4 December 2003 NYU Colloquium 1

Electronic Liquid CrystalsNovel Phases of Electrons in Two

DimensionsAlan DorseyUniversity of Florida

Collaborators:Leo Radzihovsky (U Colorado)Carlos Wexler (U Missouri)Mouneim Ettouhami (UF)

Support from the NSF

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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

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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…

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Results courtesy of Ken Cooper

http://www.its.caltech.edu/~jpelab/Ken_web_page/ferrofluid.html

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Modulated phases

Langmuir monolayer (phospholipid and

cholesterol)

Ferromagnetic film (magnetic garnet)

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Liquid crystals

T

smectic-C smectic-A nematic isotropic

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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

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Two-dimensional electron gas (2DEG)

• Created in GaAs/AlGaAs heterostructures• Magnetic field quantizes electron motion into

highly degenerate Landau levels

BBAlGaAs

EEFFN=0123

K/T 19 ),2/1( ccN NE • Magnetic length 1/26 cm/T 1056.2/ eBlb

• Experiments at FccB EETk , ,

2-11 cm 1027.2 en

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The quantum Hall effect• Filling fraction (per spin):

eB

hn

BA

ehN ee

)/(

states #

electrons #

812,25/ ,)/( 22 ehhexy

• State of the art mobility reveals interaction effectss V/cm 10 27

• No Hall effect at half filling

4 December 2003 NYU Colloquium 10

Charge density wave in 2D?

Hartree-Fock [Fogler et al. (1996)] predicts a CDW in higher LLs. Shown to be exact by Moessner and Chalker (1996).

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)?

4 December 2003 NYU Colloquium 11

Hartree-Fock treatment of CDW

)()()()()()()()(

)(ˆx)(

yxyxyVxyyxyV

xT

j

y y

j

jjj

direct or “Hartree” term exchange or “Fock” term

• Direct vs. exchange balance leads to stripes or bubbles

• Direct: repulsive long range Coulomb interaction

• Exchange: attractive short range interaction

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Experimental evidence

dc transport: Lilly et al. (1999)

Microwave conductivity: R. Lewis & L. Engel (NHMFL)

4 December 2003 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

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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)]!

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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

])( )()([ 2-12222smectic uKuBrdH xy

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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.

TEdd

dean /22

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The quantum Hall nematic

• Dislocations “melt” the smectic [Toner&Nelson (1982)].

KTiri ree /2)0(2)(2

])()()([ 223

21

2 nhnnnematic KKrdH

• Algebraic orientational order:

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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.

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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.

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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:

zL n

223

21

212( hqKqKql yxxb q)

3~ q• Note that

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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)].

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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)?

xk

xk

yk

yk

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SummaryFascinating problem of orientationally ordered point particles!