Quantum Hall physics: Introduction and current affairs Ulrich - - PowerPoint PPT Presentation

Quantum Hall physics: Introduction and current affairs

Ulrich Zuelicke

u.zuelicke@massey.ac.nz

Institute of Fundamental Sciences Massey University Palmerston North, New Zealand

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.1

Outline

Introduction to basics of the quantum Hall effect sample geometry, measurement technique incompressibility

• quantized Hall resistance

microsocpic origin of incompressibility role of disorder Overview of exotic interaction effects quasiparticles with fractional charge and statistics; composite fermions; chiral Luttinger liquids; quantum Hall ferromagnets fractional quantum Hall effect in rotating atomic gases

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.2

Introduction to the basics of the quantum Hall effect

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.3

Two–dimensional electron systems

Quantum Hall (QH) effect observed in 2D electron systems placed in a perpendicular magnetic field

B

2DES with density n filling factor = n/B ν φ

important parame- ter: filling factor

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.4

Two–dimensional electron systems

Quantum Hall (QH) effect observed in 2D electron systems placed in a perpendicular magnetic field

illustration taken from: Jeckelmann and Jeanneret ’01

typically realized in semiconductor heterostructures using band–gap engineering: (Ga,Al)As, (In,Al)As,

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.4

Two–dimensional electron systems

Quantum Hall (QH) effect observed in 2D electron systems placed in a perpendicular magnetic field

illustration taken from: Jeckelmann and Jeanneret ’01

typically realized in semiconductor heterostructures using band–gap engineering: (Ga,Al)As, (In,Al)As,

• riginal discovery of QH effect in silicon MOSFETs

Klitzing, Dorda, Pepper ’80

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.4

Setup for transport measurement

mesoscopic current and voltage probes attached to sample, measure resistances

and

I voltage probes source drain V

H

V

L

Büttiker ’86 source: PTB webpage

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.5

Setup for transport measurement

mesoscopic current and voltage probes attached to sample, measure resistances

and

I voltage probes source drain V

H

V

L

Büttiker ’86 Willett et al. ’87

• bserve plateaux in

where, with great precision (error

),

, concomitant with

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.5

Classical magnetotransport theory

electrodynamics in 2D:

simple Drude theory yields (with

):

ρ

H

ρ B

L

• quantum Hall system not a classical conductor

at plateau: both longitudinal resistivity and conductivity vanish (possible because

finite)

• quantum Hall system: perfect conductor or insulator?

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.6

Quantized

: Thermodynamic argument

reasonable hypothesis: quantum Hall system is a bulk insulator, i.e., is incompressible in the bulk:

*

µ n n

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.7

Quantized

: Thermodynamic argument

reasonable hypothesis: quantum Hall system is a bulk insulator, i.e., is incompressible in the bulk:

*

µ n n

µ + δµ µ + δµ I δ I + I

incompressibility at magnetic–field–dependent density

implies quantization of Hall resistance!

Widom ’82

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.7

Quantized

: Thermodynamic argument

reasonable hypothesis: quantum Hall system is a bulk insulator, i.e., is incompressible in the bulk:

*

µ n n

µ + δµ µ + δµ I δ I + I

incompressibility at magnetic–field–dependent density

implies quantization of Hall resistance!

Widom ’82

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.7

Quantized

: Thermodynamic argument

reasonable hypothesis: quantum Hall system is a bulk insulator, i.e., is incompressible in the bulk:

*

µ n n

I δ I δ µ µ µ + I δ I + µ µ + δµ I δ

incompressibility at magnetic–field–dependent density

implies quantization of Hall resistance!

Widom ’82

• ✄✟❥

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.7

Take–home message # 1: Quantized Hall resistance due to incompressibility (mobility gap).

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.8

Incompressibility at integer

Schrödinger eq. for 2D electrons in a perpendicular magnetic field reads

with

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.9

Incompressibility at integer

Schrödinger eq. for 2D electrons in a perpendicular magnetic field reads

with

find representation of

and angular momentum

in terms of independent harmonic–oscillator ladder operators

and

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.9

Incompressibility at integer

Schrödinger eq. for 2D electrons in a perpendicular magnetic field reads

with

M

r

find representation of

and angular momentum

in terms of independent harmonic–oscillator ladder operators

and

eigenstates of

labelled by Landau–level index

and

quantum # ; localized at

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.9

Incompressibility at integer

finite sample: external potential

confines electrons; Hamiltonain given by

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.10

Incompressibility at integer

finite sample: external potential

confines electrons; Hamiltonain given by

• scillator (Landau) levels degenerate in bulk but

bent upwards at sample boundary

M

2 1 3 4 5 6 7 8 . . .

lowest Landau level second Landau level chemical potential

quantum number M

Fermi point chiral 1D

single−particle energy E

• bulk gap at integer

• at the same time: chiral

1D edge currents flowing

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.10

Incompressibility at integer

finite sample: external potential

confines electrons; Hamiltonain given by

• scillator (Landau) levels degenerate in bulk but

bent upwards at sample boundary

M

2 1 3 4 5 6 7 8 . . .

lowest Landau level second Landau level chemical potential

quantum number M

Fermi point chiral 1D

single−particle energy E

• bulk gap at integer

• at the same time: chiral

1D edge currents flowing

Hall conductance:

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.10

Rôle of disorder

• bserve quantum–Hall effect in large (mm–sized)

samples with plateaux extending over

Tesla expect, however,

typically resolution of puzzle: samples are disordered!

