Post on 03-Oct-2020
Beyond the Quantum Hall Effect
School on Low Dimensional Nanoscopic Systems
Harish-chandra Research InstituteJanuary – February 2008
Jim EisensteinCalifornia Institute of Technology
Outline of the Lectures
I. Introduction to the Quantum Hall Regimea. Overviewb. QHE Basics: Integer and Fractional, composite fermions. c. Thermal transport in quantum Hall edge channelsd. Beyond the standard paradigm: Excited Landau Levels and Double Layers
II. 2D Electrons in Excited Landau Levelsa. Basic Observations.b. CDW scenarioc. Symmetry Breakersd. Liquid crystal scenarioe. Re-entrant insulating phasesf. N = 1 Landau level
III. Exciton Condensation in Bilayer Electron Systemsa. Overview and history. Discovery of νT = 1/2 and νT = 1 QHE states.b. Quantum Hall ferromagnetism and the νT = 1 QHE state.c. νT = 1 QHE state as an exciton condensate. d. Tunnelinge. Counterflowf. Finite temperature phase transition
Lecture 1
Introduction to the Quantum Hall Regime
Two-Dimensional Electron Gas
~100 Å
A Semiconductor Sandwich
A ABCB
VB
~100 Å
subband wave function
Quantum Confinement
CB
VB
Quantum Confinement
CB
VB
donor layers
+ +-
Perfect Registry
Molecular Beam Epitaxy
Ga As
ultra highvacuum
heated cells
high qualityGaAs substrate
hot!
AlAlGaAs
AlGaAs
GaAs100 A
allows for precisionengineering of crystal
layer by layer
Molecular Beam Epitaxy
“spray painting with atoms”
Loren Pfeiffer and Ken West
μ = 36 x 106 cm2/Vs
Mobility of Electrons in GaAs
mean free path ~ ¼ mm
10 nm
*
emτμ =
Hall Effect Measurements
VH
Vxx
B
RH = VH / I = B/ne
VH
VxxI
B
The Integer Quantum Hall Effect
jh/eR
2
Hall = = 25812.807 Ohms / integer
RH
(h/e2)
Rxx
(kΩ) VH
Vxx
I
B
DOS
Energy
Landau levels
2D Electrons in a Magnetic Field
Circular orbits + de Broglie waves ⇒discrete radii and energies
Landau levels are massively degenerate: n0 = eB/h.
Precisely fill an integer number of LLs: n = j x n0
RH = B/ne = B/jen0 = h/je2
IQHE associated with fully filled Landau levels.
Energy
DOS
Magnetic Fields Enhance Interactions
Fermi gas
DOS
Energy
Landau levels
Magnetic fields quench the KE and produce a strongly interacting system
Fractional Quantum Hall Effect
FQHE associated with partially filled Landau levels.
1/3
N= 0
Laughlin’s Wavefunction - 1983
3( )i ji j
z z<
−∏Ψ(z1,z2, … ,zn) ~
A bizarre fluid state of many electrons having fractionally charged excitations.
Laughlin’s Wavefunction
1/2
No FQHE at half-filling of lowest Landau level.
Odd-Denominator Rule
Landau Levels
( )2
0
0
1 ( )2 2
1 12 2
/// 2
B
c B
c
B
eH g V z
m
N g B E
D eB heB me m
μ
ε ω μ
ωμ
∗
∗
•+
= + +
⎛ ⎞= + ± +⎜ ⎟⎝ ⎠
=
==
p A
Ν
σ Β
gμBB
ωc
N=0
N=1
N=2
B
In GaAs: m*/m0 = 0.067, g = -0.44
ωc = 20K @ B = 1TgμBB ≈ ωc /70
Wavefunctions
Landau gauge:
2 2
, , 0
/ 2
22
( ) ( )
( ) ( )
2
/
ikxN k N k
N N
k kx
e y y z
e H
y k yL
eB
σ σ
ξ
ψ φ ζ χ
φ ξ ξ
π
−
= −
=
= Δ =
=
Symmetric gauge: A = -½(r x B)
2 / 40, , 0 ( )
( ) /
zjj z e z
z x iyσ σψ ζ χ−=
= +
ˆ= −A xyB
Integer QHE and Edge States
( )2
2
2
. 2x k N
k R L Hocc k x
L dy e eI i Vy L
ehh
ε μ μπ
⎛ ⎞∂= = = − =⎜ ⎟∂⎝ ⎠∑ ∫
