the world of 2d electrons is exciting:

• enabling ‘ballistic high mobility transport’ modulation doping

• easy electrostatic control close to surface

• unique excitations (quasiparticles)

exchange statistics in the

2d world

richer than the 3d world

exchange statistics in 3d

fermionsbosons

both

fermionsbosons

1

1BE

kT

fe

1

1FD

kT

fe

consequences…

exchange statistics in 2d abelian (Laughlin qp’s)

anyons

ie 2ie

2 4i ie e

2 4 4 2i i i ie e e e

2ie

4ie

exchange statistics in 2d non-abelian

degenerate ground

statei i

i

a a

a a

U

exchange unitary

1 2 UU

1 2 2 1UU U U

non-abelian anyons1U

2U

• basics of edge channels in IQHE

• doing physics with integer edge channels

• studies of edge transport in FQHE regime

deviations from the ‘accepted’ picture

Moty Heiblum

Edge States in QHE Regime …. their nature & use

my 2d world

high mobility 2DEG

GaAs-AlGaAs

B

VH Vxx

I

L

WX

sxy en

BR

Lorentzelectric field

B

Rxx

Rxy

classical Hall effect

xx = xx =0

quantizing the Hall effect

choice of gauges for

H

A

m=0

harmonic oscillatorEn

n n

2

Bl eBdegeneracy=B /0

0

useful in interference experiments

m = 1, 2, 3

C

eB

mBn m r lL , , ,

useful in understanding edge modes

do not carry energy / current

harmonic oscillator

most convenient (another) gauge

resembling classical orbits

= number of filled LL = number of electrons per flux quantum

0=h/e

EFc

energy

*cm

eB

Ne

e

EF

Ne

=2

)( 21 nE cn

2

Bl eB

summary

what started it… in a Si MOSFET

ns

Rxx= 0

RH = ( e 2/h )-1

B = const.

= number of electrons per flux quantum

RH=(e 2/3h)-1

continued with… in a GaAs MODFET

= number of electrons per flux quantum

plateaus in xy and minima in xx energy gap

gauge invariance E (0) = E (0)

xy= e 2/h

xx=0

Magnetic Field, B (Tesla)

/1m

eB*c EF

Ne c

localizeddoes not contribute to current

delocalizedcontributes to current

e

/h weak disorder

disordered 2DEG bulk picture of QHE

plateau

gauge invariance, =1/3…

1

2o

r t

trrJπtrIq Δ)(2=Δ)(= =e / 3

φφr εσrJ =)(

h

er

2

3

1

ohmic contact

J (r )

0

r

state of the art QHE states

few basics of

IQHE edge channels

e

e

B

near the edge skipping orbits

2d layer

approaching the edge …

approaching the edge energy curves up

finite drift velocity

vd =/B

truncated harmonic oscillator

simplistic view

B >>0

B=0

incompressible

compressible

density increasing reachingmaximum density of LL

EF

electrons move from LL n+1 to ngaining cyclotron (Zeeman) energy

charge imbalance charging energy

competing with cyclotron energy

LL

proposed edge

structure

• compressible strips separated by incompressible strips

• where does current flow?

• in the insulating strips?• in the conducting strips?

arguments continue until today…

• local measurements of current distributions still lacking

edge channels immune to back

scattering

1d edge channel carries Vh

eI

2

Ef +eV

Ef

Ef

hot spot

He4 bubble

He4 bubble

ballistic, but with energy

dissipation

is the simplistic view of non-interacting1d edge channels correct ?

inter-channel interaction

intra-channel interaction

inter-channel…

=2

gQ= 2 e2/h

e2/he2/h

e/2

e/2

e/2

-e/2

neutral charge

slow ‘neutral’ mode fast ‘charge’ mode

LL1

LL2

non interacting Landau levels

e injecting electrons >0

=0

with interactions new basis

Berg et al., PRL (2009)

electronic beam splitterQuantum Point Contact (QPC)

