H. BuhmannH. Buhmann
Hartmut Buhmann
Physikalisches Institut, EP3
Universität Würzburg
Germany
H. BuhmannH. Buhmann
Outline
• Insulators and Topological Insulators
• HgTe quantum well structures
• Two-Dimensional TI
• Quantum Spin Hall Effect
• experimental observations:
• quantized conductivity
• non-locality
• spin polarization
• Three-Dimensional TI
• Quantum Hall Effect
H. Buhmann
C.L.Kane and E.J. Mele, Science 314, 1692 (2006)
C.L.Kane and E.J.Mele, PRL 95, 146802 (2005)
C.L.Kane and E.J.Mele, PRL 95, 226801 (2005)
A.Bernevig and S.-C. Zhang, PRL 96, 106802 (2006)
The Insulating States
Topologically Generalized
H. Buhmann
Covalent Insulator
Characterized by energy gap: absence of low energy electronic excitations
The vacuumAtomic Insulator
e.g. solid Ar
Dirac
VacuumEgap ~ 10 eV
e.g. intrinsic semiconductor
Egap ~ 1 eV
3p
4s
Silicon
Egap = 2 mec2
~ 106 eV
electron
positron ~ hole
The Usual Boring Insulating State
H. Buhmann
C.L.Kane and E.J. Mele, Science 314, 1692 (2006)
C.L.Kane and E.J.Mele, PRL 95, 146802 (2005)
C.L.Kane and E.J.Mele, PRL 95, 226801 (2005)
A.Bernevig and S.-C. Zhang, PRL 96, 106802 (2006)
The Insulating State – Topologically
Generalized
H. Buhmann
2D Cyclotron Motion, Landau Levels
gap cE E
Energy gap, but NOT an insulator
The Integer Quantum Hall State
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bulk
insulating
Quantized Hall conductivity :
y xy xJ E
2
xyh
ne
Integer accurate to 10-9
Edge States
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Perfect Conductor
The QHE state spatially separates
the right and left moving states
(spinless 1D liquid)
suppressed backscattering
1d dispersionmagnetic field
k
kF-kF
E
-
-
-
-
-
-
first example of a new
Topological Insulator
H. Buhmann
Topological Insulator
H. Buhmann
21( ) ( )
2 BZn d u u
i k kk k k
The distinction between a conventional insulator and the quantum Hall state
is a topological property of the manifold of occupied states
Insulator : n = 0
IQHE state : xy = n e2/h
Classified by Chern (or TKNN) integer topological invariant (Thouless et al, 1982)
( ) :H kBloch Hamiltonians
Brillouiw
n zone (torus) ith energy gap
u(k) = Bloch wavefunction
Topological Band Theory
H. Buhmann
Geometry and Topology
Gauss and Bonnet
S
gKdA )1(22
1
S: surface
K-1: product of the two radii of curvatures
dA: area element
g: number of handles
geometric
topological
Example: Mug to Torus
H. Buhmann
Vacuum
Dn = # Chiral Edge Modes
This approach can actually be generalized to a spinfull QHE at zero magnetic field:
the Quantum Spin Hall Effect
Edge States
The TKNN invariant can only change at a phase
transition where the energy gap goes to zero
Vacuum
n = 0 n = 3 n = 0
H. Buhmann
C.L.Kane and E.J. Mele, Science 314, 1692 (2006)
C.L.Kane and E.J.Mele, PRL 95, 146802 (2005)
C.L.Kane and E.J.Mele, PRL 95, 226801 (2005)
A.Bernevig and S.-C. Zhang, PRL 96, 106802 (2006)
The Insulating State – Topologically
Generalized
H. BuhmannH. Buhmann
A two-dimensional topological insulator
B
quantum Hall effect topological invariant
broken time reversal symmetry (TRS)
H. BuhmannH. Buhmann
A two-dimensional topological insulator
B
H. BuhmannH. Buhmann
A two-dimensional topological insulator
B
helical edge states
invariant under time reversal
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Protected Edge States
suppressed backscattering:
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Protected Edge States
spin rotation:
Dj = 2
Interference is destructive
suppressed backscattering:
H. Buhmann
x
x
Quantum Spin Hall Effect
Stability of Helical Edge States:
backscattering is only possible by time reversal symmetry breaking processes
(for example external magnetic fields)
sksk ,, 21,
x
x
x
x
or if more states are present
h
eGQSHE
22
H. Buhmann
Topological Insulator :
A New B=0 Phase
H. Buhmann
There are 2 classes of 2D time reversal invariant band structures
Z2 topological invariant: n = 0,1
n=0 : Conventional Insulator n=1 : Topological Insulator
Kramers degenerate at
time reversal
invariant momenta
k* = k* + G
k*=0 k*=/a k*=0 k*=/a
Edge States for 0<k</a
even number of bands
crossing Fermi energyodd number of bands
crossing Fermi energy
Topological Insulator :
A New B=0 Phase
H. Buhmann
C.L.Kane and E.J.Mele, PRL 95, 226801 (2005)
Graphene edge states
QSHE in Graphene
• Graphene – spin-orbit coupling strength is too weak gap only about 30 meV.
