Lecture 3

Exciton Condensation in Bilayer Quantum Hall Systems

Double Layer Two-Dimensional Electron Gas

Double Quantum Well

AlxGa1-xAs AlxGa1-xAs

GaAs

Electron wavefunction

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 Double Layer 2D Systems

νT = 1

1/2

1 2Tν ν ν= +

νT = 1 = ½ + ½

νT = ½ = ¼ + ¼

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

Single Particle and Many-body Origins for νT =1 QHE

ΔSASΔSAS

E

BνT=1

N=0↑Antisymmetric State

Symmetric State

1. Single Particle Tunneling

2. Pure Many-body Effect

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

i nz z w w z w− − −∏Ψ ∼

N=0↓

6

5

4

3

2

1

00.100.080.060.040.020.00

Tunneling Strength, ΔSAS/(e2/ ε )

QHE

NO QHEd

EFE0

Phase Diagram at νT = 1

Continuous evolution of QHE

QHE

NO QHE

2( / )SAS

e εΔ

d

Simple model of phase transition at νT = 1

Competition between tunneling and direct Coulomb energy drives transition

2 2R d+

+S A

−S A

R

SS

No QHE without tunneling?

MacDonald, et al. 1990

Exchange allows QHE to survive to zero tunneling

R

SS

2 2R d+

+S A

−S A

Uex < 0

Uex = 0

QHE

NO QHE

2( / )SAS

e εΔ

d

6

5

4

3

2

1

00.100.080.060.040.020.00

Tunneling Strength, ΔSAS/(e2/ ε )

QHE

NO QHEd

EFE0

layer spacing

0Quantum critical point

νT = 1/2 + 1/2νT = 1

Phase Diagram at νT = 1

Easy-Plane Ferromagnet

z

x

yϕ

S

( )↓+↑⊗=Ψ ∏k

k eiϕ

layer index → pseudospin

↑ ↓

+

pseudospin waves (Goldstone modes)

charged vortices

Kosterlitz-Thouless transition

Exchange-driven “spontaneous interlayer phase coherence”

Elementary Excitations

x

z

y

q

ω

pseudo-spin waves

1. Neutral collective modes

Goldstone mode of broken U(1) symmetry

Elementary Excitations

2. Charged excitations

Vortices carrying topological and real charge

+e/2

+e/2

meron / anti-meron pair

-e/2

+e/2+e/2

-e/2

Four flavors of vortices (merons)

Vortices paired up below Kosterlitz-Thouless transition temperature

Quantum Fluctuations

Pseudospin is NOT a good quantum number for d>0.

2

2

2 2

eV Vr

eV Vr d

ε

ε

↑↑ ↓↓

↑↓ ↓↑

= =

= =+

layer spacing0Quantum critical point

νT = 1/2 + 1/2νT = 1

Excitonic Bose Condensate

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

Intrinsic semiconductor at T = 0

hν

Add light

Only electrons

Particle-hole transformation on valence band

electrons

holes

+ excitons!

hν

Recombination

A different way: Doped bilayer semiconductor

Layer 1 Layer 2

Discard the valence band

Layer 1 Layer 2

Add a magnetic field

conduction band Landau levels

0

1

2

N = 3

Layer 1 Layer 2

ωc

A bigger magnetic field: One filled Landau level

Layer 1 Layer 2

Analogous to intrinsic semiconductor at T = 0, but no gap!

N = 0

Layer 1 Layer 2

N = 0

Transfer charge by doping or gating

Total number of electrons unchanged

Layer 1 Layer 2

N = 0

holeselectrons

excitons!

