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