Vgate

Vsource

r

lF

t

QPC0 < t < 1

preferential backscattering of edge channels

reflected higher LLs

transmitted lower LLs

partitioned LL

Vgate

how to test ? injecting and detecting

LL1

LL2

e

inducing noise in LL1 (hot mode) via QPC

e/2

e/2

e/2

-e/2

neutral charge

projecting via QPC to LL2

(cold mode)

v

v

injecting electrons into LL1 looking for fluctuations in LL2

with no net current

current fluctuations

shot noise

hotfilament

cathode anode+-

emitted electrons

noisy current in vacuum tubes

classical shot noise

Schottky, 1918

it started with - noise in vacuum tubes

classical shot noise

large number of impinging electrons

very small escape probability

time

IDC

0

current

time

IDC

Si (0)=2 e I spectral density (A2/Hz)

Vapplied

~h/eVapplied

zero temperature ordered electrons are noiseless !

shot noise =0 …. full Fermi sea (non-partitioned electrons)

Khlus, 1987Lesovik, 1989

shot noise - single channel

t <<1poissonianS =2eISchottky formula

incoming transmitted

binomialS =2eI (1-t )

t

Khlus, 1987Lesovik, 1989

spectral density of current fluctuations*)i(

i eI)(S 2

(A2/Hz)

i

ii00Ti )t1(tgVe2)0(S

conductance and shot noise in QPC

con

du

ctan

ce (

g0)

gate voltage, Vg

3

2

1

noise

current responsiblefor noise = V g0 ti

It1=1

t2=1

2DEG

QPC

Vg

S t (1-t )

shot noise in QPC- experimental results -

0

2

4

6

0 1 2 3

curr

ent

nois

e,

S (1

0 -2

8 A2 /H

z)

current, I (nA)

T =57 mKt =0.37

I

eV

Tk2

Tk2

eVcoth)t1(eI2Tgk4)0(S B

BBi

experimental considerations

2DEG : ns=1.1x1011 cm-2 ; m=4 x106 cm2/Vs

04=)0( TGkS Bi

shot noise signal Si(0)=2e*I=10 -29 A2/Hz ...…………… T*~ 40 mK

Johnson noise……………………………………………. T ~ (10-30) mK

noise in ‘warm electronics’……….....……………………T*~ 3.5 K

“home made” (MODFET) cryogenic preamplifier (T= 4.2 K)

T*~100-200 mK at f0 = 1 - 4 MHz (above 1/f noise knee)

difficulties in measurements

QPC resistance R ~ 100 k W

coax capacitance C ~ 60 pF

fmax = 1/(2 p RC) ~ 30 kHz

\ 1/f noise is large

coaxial cable

60cm 60pF

QPC( d i)2

dV=R di

cooledpreamp

(hot, 4.2 K)(cold, 50 mK)

spectrumanalyzer

R

experimental setup

* frequency above 1/f noise corner of preamplifier;* capacitance compensated by resonant circuit;

QPC

cooledpreamp

L RC

calibration signal

spectrumanalyzer

f0 ,f0

C<<

50 ; 300 K

1 G

warmpreamp

voltage gain = 1000

coax

averaging time,

noise <i2<DC current

VDC

cryostat

‘home made’

kHzRCπ

f,MHzLCπ

f 30≈2

1=Δ4→2≈

2

1= 00

injecting and detecting

LL1

LL2

e

inducing noise in LL1 (hot mode) via QPC

e/2

e/2

e/2

-e/2

neutral charge

projecting via QPC to LL2

(‘cold ‘ channel )

open QPC …………………………... no net current no low frequency noise

partition QPC ………………………no net current low frequency noise

semi-classically:• mutual capacitance between the two edge modes

• high frequency (~GHz) noise in hot LL1 is induced n LL2

• stochastic partitioning by QPC adds low-frequency spectrum

partitioning also the ‘cold channel’

G

G

G G

T

Inoue et al, PRL (2014)

T2 dependence

current of ‘cold mode’

T1=0.5

no net current

T1 dependence~T2(1-T2)

intra-channel…

EfEfEf DDD 21 1

cold channel

hot channel

VD1

warm

cha

nnel

VD2

QD

QD – narrow BPF

EfEfEf DDD 21 1

energy distribution (QD)

=1

=0

VμVV DD 36=21

-

5.0

36

VVD

energy equilibration with distance

equilibration due to e - e interactions

simplistic view of integer edge channels fails

interactions are dominant

we make use of it…