+

Particle-hole transformation on lowest Landau level in layer 2

Two Transport Channels

1. Parallel Transport

• vortex charged excitations above gap

• edge states

• quantized Hall effect

• dissipation ~ exp (-Δ/T)

I1

I2

+e/2

+e/2

meron / anti-meron 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

Experimental Issues

Need weak tunneling but strong interlayer interactions

Want:

Narrow quantum wells

High Barriers

Low Density

VB

W

Experimental Issues

Need weak tunneling but strong interlayer interactions

Want:

Narrow quantum wells

High Barriers

Low Density

VB

W

Low mobility μ ~ W-6

High Al concentration, oxidation

Low mobility, difficult to contactand gate.

Zero Tunneling Limit

layer spacing0Quantum critical point

νT = 1/2 + 1/2νT = 1

6

5

4

3

2

1

0

d/

10-7 10-6 10-5 10-4 10-3 10-2 10-1

Tunneling Strength, ΔSAS/(e2/ε )

NO QHE

QHE?

νT = 1

18nm

10nm

Al0.9Ga0.1As

N1 ~ N2 ~ 5x1010cm-2

3

2

1

0

Rxx

(kO

hms)

2.52.01.51.00.50.0Magnetic Field (T)

10

2

46

100

2

46

1000

2

Rxx

(O

hms)

30201001/Temperature (K-1)

Bilayer νtot = 1 QHE with Nearly Zero Tunneling

−Δ∝ /2TxxR e

Δ = 0.34K

νtot = 1

estimated ΔSAS ≈ 10μK⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

−×≈ ε7 2

1 3 10 e.

T=25mK

Energy gap overwhelmingly dominated by interaction effects

Δ ≈ ×ΔSAS30,000

Tunneling Experiment

IV

gate

gate

Symmetric gating allows continuous tuning of effective layer

spacing in a single sample.

Tunneling between Weakly-Coupled 2DEGs

B=0 Narrow resonance, created by momentum and energy conservation; a single particle effect. Γ∼100μeV

B>>0 Strongly suppressed tunneling around zero bias, broad high energy response; a many-body-effect.

4

3

2

1

0

-1

-15 -10 -5 0 5 10 15Interlayer Voltage (mV)

B=0

B=13T, d/l=5.3

dI/d

V

(10-

6Ω

-1)

Lowest Landau Level is a Strongly Correlated System

Vacancy + interstitial

Lowest Landau Level is a Strongly Correlated System

Tunneling is fast compared to relaxation time of charge defects.

Result: Tunneling is suppressed at voltages below mean Coulomb energy.

2

gapee V 0.3ε

Δ ≈

-5 0 5

NT = 5.4NT =10.9 x 10 10cm-2

NT = 6.9

-5 0 5

NT = 6.4

Crossing the νT =1 Phase Boundary

T=40mKTu

nnel

ing

Con

duct

ance

(10-

7 Ω

-1)

Interlayer Voltage (mV)

2.01.51.00.50.0Temperature (K)

1.0x10-6

0.5

0.0G

0 (m

ho)

low N high N

Magnetic Field and Temperature Dependences

1.5

1.0

0.5

0.0

G0

(10

-7 Ω

-1)

43210Magnetic Field (Tesla)

νT =1

x200

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.

Collective Modes

q

ω

pseudo-spin waves

x

z

y

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

6

5

4

3

2

1

0

dI/d

V

(10

-6Ω−1)

-200 0 200V (μV)

B|| = 0

B|| = 0.6T

-200 0 200V (μV)

10-7 Ω−1

Peak is suppressed and satellites appear.

B||

B⊥

q = eB|| d /

Parallel Field Spectroscopy

0.20

0.15

0.10

0.05

0.00

E /

(e2 /ε

)

2.01.51.00.50.0 q

d/ = 0.8

1.0

1.2

1.4

Measured Collective Mode Dispersion

Direct observation of linearly dispersing Goldstone mode.

- red curves: A.H. MacDonald

v ~ 15 km/sec

Temperature dependence of dI/dV resonance

10-3

10-2

10-1

100

Zero

-bia

s C

ondu

ctan

ce (

e2 /h)

d/l = 1.5

80

60

40

20

0

Line

wid

th (

μV)

0.30.20.10Temperature (K)

kBT

10

5

00.10

min 2 V0 3ns

mf mp v 5.

