Electronic Liquid Crystals Novel Phases of Electrons in Two Dimensions

1. Electronic Liquid CrystalsNovel 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 NYU Colloquium

2. Competing interactions • Long range repulsive force: uniform phase • Short range attractive force: compact structures • Competition between forcesinhomogeneous phase. • Ferromagnetic films, ferrofluids, type-I superconductors, block copolymers NYU Colloquium

3. 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… NYU Colloquium

4. Results courtesy of Ken Cooper http://www.its.caltech.edu/~jpelab/Ken_web_page/ferrofluid.html NYU Colloquium

5. Modulated phases Langmuir monolayer (phospholipid and cholesterol) Ferromagnetic film (magnetic garnet) NYU Colloquium

6. Liquid crystals T smectic-C smectic-A nematic isotropic NYU Colloquium

7. 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 NYU Colloquium

8. AlGaAs B 3 2 E 1 F N=0 Two-dimensional electron gas (2DEG) • Created in GaAs/AlGaAs heterostructures • Magnetic field quantizes electron motion into highly degenerate Landau levels • Magnetic length • Experiments at NYU Colloquium

9. The quantum Hall effect • Filling fraction (per spin): • State of the art mobility reveals interaction effects • No Hall effect at half filling NYU Colloquium

10. 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). NYU Colloquium

11. Hartree-Fock treatment of CDW • Direct vs. exchange balance leads to stripes or bubbles direct or “Hartree” term exchange or “Fock” term • Direct: repulsive long range Coulomb interaction • Exchange: attractive short range interaction NYU Colloquium

12. Experimental evidence Microwave conductivity: R. Lewis & L. Engel (NHMFL) dc transport: Lilly et al. (1999) NYU Colloquium

13. Experimental details • Anisotropy can be reoriented with an in-plane 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 NYU Colloquium

14. 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)]! NYU Colloquium

15. 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 NYU Colloquium

16. 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. • Dislocations further reduce the orientational order. NYU Colloquium

17. The quantum Hall nematic • Dislocations “melt” the smectic [Toner&Nelson (1982)]. • Algebraic orientational order: NYU Colloquium

18. 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. NYU Colloquium

19. 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. NYU Colloquium

20. Theoretical digression… • The collective degrees of freedom are the rotations of the dislocation-free domains (nematogens). Their angular momenta and directors 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: • Note that NYU Colloquium

21. 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)]. NYU Colloquium

22. 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)? NYU Colloquium

23. Summary Fascinating problem of orientationally ordered point particles! NYU Colloquium