) longitudinal resistivity at B=0 displays a strong temperature dependence, consistent with predictions that large valley degeneracy should enhance screening[1] and 3) the Hall coefficient near B=0 is modified by the presence of multiple valleys, and we can use this correction to measure the intervalley Coulomb drag and its temperature dependence. [1] E. H. Hwang and S. Das Sarma. PRB 75, 073301 (2007)