![Page 1: Lecture 19-20 Quantum Hall effect in 2D electron systems and its interpretation in terms of edge states of Landau levels Two-dimensional electrons in GaAs/AlGaAs](https://reader036.vdocument.in/reader036/viewer/2022062409/56649ef45503460f94c073f8/html5/thumbnails/1.jpg)
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
![Page 2: Lecture 19-20 Quantum Hall effect in 2D electron systems and its interpretation in terms of edge states of Landau levels Two-dimensional electrons in GaAs/AlGaAs](https://reader036.vdocument.in/reader036/viewer/2022062409/56649ef45503460f94c073f8/html5/thumbnails/2.jpg)
eH e
HR
0
weak magnetic
fieldregime
![Page 3: Lecture 19-20 Quantum Hall effect in 2D electron systems and its interpretation in terms of edge states of Landau levels Two-dimensional electrons in GaAs/AlGaAs](https://reader036.vdocument.in/reader036/viewer/2022062409/56649ef45503460f94c073f8/html5/thumbnails/3.jpg)
Landau levels for electrons in a magnetic field
mev6.0 mev8.1
![Page 4: Lecture 19-20 Quantum Hall effect in 2D electron systems and its interpretation in terms of edge states of Landau levels Two-dimensional electrons in GaAs/AlGaAs](https://reader036.vdocument.in/reader036/viewer/2022062409/56649ef45503460f94c073f8/html5/thumbnails/4.jpg)
Edge states of Landau levels and edge currents
eBqxosc /
)(
)(
qd
qdEvy
![Page 5: Lecture 19-20 Quantum Hall effect in 2D electron systems and its interpretation in terms of edge states of Landau levels Two-dimensional electrons in GaAs/AlGaAs](https://reader036.vdocument.in/reader036/viewer/2022062409/56649ef45503460f94c073f8/html5/thumbnails/5.jpg)
an ideal ‘ballistic quantum wire‘
2
filling factor
(both spin states are
filled int he lowest
Landau level)
![Page 6: Lecture 19-20 Quantum Hall effect in 2D electron systems and its interpretation in terms of edge states of Landau levels Two-dimensional electrons in GaAs/AlGaAs](https://reader036.vdocument.in/reader036/viewer/2022062409/56649ef45503460f94c073f8/html5/thumbnails/6.jpg)
![Page 7: Lecture 19-20 Quantum Hall effect in 2D electron systems and its interpretation in terms of edge states of Landau levels Two-dimensional electrons in GaAs/AlGaAs](https://reader036.vdocument.in/reader036/viewer/2022062409/56649ef45503460f94c073f8/html5/thumbnails/7.jpg)
‘ballistic quantum wire‘due to edge states: edge current
222
spindegeneracy
2 2 2
2 2
![Page 8: Lecture 19-20 Quantum Hall effect in 2D electron systems and its interpretation in terms of edge states of Landau levels Two-dimensional electrons in GaAs/AlGaAs](https://reader036.vdocument.in/reader036/viewer/2022062409/56649ef45503460f94c073f8/html5/thumbnails/8.jpg)
Quantum resistance standard
(1 Klitzing)
2
2
2
2
![Page 9: Lecture 19-20 Quantum Hall effect in 2D electron systems and its interpretation in terms of edge states of Landau levels Two-dimensional electrons in GaAs/AlGaAs](https://reader036.vdocument.in/reader036/viewer/2022062409/56649ef45503460f94c073f8/html5/thumbnails/9.jpg)
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
![Page 10: Lecture 19-20 Quantum Hall effect in 2D electron systems and its interpretation in terms of edge states of Landau levels Two-dimensional electrons in GaAs/AlGaAs](https://reader036.vdocument.in/reader036/viewer/2022062409/56649ef45503460f94c073f8/html5/thumbnails/10.jpg)
)1980(2
1
e
h
nRH
n = number of filled spin-polarised Landau levels
![Page 11: Lecture 19-20 Quantum Hall effect in 2D electron systems and its interpretation in terms of edge states of Landau levels Two-dimensional electrons in GaAs/AlGaAs](https://reader036.vdocument.in/reader036/viewer/2022062409/56649ef45503460f94c073f8/html5/thumbnails/11.jpg)
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
![Page 12: Lecture 19-20 Quantum Hall effect in 2D electron systems and its interpretation in terms of edge states of Landau levels Two-dimensional electrons in GaAs/AlGaAs](https://reader036.vdocument.in/reader036/viewer/2022062409/56649ef45503460f94c073f8/html5/thumbnails/12.jpg)
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)
![Page 13: Lecture 19-20 Quantum Hall effect in 2D electron systems and its interpretation in terms of edge states of Landau levels Two-dimensional electrons in GaAs/AlGaAs](https://reader036.vdocument.in/reader036/viewer/2022062409/56649ef45503460f94c073f8/html5/thumbnails/13.jpg)
![Page 14: Lecture 19-20 Quantum Hall effect in 2D electron systems and its interpretation in terms of edge states of Landau levels Two-dimensional electrons in GaAs/AlGaAs](https://reader036.vdocument.in/reader036/viewer/2022062409/56649ef45503460f94c073f8/html5/thumbnails/14.jpg)
![Page 15: Lecture 19-20 Quantum Hall effect in 2D electron systems and its interpretation in terms of edge states of Landau levels Two-dimensional electrons in GaAs/AlGaAs](https://reader036.vdocument.in/reader036/viewer/2022062409/56649ef45503460f94c073f8/html5/thumbnails/15.jpg)
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
![Page 16: Lecture 19-20 Quantum Hall effect in 2D electron systems and its interpretation in terms of edge states of Landau levels Two-dimensional electrons in GaAs/AlGaAs](https://reader036.vdocument.in/reader036/viewer/2022062409/56649ef45503460f94c073f8/html5/thumbnails/16.jpg)
Caustics in the electron skipping motion
C. Beenakker, 1992(theory)
B. van Wees (exp)