I δ I δ µ µ µ + δµ µ + δµ I δ I + I

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.11

Take–home message # 2: Finite width of Hall plateaux in large samples due to disorder.

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.12

Incompressibility at fractional

Tsui, Gossard, Störmer ’82

fractional filling factor

• ccupy only states in lowest

Landau level (LLL), these are labelled by and (except close to edge) have the same (kinetic) energy

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.13

Incompressibility at fractional

Tsui, Gossard, Störmer ’82

fractional filling factor

• ccupy only states in lowest

Landau level (LLL), these are labelled by and (except close to edge) have the same (kinetic) energy wave function:

with

, general LLL wave function:

where

polynomial with, at most,

zeroes

M

r

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.13

Incompressibility at fractional

two–electron wave functions classified by relative and center–of–mass angular momenta

and are uniquely determined by analyticity requirement:

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.14

Incompressibility at fractional

two–electron wave functions classified by relative and center–of–mass angular momenta

and are uniquely determined by analyticity requirement:

are eigenstates for arbitrary interaction

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.14

Incompressibility at fractional

two–electron wave functions classified by relative and center–of–mass angular momenta

and are uniquely determined by analyticity requirement:

are eigenstates for arbitrary interaction

many-body Hamiltonian for electrons in LLL:

• has discrete matrix elements in mM representation!

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.14

Incompressibility at fractional

• electron wave functions characterized by their zeroes!

consider finite sample with

electrons,

flux quanta

• many–body wave function

for electrons changes sign upon particle exchange

Laughlin ’83

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.15

Incompressibility at fractional

• electron wave functions characterized by their zeroes!

consider finite sample with

electrons,

flux quanta

• must

be polynomial in each particle coordinate with

zeroes: sample size!

Laughlin ’83

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.15

Incompressibility at fractional

• electron wave functions characterized by their zeroes!

consider finite sample with

electrons,

flux quanta

• at least one zero at posi-

tion of every other electron, due to antisymmetry

Laughlin ’83

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.15

Incompressibility at fractional

• electron wave functions characterized by their zeroes!

consider finite sample with

electrons,

flux quanta

• position of remaining ze-

roes arbitrary in the absence

• f interactions

Laughlin ’83

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.15

Incompressibility at fractional

• electron wave functions characterized by their zeroes!

consider finite sample with

electrons,

flux quanta

• ✁

: to minimize inter- action energy, nucleate more zeroes at other electrons

Laughlin ’83

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.15

Incompressibility at fractional

• electron wave functions characterized by their zeroes!

consider finite sample with

electrons,

flux quanta

• not enough zeroes when

!!

• rigin of incom-

pressibility at

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.15

Incompressibility at fractional

• electron wave functions characterized by their zeroes!

consider finite sample with

electrons,

flux quanta

• stray/missing zeroes are

quasiholes/electrons w/ frac- tional charge and statistics

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.15

Composite bosons / fermions

Source: Website of Bell Labs

• zeroes of many–body wave

function can be understood as flux vortices (phase winding!)

• singular gauge transformation

eliminates winding phase, intro- duces fictitious magnetic field many–electron wave function for

:

many–boson wave function in zero magnetic field

Zhang, Hansson, Kivelson ’89

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.16

Composite bosons / fermions

Source: Website of Bell Labs

• zeroes of many–body wave

function can be understood as flux vortices (phase winding!)

• singular gauge transformation

eliminates winding phase, intro- duces fictitious magnetic field many–electron wave function for

:

many–fermion wave function at

Jain ’89

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.16

Composite bosons / fermions

Source: Website of Bell Labs

• zeroes of many–body wave

function can be understood as flux vortices (phase winding!)

• singular gauge transformation

eliminates winding phase, intro- duces fictitious magnetic field many–electron wave function for

:

many–fermion wave function at

Jain ’89

more generally:

integer

• quantum Hall effect at

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.16

Composite bosons / fermions

more generally:

integer

• quantum Hall effect at

: composite–fermion liquid

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.16

Overview of nontrivial interaction physics in the quantum Hall regime

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.17

Electron fractionalization

• Laughlin state has all

electrons saturated with three flux tubes

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.18

Electron fractionalization

• make hole excitation,

i.e., physically remove an electron: leaves 3 zeroes

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.18

Electron fractionalization

• three

independent fractionally charged quasi- holes generated!

