Electronic Liquid Crystals Novel Phases of Electrons in Two Dimensions Alan Dorsey University of Florida Collaborators: Leo Radzihovsky (U Colorado) Carlos Wexler (U Missouri) Mouneim Ettouhami (UF) Support.
Download ReportTranscript Electronic Liquid Crystals Novel Phases of Electrons in Two Dimensions Alan Dorsey University of Florida Collaborators: Leo Radzihovsky (U Colorado) Carlos Wexler (U Missouri) Mouneim Ettouhami (UF) Support.
Electronic Liquid Crystals Novel Phases of Electrons in Two Dimensions Alan Dorsey University of Florida Collaborators: Leo Radzihovsky (U Colorado) Carlos Wexler (U Missouri) Mouneim Ettouhami (UF) Support from the NSF 4 December 2003 NYU Colloquium 1 Competing interactions •Long range repulsive force: uniform phase •Short range attractive force: compact structures •Competition between forcesinhomogeneous phase. •Ferromagnetic films, ferrofluids, type-I superconductors, block copolymers 4 December 2003 NYU Colloquium 2 Ferrofluid in a Hele-Shaw cell •Ferrofluid: colloid of 1 micron spheres. Fluid becomes magnetized in an applied field. •Hele-Shaw cell: ferrofluid between two glass plates Surface tension competes with dipole-dipole interaction… 4 December 2003 NYU Colloquium 3 Results courtesy of Ken Cooper http://www.its.caltech.edu/~jpelab/Ken_web_page/ferrofluid.html 4 December 2003 NYU Colloquium 4 Modulated phases Langmuir monolayer (phospholipid and cholesterol) 4 December 2003 NYU Colloquium Ferromagnetic film (magnetic garnet) 5 Liquid crystals T smectic-C 4 December 2003 smectic-A nematic NYU Colloquium isotropic 6 Outline •Overview of the two dimensional electron gas and the quantum Hall effect •Theoretical and experimental evidence for a charge density wave? •Liquid crystal physics in quantum Hall systems—smectics and nematics •Quantum theory of the nematic phase 4 December 2003 NYU Colloquium 7 Two-dimensional electron gas (2DEG) AlGaAs B ne 2.271011 cm-2 3 2 1 N=0 EF • Created in GaAs/AlGaAs heterostructures • Magnetic field quantizes electron motion into highly degenerate Landau levels EN c ( N 1/ 2), c 19 K/T • Magnetic length lb / eB 2.56106 cm/T1/2 • Experiments at kBT c , Ec , EF 4 December 2003 NYU Colloquium 8 The quantum Hall effect • Filling fraction (per spin): # electrons Ne(h / e) hne # states BA eB xy (e / h) , h / e2 25,812 2 • State of the art mobility reveals interaction effects 107 cm2 / V s • No Hall effect at half filling 4 December 2003 NYU Colloquium 9 Charge density wave in 2D? CDWs proposed by Fukuyama et al. (1979) as the ground state of a partially filled LL, but the Laughlin liquid has a lower energy. What happens in higher LLs (lower magnetic fields)? Hartree-Fock [Fogler et al. (1996)] predicts a CDW in higher LLs. Shown to be exact by Moessner and Chalker (1996). 4 December 2003 NYU Colloquium 10 Hartree-Fock treatment of CDW • Direct vs. exchange balance leads to stripes or bubbles j j ( x) Tˆ j ( x) V ( y x) ( y) ( y) j ( x) V ( y x) ( y) ( x) j ( y) y y direct or “Hartree” term exchange or “Fock” term • Direct: repulsive long range Coulomb interaction • Exchange: attractive short range interaction 4 December 2003 NYU Colloquium 11 Experimental evidence dc transport: Lilly et al. (1999) 4 December 2003 Microwave conductivity: R. Lewis & L. Engel (NHMFL) NYU Colloquium 12 Experimental details • Anisotropy can be reoriented with an inplane field (new features at 5/2, 7/2) • Transition at 100 mK • “Easy” direction [110] • “Native” anisotropy energy about 1 mK • No QHE: “compressible” state 4 December 2003 NYU Colloquium 13 A charge density wave? • Transport anisotropy consistent with CDW state • BUT: • Transport in static CDW would be too anisotropic •Formation energy of several K, not mK •Data also consistent with an anisotropic liquid Fluctuations must be important [Fradkin&Kivelson (1999), MacDonald&Fisher (2000)]! 4 December 2003 NYU Colloquium 14 The quantum Hall smectic • Classical smectic is a “layered liquid” •Stripe fluctuations lead to a “quantum Hall smectic” • Wexler&ATD (2001): find elastic properties from HFA H smectic d 2 r [B( y u ) 2 K ( 2x u ) 2 -1 ( ) 2 ] 4 December 2003 NYU Colloquium 15 Order in two dimensions Problem: in 2D phonons destroy the positional order but preserve the orientational order. However, this ignores dislocations (=half a layer inserted into crystal). • Topological character. • Dislocation energy in a smectic is finite, there will be a nonzero density. nd d2 a 2e Ed / T • Dislocations further reduce the orientational order. 4 December 2003 NYU Colloquium 16 The quantum Hall nematic • Dislocations “melt” the smectic [Toner&Nelson (1982)]. H nematic d 2 r[ K1 ( n) 2 K 3 ( n) 2 (h n) 2 ] • Algebraic orientational order: 4 December 2003 e NYU Colloquium i 2 ( r ) i 2 ( 0) e r 2T / K 17 Nematic to isotropic transition •Low temperature phase is better described as a nematic [Cooper et al (2001)]. Local stripe order persists at high temperatures. •Nematic to isotropic transition occurs via a disclination unbinding (Kosterlitz-Thouless) transition. • Wexler&ATD: start from HFA and find transition at 200 mK, vs. 70-100 mK in experiments. 4 December 2003 NYU Colloquium 18 Quantum theory of the QHN • Classical theory overestimates anisotropy below 20 mK. Are quantum fluctuations the culprit? • Quantum fluctuations can unbind dislocations at T=0. Radzihovsky&ATD (PRL, 2002): use dynamics of local smectic layers as a guide. Make contact with hydrodynamics. 4 December 2003 NYU Colloquium 19 Theoretical digression… • The collective degrees of freedom are the rotations of the dislocation-free domains (nematogens). Their angular momenta Lz and directors n are conjugate. • Commutation relations are derived in the high field limit, and lead to an unusual quantum rotor model. • Broken rotational symmetry leads to a Goldstone mode with anisotropic dispersion: (q) lb2 1qx2 K1qx2 K3q y2 h 2 3 • Note that ~ q 4 December 2003 NYU Colloquium 20 Predictions • QHN exhibits true long range order at zero temperature; quantum fluctuations important below 20 mK. • QHN unstable to weak disorder. Glass phase? • Tunneling probes low energy excitations. See a pseudogap at low bias. • Damping of Goldstone mode due to coupling to quasiparticles. • Resistivity anisotropy proportional to nematic order parameter [conjectured by Fradkin et al. (2000)]. 4 December 2003 NYU Colloquium 21 New directions • Start from half-filled fermi liquid state. Can interactions cause the FS to spontaneously deform? • Variational wavefunctions? • Experimental probes: tunneling, magnetic focusing, surface acoustic waves. • Relation to nanoscale phase separation in other systems (e.g., cuprate superconductors)? 4 December 2003 NYU Colloquium ky kx ky kx 22 Summary Fascinating problem of orientationally ordered point particles! 4 December 2003 NYU Colloquium 23