lecture 3 exciton condensation in bilayer quantum hall...
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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