Γ ≈ μ ≈ τ⇒τ≈⇒ = τ≈ μ

-0.2

-0.1

0

0.1

2

3

4

5

dI/d

V (1

0-6Ω

-1)

-20

-10

0

10

20

Tunn

elin

g cu

rren

t (pA

)

-4 -2 0 2 4Interlayer voltage (mV)

Josephson Effect in a Non-Superconductor?

T cJ J sin ϕ= Δ

T SAS2eJ sin

4ϕ

π= Δ

c SAS2eAI 5 A

40μ

π= Δ ≈

dc Josephson effect in a superconductor:

Interlayer tunneling in νT = 1 bilayer QHE:

QHE critical current:

Suggests that pseudospin field is heavily disordered on macroscopic scales

cf. Fertig and Straley, cond-mat/0301128

V

I

Counterflow Experiment

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?

Parallel vs. Counterflow Transport

0 0CF CFxx xyR and R→ →

|| ||0 /xyxxR and R h e→ → 2 ||xx 0σ →

CFxxσ → ?

2 2xx

xx xx xy

RR R

σ+

=

30201001/T (K-1)

0.01

0.1

1

10

100R

xx a

nd R

xy (k

Ω)

30201001/T (K-1)

Temperature Dependences

Parallel Counterflow

d/ = 1.5

Rxx

Rxy

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.

I

V

d

A quantum phase transition

layer spacing0Quantum critical point

νT = 1/2 + 1/2νT = 1

A quantum phase transition

layer spacing

νT = 1/2 + 1/2νT = 1

A quantum phase transition

layer spacing

νT = 1/2 + 1/2νT = 1

density imbalance

Zeeman energy tunneling energy

in-plane magnetic field

A finite temperature transition?

layer spacing

νT = 1/2 + 1/2νT = 1

temperature

Kosterlitz-Thouless?First-order?

?

d/

T

0.2

0.1

0

dI/d

V (μ

S)

Interlayer Voltage

x10

T = 55 mK65

75 85 100 160125

d/ = 1.71

2

1

0

dI/d

V (μ

S)

d/ = 1.56

T = 55 mK

x10x100

1.621.65 1.68

1.711.76

1.80

Entering the coherent phase

No obvious thermal smearingCurves shift with temperature

10-4

10-3

10-2

10-1

100G

(0)

(μS

)

1.81.71.6d/

55 mK

250200 160 125 85

Onset of tunneling is “universal”

10-4

10-3

10-2

10-1

100G

(0)

(μS

)

1.81.71.6d/

55 mK

250200 160 125 85

Onset of tunneling is “universal”

⎡ ⎤⎛ ⎞ ⎛ ⎞= −⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎣ ⎦

(0)p

c

d dG Kempirical fitting function:

1.85

1.80

1.75

1.70

(d/

) c

250200150100500

T (mK)

Coherent Bilayer

Incoherent Bilayer

4

3

2p

Results of Fitting

A finite temperature phase transition.

1.85

1.80

1.75

1.70

(d/

) c

250200150100500

T (mK)

Coherent Bilayer

Incoherent Bilayer

4

3

2p

Results of Fitting

K-T “Theory”

Is it the K-T transition? Maybe.

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

Counterflow and Coulomb drag reveal nearly lossless collective transport of excitons.

Results

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

These are the main ingredients of excitonic ``BEC``

Puzzles

• tunneling conductance peak too small

• B||-dependence of tunneling not understood

• no non-linearity observed in counterflow transport

• thermally-activated dissipation suggests T>TKT

Disorder induced free vortices

present at all temperatures.

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

Add a magnetic field

Start with a double layer 2D electron gas

+_

+_

+_

+_

+_

+_

Quantum Hall Excitonic Condensate