BCS Quantum Hall States

Ian Spielman

Mindy Kellogg

Alex Champagne

Loren Pfeiffer

Ken West

Jim Eisenstein

2D electron gas

B

Loren Pfeiffer and Ken West

µ = 31 x 106 cm2/Vs

Mobility of electrons in GaAs

mean free path ~ ¼ mm

10 nm

Quantum Hall effect

FQHE associated with partially filled Landau levels.

1/3

N= 0

VH

Vxx

I

B

Laughlin’s wavefunction

3( )i ji j

z z<

−∏Ψ(z1,z2, … ,zN) ~

A bizarre fluid state of many electrons having fractionally charged excitations.

Laughlin’s wavefunction

BCSQHE

Square peg, round hole?

800

600

400

200

0

5.04.54.0

T=20, 40, 100mK

5/2

7/3

Magnetic Field (T)

Long

itudi

nal R

esis

tanc

e (O

hms)

5/2 FQHE State

1/2 state: Composite Fermion Fermi Liquid - No QHE

5/2 state: BCS-like p-wave paired CF Liquid - QHE

Halperin-Lee-Read ‘93

Moore-Read ’91

Rezayi-Haldane ‘00

Analogous to 3He

Theory Conjectures

Bilayer BCS quantum Hall state

Presto! A BCS-like superfluid comprised of interlayer excitons.

Add a magnetic field

Start with a double layer 2D electron gas

+_

+_

+_

+_

+_

+_

1/2

No QHE at half-filling of the lowest Landau level

QHE in a single layer 2D system

N = 0

N = 1

ν = ½

QHE in a double layer 2D system

νT = 1 = ½ + ½

2.5

2.0

1.5

1.0

0.5

0.0

Hal

l Res

ista

nce

(h/

e2 )

1086420Magnetic Field (Tesla)

150

100

50

0

Diagonal R

esistance (kΩ)

x10

1/2+1/2

+

Pure Many-body Effect

,..,( ) ( ) ( )i j k l m n

i nz z w w z w− − −∏Ψ ∼

Origin of νT =1 QHE

layer spacing0Quantum critical point

νT = 1/2 + 1/2νT = 1

Particle-Hole Commensurability

Easy-plane ferromagnet

z

x

yφ

S

layer index → pseudospin

↑ ↓

+

pseudo-spin waves, charged vortices, etc.

Spontaneous interlayer phase coherence and “which layer” uncertainty.

12k

k φ⎡ ⎤Ψ = ⊗ ↑ + ↓⎢ ⎥⎣ ⎦∏ ie

LAYER 1 LAYER 2

∇φ dissipationless pseudo-spin currents

Excitonic BCS state

electrons

+electrons

LAYER 1 LAYER 2

electrons holes filled Landau level

+ +

LAYER 2

LAYER 1

†,1 ,2

1 1 '2 k k

k

φ⎡ ⎤Ψ = +⎢ ⎥⎣ ⎦∏ i c c vace

exciton creation operator

∇φ excitonic supercurrents

Two transport channels

1. Parallel Transport

• vortex charged excitations above gap

• edge states

• quantized Hall effect

• dissipation ~ exp (-∆/T)

I1

I2

+e/2

+e/2

vortex / anti-vortex pair

Two transport channels

2. Counterflow Transport

• collective exciton transport in condensate

• bulk flow

• zero dissipation for T < TKT

∇φ = constant

Jex = ρs ∇φ

+

_

Jex

I1

I2

counter-flow transport

The most interesting physics is in the antisymmetric channel

interlayer tunneling

It

separate layer contacts are essential

+

_

Jex

I1

I2

Tunneling experiment

IV

gate

gate

Symmetric gating allows continuous tuning of effective layer

spacing in a single sample.

Crossing the phase boundary

300

200

100

0Tunn

elin

g C

ondu

ctan

ce

(10-9

Ω-1

)

-5 0 5Interlayer Voltage (mV)

Zero bias tunneling heavily suppressed by intra-layer correlations.

layer spacing0

νT = 1/2 + 1/2

300

200

100

0Tunn

elin

g C

ondu

ctan

ce

(10-9

Ω-1

)

-5 0 5Interlayer Voltage (mV)

Crossing the phase boundarylayer spacing0

νT = 1/2 + 1/2νT = 1

Coulomb gap replaced by resonant enhancement.

0.4

0.3

0.2

0.1

0.0

dI/d

V

(1

0-6Ω

-1)

420-2-4Interlayer Voltage (mV)

Γ~ 2 µeV

Extremely narrow resonance

Counter-flow superfluidity rapidly relaxes charge defects created by

tunneling.

T = 25mK

Collective Modes

q

ω

pseudo-spin waves

x

z

y

Tunneling detects the Goldstone mode of the broken symmetry ground state.

Fertig ‘89

Wen & Zee ‘92

I

V

d

DC Josephson effect?

A finite temperature transitionG

0 (n

S)

1.81.71.6d/

55 mK

103

102

101

100

10-1

250 mK

0c

pd dG A

⎡ ⎤⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

∼

p = 3.0 ± 0.3

1.9

1.8

1.7

(d/

) c

250200150100500T (mK)

Excitonic Phase

Weakly Coupled Phase

V

I Prediction:pV I∝

where:

p = 1 for T > TKT

p ≥ 3 for T ≤ TKT

TKT ~ 0.2 K

What about superfluidity?

Counterflow Experiment

I

Counterflow Experiment

-10

-5

0

5

10

Hal

l Vol

tage

1086420Magnetic Field

Counterflow Experiment - cartoon

I

VH

-10

-5

0

5

10

Hal

l Vol

tage

1086420Magnetic Field

Counterflow Experiment - cartoon

I

VH

-10

-5

0

5

10

Hal

l Vol

tage

1086420Magnetic Field

Charge neutral excitons should feel no Lorentz force!

νT = 1

Counterflow Experiment - cartoon

I

VH

Counterflow Experiment - reality

νT = 1

2.01.51.00.50Magnetic Field (T)

1.5

1.0

0.5

0.0

Rxy

(h

/e2 )

T = 30 mK

1 0 0CFxyAt R as T= → →νT

I

VH

exciton transport dominates counterflow

Counterflow Experiment - reality

1: 0 0CF CFxy xxT aR ndAt Rν = → →

νT = 1

2.01.51.00.50Magnetic Field (T)

1.5

1.0

0.5

0.0

Rxy

(h

/e2 )

2.0

1.0

0.0

Rxx (kO

hms)

T = 30 mK

I

VH

Vxx

Excitonic superfluidity?

10-4

10-3

10-2

10-1

100

101

102

103

30201001/T (K-1)

Conductivities

CFxxσ

||xxσ

( )2e hxxσ

|| ( )B 0xxσ =

d/ = 1.5

Counterflow dissipation small but non-zero at all finite T.

Exciton diffusivity reaches ~5000 cm2/sec.

Tunneling reveals a phase transition to an interlayer phase coherent state and unveils the Goldstone mode.

Counterflow experiments reveal nearly lossless collective transport of excitons.

Summary

In closely-spaced bilayer 2D electron systems at νT = 1:

An excitonic BCS QHE state

Where to go next?

Persistent counter-currents in rings. How to detect?

Excitonic Josephson junctions. How to fabricate?

+

_I1

I2

+

_I1

I2