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)
• Filippos Klironomos (UF)

Support from the NSF and National High Magnetic
Field Laboratory
2
Welcome to Florida!
Gainesville
3
Outline

• Physics of modulated phases
• The two dimensional electron gas (2DEG)
• Evidence for a charge density wave?
• Liquid crystal physics in quantum Hall systems
  smectics and nematics
• Quantum theory of the nematic phase

4
Competing interactions

• Long range repulsive force uniform phase
• Short range attractive force compact structures
• Competition between forces?inhomogeneous (meso)
  phase
• Ferromagnetic films, ferrofluids, type-I
  superconductors, block copolymers

5
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
6
Results courtesy of Ken Cooper
ferromovie.mov
http//www.its.caltech.edu/jpelab/Ken_web_page/fe
rrofluid.html
7
Modulated phases
Langmuir monolayer (phospholipid and cholesterol)
Intermediate state of type-I superconductor
8
Two-dimensional electron gas (2DEG)

• Created in GaAs/AlGaAs heterostructures
• Magnetic field quantizes electron motion into
  highly degenerate Landau levels (degeneracy
  scales as area)

• Magnetic length

• Experiments at

9
The quantum Hall effect
Ex rxy Jy , Ex rxx Jx
Vx
Ix
Iy

• Filling fraction (per spin)

Clean samples!
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).
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

12
Transport anisotropy

• M. Lilly et al. (1999)
• huge anisotropy below
• 100 mK
• n4 separates
• transport regimes
• Anisotropy aligns with
• GaAs crystal axes
• Requires high mobility
• samples

Magnetic field (Tesla)
N0,1
N2,3,
13
Temperature dependence

• Easy direction lt110gt
• Native anisotropy energy about 1 mK
• No QHE compressible state

?????
lt110gt
-
lt110gt
14
Hall and longitudinal resistance
15
Reorientation with in-plane field
16
Microwave conductivity
R. Lewis L. Engel (NHMFL)
Wigner/bubble crystal
Electron stripes
Wigner/bubble crystal
17
Interpretation of microwave data

• Collectively pinned CDW, with Larkin domains of
  size L

?

• Pinning frequency (zero field)

18
Phase diagram
Wigner/bubble crystal
Stripes
Wigner/bubble crystal
19
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 FradkinKivelson
(1999), MacDonaldFisher (2000)!
20
Liquid crystals
T
smectic-C
smectic-A
nematic
isotropic
21
The quantum Hall smectic

• Classical smectic is a layered liquid
• Stripe fluctuations lead to a quantum Hall
  smectic
• WexlerATD (2001) find elastic properties from
  HFA

22
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 (Burgers vector).
• Dislocation energy in a smectic is finite, there
  will be a nonzero density.
• Dislocations further reduce the orientational
  order.

23
The quantum Hall nematic

• Dislocations melt the smectic TonerNelson
  (1982).

• Algebraic orientational order

24
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.
• WexlerATD start from HFA and find transition
  at 200 mK, vs. 70-100 mK in experiments.

25
Quantum theory of the QHN

• Classical theory overestimates anisotropy below
  20 mK. Are quantum fluctuations the culprit?
• Quantum fluctuations can unbind dislocations at
  T0.

RadzihovskyATD (PRL, 2002) use dynamics of
local smectic layers as a guide. Make contact
with hydrodynamics.
26
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

27
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).

28
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.
• Transitions between solid phases.
• Relation to nanoscale phase separation in other
  systems (e.g., high Tc superconductors, CMR
  materials)?

Pomeranchuk instability
29
Summary
Marriage of correlated electron and soft matter
physics
Fascinating problem of orientationally ordered
point particles!