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.18

Electron fractionalization

• experimental verification: shot–noise measurements

Saminadayar et al. ’97; Reznikov et al. ’97

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.18

Quantum–Hall edge excitations

integer

: energy gap due to Landau quantization E k z y x edge potential

• low–lying 1D edge excitations

k E µ x y z W kF

F

−k

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.19

Chiral Luttinger liquids

At fractional filling factors, e.g.,

: energy gap due to interactions

Laughlin ’83

branches of chiral edge excitations predicted that form chiral Luttinger liquids

Wen ’91

tunneling from 3D contact into edge: expect

• bserve

n

• vergrowth

+

B

cleaved edge barrier quantum well 2DEG Grayson et al. ’98 Grayson et al. ’01 Chang et al. 01 Hilke et al. 01

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.20

New tunneling geometry: 2D contact

V I p

z y x y

B

conserved in tunneling!

(Huber et al. ’02)

eV k=X/l E k=X/l E

Landau–level spectroscopy chiral Luttinger liquids at

UZ, Shimshoni, Governale ’02

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.21

Quantum Hall ferromagnets

for 2D electrons in GaAs heterostructures in a high (

Tesla) magnetic field, Zeeman splitting is small cyclotron energy:

K

• Tesla

200 K Coulomb exchange:

K

• Tesla

150 K Zeeman splitting:

• Tesla

5 K

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.22

Quantum Hall ferromagnets

for 2D electrons in GaAs heterostructures in a high (

Tesla) magnetic field, Zeeman splitting is small cyclotron energy:

K

• Tesla

200 K Coulomb exchange:

K

• Tesla

150 K Zeeman splitting:

• Tesla

5 K electron system at

is 100% spin polarized even in the absence of Zeeman splitting: ideal ferromagnet

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.22

Quantum Hall ferromagnets

low–lying excitation in typical ferromagnets: spin waves with wave vector

; created by

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.23

Quantum Hall ferromagnets

low–lying excitation in typical ferromagnets: spin waves with wave vector

; created by

generalization to the LLL quantum Hall ferromagnet: 1D wave vector

related to real–space distance

q E k

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.23

Quantum Hall ferromagnets

low–lying excitation in typical ferromagnets: spin waves with wave vector

; created by

generalization to the LLL quantum Hall ferromagnet: 1D wave vector

related to real–space distance

q E k

Barret et al. ’95

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.23

Skyrmions

low–lying charged excitations in quantum Hall ferro– magnets: skyrmion spin textures

Sondhi et al. ’93

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.24

Skyrmions

low–lying charged excitations in quantum Hall ferro– magnets: skyrmion spin textures

Sondhi et al. ’93

spin texture results in additional (topological, Berry–phase) vector potential

such that

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.24

Skyrmions

low–lying charged excitations in quantum Hall ferro– magnets: skyrmion spin textures

Sondhi et al. ’93

spin texture results in additional (topological, Berry–phase) vector potential

such that

skyrmion texture reduces effective magnetic flux by

• ✕✗✮

integer, hence quantum Hall gap occurs at electron number shifted by

• ✕✗✮

skyrmion carries electric charge

• ✕

• International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP

, University of Tokyo, August 13 – 21, 2003 – p.24

QHE in ultracold bose gases

interacting atoms confined in a 2D harmonic trap:

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.25

QHE in ultracold bose gases

interacting atoms confined in a 2D harmonic trap:

when trap is rotating with frequency

: additional term; for

: recover quantum-Hall Hamiltonian (

)

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.25

QHE in ultracold bose gases

interacting atoms confined in a 2D harmonic trap:

when trap is rotating with frequency

: additional term; for

: recover quantum-Hall Hamiltonian (

) bose system: incompressible at

(minimization

• f contact–interaction energy possible only for

)

analog of fractional quantum Hall effect

• Cooper, Wilkin ’99; Wilkin, Gunn ’00

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.25

Creation of Laughlin quasiparticles

intense Laser beam simulates disorder potential; create fractionally charged quasiparticle

Paredes et al. ’01

direct spatial control: something condensed–matter experimentalists do not have!

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.26

Measurement of fractional statistics

at intermediate Laser intensities: create superposition

• f initial state and state with one more quasiparticle

use this to directly measure fractional statistics of Laughlin quasiparticles

Paredes et al. ’01

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.27

Conclusions

Basics of the quantum Hall effect incompressibility

quantized Hall resistance integer effect: Landau quantization fractional effect: interactions finite plateau width due to disorder quenching of kinetic energy in the lowest Landau level gives rise to novel interaction effects: laboratory for multitude of correlated–electron states new avenues for the study of such effects in trapped cold atomic gases

International Summer School ”Quantum Transport in Mesoscopic Scale and Low Dimensions”, ISSP , University of Tokyo, August 13 – 21, 2003 – p.28