• not accessible in experiments
gap
H. Buhmann
QSHE in HgTe
Helical edge states
for inverted HgTe QW
B.A Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006)
k
E
H1
E1
0
H. BuhmannH. Buhmann
H. Buhmann
band structure
D.J. Chadi et al. PRB, 3058 (1972)
semiconductor
fundamental energy gap
meV 30086 EE
semi-metal
HgTe
-1.0 -0.5 0.0 0.5 1.0
k (0.01 )
-1500
-1000
-500
0
500
1000
E(m
eV) 8
6
7
Eg
D
H. Buhmann
CdTe and HgTe
H = H1 + HD + Hmv + Hso
CdTe HgTe
H. Buhmann
-1.0 -0.5 0.0 0.5 1.0
k (0.01 )
-1500
-1000
-500
0
500
1000
E(m
eV)
HgTe
8
6
7
-1.0 -0.5 0.0 0.5 1.0
k (0.01 )
Hg0.32Cd0.68Te
-1500
-1000
-500
0
500
1000
E(m
eV)
6
8
7
VBO
BarrierQW
VBO = 570 meV
HgTe-Quantum Wells
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Typ-III QW
band structure
HgCdTeHgTe
6
8
inverted
HgTe-Quantum Wells
HgCdTe
HH1E1
d
normal
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Bandstructure
HgCdTe
HgTe
6
8
HgCdTe
HgTe
6
8
Zero Gap for 6.3 nm QW structures
H. Buhmann
4 6 8 10 12 14
-80
-40
0
40
80 E2
H4
H3
H2
H1
E [
me
V]
d [nm]
E1
dc
0.0 0.1 0.2 0.3 0.4 0.5
-100
-50
0
50
100
in-plane direction
(kx,k
y)=(1,0)
(kx,k
y)=(1,1)
H2
H1
E1
E [
me
V]
k [nm-1]
Dirac Band Structure
H. Buhmann
Zero Gap
-1 0 1 2 3-2
-1
0
1
2
3
4
5
6
7
xy [
e²/
h]
VG - V
Dirac [V]
B = 5 T
0
1k
2k
3k
xx [
-0,5 0,0 0,5 1,0
0
2
4
6
8
10
12
14
B = 1 T
xx [
VG - V
Dirac [V]
xy [
e²/
h]
0
10k
20k
30k
40k
50k
60k
70k
80k
B. Büttner et al., Nature Physics 7, 418 (2011)
H. Buhmann
Kane model
effective model
Parameters:
M ~ 0 (-0.035 meV)
D = -682.2 meV nm²
G = -857.3 meV nm²
A = 372.9 meV nm
LLchart
E(VG) conversion
taken from
appropriate
Kane model
LL-Dispersion
B. Büttner et al., Nature Physics 7, 418 (2011)
H. Buhmann
Finite Mobility
4 8 12 16 20 240.2
0.4
0.6
0.8
1.0
1.2
1.4 measurement
theory
xx in
e2/h
T [K]
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.60
20k
40k
60k
80k
xx
VG-V
Dirac [V]
xx,min
= 0.36 e²/h
DVG
B. Büttner et al., Nature Physics 7, 418 (2011)
H. Buhmann
Bandstructure
HgCdTe
HgTe
6
8
HgCdTe
HgTe
6
8
B.A Bernevig et al., Science 314, 1757 (2006)
