Lecture 19-20

Quantum Hall effect in 2D electron systems and its interpretation in terms of edge states of Landau levels

zLorentz lvHeF

0

Two-dimensional electrons in GaAs/AlGaAs heterostructure,

or in a Si/SiO2 field-effect transistor

eH e

HR

0

weak magnetic

fieldregime

Landau levels for electrons in a magnetic field

mev6.0 mev8.1

Edge states of Landau levels and edge currents

eBqxosc /

)(

)(

qd

qdEvy

an ideal ‘ballistic quantum wire‘

2

filling factor

(both spin states are

filled int he lowest

Landau level)

‘ballistic quantum wire‘due to edge states: edge current

222

spindegeneracy

2 2 2

2 2

Quantum resistance standard

(1 Klitzing)

2

2

2

2

Integer quantum Hall effect

cnce

nEm

AiH

)(2

)(21

2

density of

states

h

geofunitsin

2

2 31

3

1

2

eeB

gh

xx

Hxy R

1

degeneracy factor, g

(g=2 for spin)

E

)2(FE

localised states in the 2D bulk,

current carried only by edge states

)1(FE

eeB

gh

current carried by states in the bulk, from one edge to

the other

)1980(2

1

e

h

nRH

n = number of filled spin-polarised Landau levels

filling factor

xpyp

eB

cr

nv

n

cB

Bn

)0(

...3,2,1,02

‘relativistic-type’ Landau level spectrum

vpcond

sec/10~ 8cmv

vpvalence

-2-4 40

ne (1012 cm-2)

2

2

4

6

0

xx

(k

) x

y (4e2

/h)

1

2

-1

-2

-4

0

-3

4

3

magnetic length

eB

hcnn e

LL

e

Graphene synthesised on SiCflake growth and lies as a carpet over SiC substrate.

Seyller (Erlangen), Yakimova (Linkoping)

Development of QHE resistance standard using SiC-synthesised

graphene. Currently: 9 digit accuracy

(Chalmers, NPL-UK, Lancaster)

Skipping orbits and electron ‘focusing‘

max2 1323 ILRN c

NeL

pB FN

13

2

F

N

F

c

c mv

eB

vRL

N

12

23

13I

B

cc

vR

B. van Wees 1989

12V 13I

Caustics in the electron skipping motion

C. Beenakker, 1992(theory)

B. van Wees (exp)