E
y
x
Disorder Determines Plateau Widths
EF
y
B
Rxy
E
y
x
Disorder Determines Plateau Widths
EF
y
B
Rxy
E
y
x
Disorder Determines Plateau Widths
EF
y
B
Rxy
E
y
x
Disorder Determines Plateau Widths
EF
y
B
Rxy
Laughlin’s Wavefunction
Single Electron in Lowest Landau Level
Symmetric gauge: A = -½(r x B)
2 / 40, , 0
2
( )
0
zjj z e z
eBj n Rh
σ σ
φ
ψ ζ χ
π
−=
≤ < =
Single Electron in Lowest Landau Level
2
1
1
1 10 4
( )n
jj
j
zz a z expφ
ψ−
=
−⎛ ⎞⎜ ⎟⎝ ⎠
∑∼
1 2( , , , ) ( )n i ji j
z z z z z<
Ψ −∏… ∼
Filled Lowest Landau Level
2
1 21 2
11 1 1
1 2
1 1 1
( , , , ) exp4
njn
nj
n n nn
zz z zz z z
z z z=
− − −
⎡ ⎤⎢ ⎥Ψ = × −⎢ ⎥⎣ ⎦∑…
A unique Slater determinant:
1nnφ
ν = =
1 2 2
1 2 1 3 1
1
1
( , , , ) ( , , )
(
( )
( ( ( )) ) )
n n
n
z z z z z
z z z z z
z
z z
ϕ χ
ϕ
Ψ =
= − − −
… …
Visualizing ν = 1
Uncorrelated ν < 1 State
n nφ<
P(z1) is an undetermined polynomial with nφ-n zeros
1 1 2 1 3 1 1( ) ( )( ) ( ) ( )nz z z z z z z P zϕ − − − ×∼
31 2( , , , ) ( )n i j
i jz z z z z
<
Ψ −∏… ∼
Laughlin correlated ν = 1/3 State
2 2 21 1 2 1 3 1( ) ( ) ( ) ( )nP z z z z z z z− − −∼
Excitations of the Laughlin Liquid
Add one more flux quantum:
31 2( , , , ) ( ) ( )n k i j
k i jz z z z w z z∗
<
Ψ − −∏ ∏… ∼
Φ0
w
A quasi-hole with charge q = +e/3
Excitations of the Laughlin Liquid
B < B1/3
B > B1/3
quasiholesq = + e/3
quasielectronsq = - e/3
nφ
Etot
ν = 1/3
q-eq-h A gap to charged excitations
2
0.1 eε
Δ ≈
Many Fractional Quantum Hall States
Laughlin states:ν = 1/m = 1/3, 1/5, ... ν = 1 - 1/m = 2/3, 4/5, ...
Whence 2/5, 3/7 etc.?
Hierarchy Model
5/135/17
2/52/7
3/73/11
1/3
Interactions amongst quasiparticles produce new condensates
Most hierarchy states are not observed
Composite Fermions
Chern-Simons singular gauge transformation:
Attach an even number of fictitious flux quanta to
each electron
B* = B - 2φ0n1 1 2CFν ν
= −
1, 2, 3,CF jν = = … 1 2 3, , ,2 1 3 5 7
jj
ν = =+
…
Jain, others
Half-filled Landau Level
CFν = ∞12
ν = B* = 0
Fermi sea of CFs
At ν = 1/2 quasiparticles move in straight lines.
11/ 2cR
ν −∼
02F Fk k=
Halperin, Lee, Read, others
Semi-classical transport of CFs
Dimensional resonances in an anti-dot lattice
Kang, et al. 1993
High Landau Levels
lowest Landau level
1987: Even-Denominator FQHE
5/2N=1
N=0
Willett, et al.
Transport Anisotropy in High Landau Levels
ν = 4 is a boundary between different transport regimes.
N = 0 & 1N = 2, 3, ...
Magnetic Field (Tesla)
Rxx
& R
yy(O
hms)
ν = 9/2
7/2
11/2
5/2
13/2
ν=4
T=25mK
<110>
<110>B1200
1000
800
600
400
200
0543210
Electronic Liquid Crystals
1000
800
600
400
200
0Lo
ngitu
dina
l Res
ista
nces
(Ω
)2001000
Temperature (mK)
A nematic to isotropic transition?
Double Layer Two-Dimensional Electron Gas
1086420Magnetic Field (Tesla)
2.5
2.0
1.5
1.0
0.5
0.0
Hal
l Res
ista
nce
(h/
e2 )
150
100
50
0
Diagonal R
esistance (kΩ)
x10
QHE in Double Layer 2D Systems
νT = 1
1/2
νT = 1 = ½ + ½
νT = ½ = ¼ + ¼
A BCS-like superfluid comprised of interlayer excitons.
Add a magnetic field
Start with a double layer 2D electron gas
+_
+_
+_
+_
+_
+_
Quantum Hall Superfluid