H. Buhmann
Effective Tight Binding Model
square lattice with 4 orbitals per site
, , , , ( ), , ( ),x y x ys s p ip p ip
nearest neighbor hopping integral
basis: 1 1 1 1, , ,E H E H
*
( ) 0( , )
0 ( )eff x y
h kH k k
h k
H. Buhmann
Effective Tight Binding Model
square lattice with 4 orbitals per site
, , , , ( ), , ( ),x y x ys s p ip p ip
nearest neighbor hopping integral
basis: 1 1 1 1, , ,E H E H
*
( ) 0( , )
0 ( )eff x y
h kH k k
h k
relativistic Dirac equation with tunable parameter M (band-gap)
at the -pointHgTe
2
2
)(sinsin
sinsin)()(
kkMkikA
kikAkkMkh
xx
xx
)(
)(0
kMikkA
ikkAkMk
xx
xx
QW inverted
QW normalfor
0
0
M
M
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Mass domain wall
states localized on the domain wall
which disperse along the x-direction
M
y
0
M>0
M<0
M>0
y
x
x
M>0M<0
inverted
band structure
helical edge states
k
E
H1
E1
0
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Quantum Spin Hall Effect
normal
insulator
bulk
bulk
insulating
entire sample
insulating
M > 0 M < 0
QSHE
H. BuhmannH. Buhmann
H. BuhmannH. Buhmann
x
x
h
eG
22
H. Buhmann
Experiment
k
e
k
e
H. Buhmann
Small Samples
L
W
(L x W) mm
2.0 x 1.0 mm
1.0 x 1.0 mm
1.0 x 0.5 mm
H. Buhmann
-1.0 -0.5 0.0 0.5 1.0 1.5 2.010
3
104
105
106
G = 2 e2/h
Rxx /
(VGate
- Vthr
) / V
QSHE Size Dependence
(1 x 0.5) mm2
(1 x 1) mm2(2 x 1) mm2
(1 x 1) mm2
non-inverted
König et al., Science 318, 766 (2007)
H. Buhmann
QSHE Temperature Dependence
-2.0 -1.6 -1.2 -0.8 -0.4 0.0-2.0
0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
T = 1.8 K
T = 2.5 K
T = 4.2 K
T = 6.4 K
T = 9.8 K
T = 19 K
T = 28 K
T = 48 K
R2
-8,
9-1
1(k
)
UG(V)
Q2367 Microhallbar 1 x 1 mm2
I(2-8), U(9-11); B=0 T; T= variabel;
H. Buhmann
QSHE Excitation Voltage Dependence
-2.0 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.20
2
4
6
8
10
12
14
16
18
1600mV
1400mV
1200mV
1000mV
800mV
600mV
400mV
200mV
R (
kO
hm
)
Ug (V)
H. Buhmann
-1.0 -0.5 0.0 0.5 1.0 1.5 2.00
2
4
6
8
10
12
14
16
18
20
G = 2 e2/h
Rxx / k
(VGate
- Vthr
) / V
Conductance Quantization
in 4-terminal geometry!?
V
I
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Small Samples
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Role of Ohmic Contacts
n-type
metallic
contact areas
and leads
areas
where scattering
takes place
spin relaxation
H. Buhmann
Multi-Terminal Probe
210001
121000
012100
001210
000121
100012
T
2
144
3 2
2
142
4 1
2
2
3
t
t
I eG
h
I eG
h
m m
m m
generally22
2
)1(
e
hnR t
3
exp4
2 t
t
R
R
h
eG t
2
exp,4 2
Landauer-Büttiker Formalism normal
conducting contacts
no QSHE
ij
jijiiiii TRMh
eI mm
2
H. Buhmann
QSHE in inverted HgTe-QWs
34
2 t
t
R
R-1.0 -0.5 0.0 0.5 1.0 1.5 2.0
0
5
10
15
20
25
30
35
40
G = 2/3 e2/h
G = 2 e2/h
Rxx (
k
(VGate- Vthr) (V)
I1
2 3
4
56
V
(1 x 0.5) mm2
(2 x 1) mm2
I1
2 3
4
56
V
H. Buhmann
Non-locality
1 mm2 mm
1 2 3
6 5 4
H. Buhmann
-5.5 -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00
2
4
6
8
10
12
14
16
18
20
22
0
2
4
6
8
10
12
14
@6 mK
Rn
l (k
)I (n
A)
UGate
[ V ]
Q2308-02
Non-locality
I: 1-4
Vnl:2-3
1 2 3
6 5 4
V
I1
2 3
4
56
V
A. Roth, HB et al.,
Science 325,295 (2009)
H. Buhmann
Non-locality
-1 0 1 2 3 40
2
4
6
8
10
12
R (
k
)
V* (V)
G=3 e2/h
I: 1-3
Vnl:5-4
1 2 3
6 5 4
V
4-terminal
LB: 8.6 k
A. Roth, HB et al.,
Science 325,295 (2009)
H. Buhmann
Non-locality
A. Roth, HB et al.,
Science 325,295 (2009)
G=4 e2/h
I: 1-4
Vnl:2-3
1
3
2
4
V
-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00
1
2
3
4
5
6
7
R14,2
3 / k
Vgate / V
LB: 6.4 k
H. Buhmann
-6,0 -5,5 -5,0 -4,5 -4,0 -3,5 -3,00
1
2
3
4
5
6 R
nl (
k
)
V (V)
LB: 4.3 k
Non-locality
1 2 3
6 5 4
V
A. Roth, HB et al.,
Science 325,295 (2009)
H. Buhmann
Back Scattering
potential fluctuations introduce areas of normal metallic (n- or p-) conductance
in which back scattering becomes possible
QSHE
The potential landscape is modified by gate (density) sweeps!
H. Buhmann
Back Scattering
area of raised or lowered chem. Pot.the probalility for back scatterring increases
strength
length additional ohmic contact
transition from a two to a three terminal device
H. Buhmann
Additional Scattering Potential
A. Roth, HB et al.,
Science 325,295 (2009)
Transition from
2 e2/h
scattering strength
3/2 e2/h
H. Buhmann
Additional Scattering Potential
A. Roth, HB et al.,
Science 325,295 (2009)
-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00
1
2
3
4
5
6
7
R14,2
3 / k
Vgate / V
H. Buhmann
-6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.00
1
2
3
4
5
6 R
nl (k
W)
LB: 4.3 k
one additional contact:
• between 2 and 3
3.7 k
Potential Fluctuations
1 2 3
6 5 4
V
1 2 3
6 5 4
V
x
A. Roth, HB et al.,
Science 325,295 (2009)
H. Buhmann
1 2 3
6 5 4
V
-5.5 -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00
5
10
15
20
Rnl /
k
Vg / V
LB: 12.9 k
different gate sweep direction
1 2 3
6 5 4
V x
LB: 22.1 k
Potential Fluctuations
• Hysteresis effects due to charging of trap states at the SC-insulator interface
J. Hinz, HB et al., Semicond. Sci. Technol. 21 (2006) 501–506
H. Buhmann
HgTe Quantum Well Structures
gate
insulator
cap layer
doping layer
barrier
barrierquantum well
doping layer
buffer
substrate
Au
100 nm Si N /SiO
3 4 2
25 nm HgCdTe x = 0.7
CdZnTe(001)
25 nm CdTe
10 nm HgCdTe x = 0.7
9 nm HgCdTe with I
10 nm HgCdTe x = 0.7
4 - 12 nm HgTe
10 nm HgCdTe x = 0.7
9 nm HgCdTe with I
10 nm HgCdTe x = 0.7
layer structure
meta
l insula
tor
QW
impurity statesdischarging charging
hysteresis effects:
J. Hinz et al., Semicond. Sci. Technol. 21 (2006) 501–506
H. BuhmannH. Buhmann
H. Buhmann
Spin Polarizer
H. Buhmann
Spin Polarizer
50%
50%1 : 4
H. Buhmann
Spin Polarizer
50%
100 %
100%
100%
H. Buhmann
Spin Injector
100 %100%
spin
injection
even with backscattering!
H. Buhmann
Spin Detection
100 %100%
spin
detector
Dm
H. BuhmannH. Buhmann
SHE-1
QSHE
Spin Hall Effect
as creator and analyzer of
spin polarized electrons
SHE
QSHE
H. Buhmann
Split Gate H-Bar
H. Buhmann
QSHE Spin-Detector
-4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00.00
0.02
0.04
0.06
0.08
0.10
-4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00.0
0.2
0.4
0.6
0.8
1.0
Vdetector-gate / V
5x
50x
-4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.50
1
2
3
4
5
R34
,12/ k
Vinjector-gate
/ V
n-type injector
p-type injector
QSHI-type detector
SHE
QSHE
QSHI-type detector
injector sweep
detector sweep
H. BuhmannH. Buhmann
0.0 0.1 0.2 0.3 0.4 0.5 0.6-75
-50
-25
0
25
50
75
0.0 0.1 0.2 0.3 0.4 0.5 0.6-75
-50
-25
0
25
50
75
Q2198
H2
E1
H1
EF
n =4.42·1011
cm-2
ndl=1.0·10
11 cm
-2
E /
me
V
k / nm-1
Dk
H2
H1
E1
EF
p =3.26·1011
cm-2
ndl=1.0·10
11 cm
-2
E /
me
Vk / nm
-1
n-regime p-regime
,16
xs y F F
eEj p p k
m
D
J.Sinova et al.,
Phys. Rev. Lett. 92, 126603 (2004)
Intrinsic Spin Hall Effekt
Nature Physics 6, 448 (2010)
H. Buhmann
QSHE Spin-Injector
-4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.50
1
2
3
4
5
R12
,34/ k
Vdetector-gate / V
5x
50x
n-type detector
p-type detector
QSHI-type injector
-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00.00
0.02
0.04
0.06
0.08
0.10
-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00.0
0.2
0.4
0.6
0.8
1.0
SHE-1
QSHE
QSHI-type detector
detector sweep
injector sweep
H. BuhmannH. Buhmann
V
-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00
20
40
60
80
100
Rn
on
loc/
Ugate
/ V
-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00
200
400
600
800
1000
Rn
on
loc/
Ugate
/ V
detector sweep
p-type
n-type
V
-4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00
200
400
600
800
1000
Rn
on
loc/
Ugate
/ V
-4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00
20
40
60
80
100
Rn
on
loc/
Ugate
/ V
H. Buhmann
Spin-Current Switch
I2 I1
H. Buhmann
Spin-Current Switch
I2 I1
H. Buhmann
Spin-Current Switch
I2 I1
QSHE
H. Buhmann
Spin-Current Switch
I2 I1
ballistic
injection
H. BuhmannH. Buhmann
9/10 11/12 1/2
6/58/7 4/3
B E
ACDF
I I
-3,0 -2,5 -2,0 -1,5 -1,0 -0,5 0,0
1,0
1,5
2,0
2,5
3,0
3,5
4,0
4,5
5,0
I [n
A]
U (Gate ACDF) [V]
QSHE
ballstic
H. BuhmannH. Buhmann
9/10 11/12 1/2
6/58/7 4/3
B E
ACDF
I I
-3,0 -2,5 -2,0 -1,5 -1,0 -0,5 0,0
1,0
1,5
2,0
2,5
3,0
3,5
4,0
4,5
5,0
I [n
A]
U (Gate ACDF) [V]
-3,0 -2,5 -2,0 -1,5 -1,0 -0,5 0,09
10
11
12
13
14
15
I [n
A]
U (Gate ACDF) [V]
H. BuhmannH. Buhmann
H. BuhmannH. Buhmann
H. BuhmannH. Buhmann
H. BuhmannH. Buhmann
¼ Graphene
H. BuhmannH. Buhmann
H. Buhmann
-0,3 -0,2 -0,1 0,0 0,1 0,2 0,3
-500
-400
-300
-200
-100
0
100
200
EF
E, m
eV
k, nm-1
Bulk HgTe
1670 1680 1690 1700 1710 1720 1730Qx*10000(rlu)
5910
5920
5930
5940
5950
5960
5970Qy*10000(rlu) Q2424-map_1.xrdml
1.6
2.9
5.2
9.3
16.6
29.7
53.3
95.4
171.0
306.3
548.7
982.9
1760.7
3154.0
5650.0
10121.3
18131.1
32479.6
58183.3
104228.3
186712.2
x-ray diffraction (115 reflex)
fully strained
70 nm HgTe
on CdTe substrate
band structure
- gap openning
- two Dirac cones on
different surfaces
H. BuhmannH. Buhmann
0 2 4 6 8 10 12 14 160
2000
4000
6000
8000
10000
12000
14000
16000
0
2000
4000
6000
8000
10000
12000
14000
Rxx
(SdH)
Rxx in O
hm
B in Tesla
n=4
n=3
Rxy
(Hall)
Rxy in O
hm
n=2
H. BuhmannH. Buhmann
2 4 6 8 10 12 14 160
2
4
6
8
10
12
14
xy [e
2/h
]
B [T]
0 2 4 6 8 10 12 14 160
2
4
6
8
10
12
14
16
0
2
4
6
8
10
12
14
Rxx [
k
]
B [T]
Rxy [
k
]
Brüne et al., Phys. Rev. Lett. 106, 126803 (2011)
H. BuhmannH. Buhmann
2 4 6 8 10 12 14 160
2000
4000
6000
8000
10000
Rxx in O
hm
B in Tesla
DO
S
Density of states for two independent surfaces with slightly different density
2-11
2
-211
1
cm 1085.4
cm 107.3
n
n
H. BuhmannH. Buhmann
Brüne et al., Phys. Rev. Lett. 106, 126803 (2011)
H. Buhmann
Summary
• the QSH effect which consists of
– an insulating bulk and
– two counter propagating spin polarized edge channels
(Kramers doublet)
• quantized edge channel transport
• the QSH effect can be used as an effective
– spin injector and
– spin detector
with 100 % spin polarization properties
• Strained HgTe bulk exhibits two-dimensional surface
states
H. Buhmann
Acknowledgements
Quantum Transport Group (Würzburg, H. Buhmann)
C. Brüne
C. Ames
P. Leubner
L. Maier
M. Mühlbauer
A. Friedel
C. Thienel
H. Thierschmann
R. Schaller
J. Mutterer
A. Astakhova
CX Liu
Ex-QT:C.R. Becker
T. Beringer
M. Lebrecht
J. Schneider
S. Wiedmann
N. Eikenberg
R. Rommel
F. Gerhard
B. Krefft
A. Roth
B. Büttner
R. Pfeuffer
Stanford University
S.-C. Zhang
X.L. Qi
J. Maciejko
T. L. Hughes
M. König
Texas A&M University
J. Sinova
Lehrstuhl für Experimentelle Physik 3: L.W. Molenkamp
Univ. Würzburg
Inst. f. Theoretische Physik
E.M. Hankiewicz
B. Trautzettel
Collaborations:
H. BuhmannH. Buhmann
Quantum Spin Hall Effekt
Science 318, 766 (2007)
The Quantum Spin Hall Effect:
Theory and Experiment
J. Phys. Soc. Jap.
Vol. 77, 31007 (2008)
Intrinsic Spin Hall Effekt
Nature Physics 6, 448 (2010)
Nonlocal edge state transport
in the quantum spin Hall state
Science 325, 294 (2009)
Quantum Hall Effect from the
Topological Surface States of
Strained Bulk HgTe
Phys. Rev. Lett. 106, 126803 (2011)