H. BuhmannH. Buhmann
Hartmut Buhmann
Physikalisches Institut EP3Physikalisches Institut, EP3Universität Würzburg
Germany
H. BuhmannQuantum Hall Effect
1 h
Nobel Prize K. von Klitzing 1985
Hall resistance 2
1eh
ienB
sxy ==ρ
ρxy
semiconductor 2DEG
ρxx
zero longitudinal resistance
H. BuhmannEdge Channels
magnetic field
-
g
-
-
- -
-
hi l d t tchiral edge states
H. BuhmannQuantum Spin Hall Effect
• The QSH state can be thought of as two copies of QH states one forB two copies of QH states, one for each spin component, each seeing the opposite magnetic field. (Bernevig and Zhang PRL 2006)
effB
(Bernevig and Zhang, PRL, 2006)
B effB
H. BuhmannQuantum Spin Hall Effect
• The QSH state can be thought of as two copies of QH states one fortwo copies of QH states, one for each spin component, each seeing the opposite magnetic field. (Bernevig and Zhang PRL 2006)(Bernevig and Zhang, PRL, 2006)
• The QSH state does not break the time reversal symmetry, and can exist without any external magnetic field.
−−+ sksk ,, ,ψψ
backscattering between Kramers’ doublets is forbidden
insulating bulk
H. BuhmannQuantum Spin Hall Effect
no magnetic fieldmagnetic field
- -- spin up
g
-
-
-
--
-
- - --spin down
chiral edge states helical edge states
H. BuhmannQSHE in Graphene
Graphene edge states
C.L.Kane and E.J.Mele, PRL 95, 226801 (2005)
• Graphene – spin-orbit coupling strength is too weak gap only about 10-3 meV. p p p g g g p y
• not accessible in experiments
H. BuhmannQSHE in HgTe
Helical edge statesfor inverted HgTe QW
E
H1
E1
kπ 0 π
B.A Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006)
H. BuhmannH. Buhmann
H. BuhmannHgTe Quantum Well Structures
Carrier densities: ns = 1x1011 ... 2x1012 cm-2layer structure
gate Au
Carrier mobilities: μ = 1x105 ... 1.5x106 cm2/Vs
gate
insulator
Au
100 nm Si N /SiO3 4 2
cap layer
barrierdoping layer
25 nm HgCdTe x = 0.7
10 nm HgCdTe x = 0 7 9 nm HgCdTe with I10 nm HgCdTe x = 0.7
doping layer
barrier
barrierquantum well
9 nm HgCdTe with I10 nm HgCdTe x = 0.74 - 12 nm HgTe10 nm HgCdTe x = 0.7
symmetric or asymmetricd i
buffer
substrate CdZnTe(001)
25 nm CdTe10 nm HgCdTe x = 0.7 doping
( )
H. Buhmannn to p Transitions
nmax = 1.35 x 1012 cm-2
100
150
H1
pmax = 3.2 x 1011 cm-20
50
E /
meV H2
150
-100
-50
E130000
T = 1.5 K
0,0 0,1 0,2 0,3 0,4 0,5-150
k / nm-1
10000
20000VGate
+5,0 V +4,5 V+4,0 V
Q21638 nm QW 0V
10000
0
+3,5 V +3,0 V +2,5 V +2,0 V+1,5 V
Rxy
/ Ω +5V
gate
insulator
cap layer
doping layer
-20000
-10000 +1,0 V +0,5 V 0 V -2,0 V-2V
doping layer
barrier
barrierquantum well
doping layer
buffer
0 2 4 6 8-30000
B / T
substrate
H. BuhmannHgTe
band structuresemi-metal or semiconductorsemi metal or semiconductor
1000
0
500
eV) 8
E
-500E(m
e
6
Eg
-1 0 -0 5 0 0 0 5 1 0-1500
-1000
7
fundamental energy gap
1.0 0.5 0.0 0.5 1.0k (0.01 )
D.J. Chadi et al. PRB, 3058 (1972)
fundamental energy gap
meV 30086 −≈− ΓΓ EE
H. BuhmannHgTe-Quantum Wells
BarrierQW
1000HgTe Hg0.32Cd0.68Te
1000
500
8
5006
-500
0
E(m
eV) 8
6 -500
0
E(m
eV)
8VBO
-1000
7
-1000
7
-1.0 -0.5 0.0 0.5 1.0k (0.01 )
-1500
7
-1.0 -0.5 0.0 0.5 1.0k (0.01 )
-1500
7
VBO = 570 meVVBO = 570 meV
H. BuhmannHgTe-Quantum Wells
Typ-III QW
d
HgTe
Γ6
HgCdTeHgCdTe
g
HH1E1
Γ8
band structure
invertednormal
H. BuhmannBand Gap Engineering
ΓH CdTHgTe
Γ6
Zero Gap for 6.3 nm QW structures
HgCdTeHgTe
Γ6
H1E1
HgCdTe
Γ8
E1H1
Γ8
Γ8
normal
inverted
direct band gaps between80 ... 0 and 0 ... - 30 meV
H. BuhmannQSHE in HgTe
Helical edge statesfor inverted HgTe QW
E
H1
E1
kπ 0 π
B.A Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006)
H. BuhmannMass domain wall
invertedband structure
EH1
states localized on the domain wall which disperse along the x-direction
E1
y
m>0m<0
k3π 0 π
m
m>0y
x
m 0m 0
m0
m<0
x
m>0
x
helical edge states
H. BuhmannSimplified Picture
normal
m > 0 m < 0
insulator QSHE
bulkbulk
bulkinsulating
entire sampleinsulating
H. BuhmannH. Buhmann
H. Buhmann
Experiment
ε ε
k k
H. BuhmannSmaller Samples
LL
W
(L x W) μm
2.0 x 1.0 μm1.0 x 1.0 μm1.0 x 0.5 μm
H. BuhmannQSHE Size Dependence
106 (1 x 1) μm2
non-inverted10
Ω
non inverted
105
2
Rxx
/ Ω
104G = 2 e2/h
(1 x 1) μm2(2 x 1) μm2
-1.0 -0.5 0.0 0.5 1.0 1.5 2.0103 (1 x 0.5) μm2
(VGate- Vthr) / V
H. BuhmannQSHE in inverted HgTe-QWs
20
in 4-terminal geometry!?
V
16
18
0 V
12
14 G = 2 e2/h
kΩ
I
8
10
Rxx
/ k
2
4
6
-1.0 -0.5 0.0 0.5 1.0 1.5 2.00
2
(V V ) / V(VGate- Vthr) / V
H. BuhmannQSHE in inverted HgTe-QWs
40
30
35G = 2/3 e2/h
I1
2 3
4
V
25
30
(kΩ
) 2 3V 56
15
20
G = 2 e2/h
Rxx
( I1 4
56
5
10(2 x 1) μm2
32 ≈t
RR-1.0 -0.5 0.0 0.5 1.0 1.5 2.0
0
(V V ) (V)
(1 x 0.5) μm2
4tR(VGate- Vthr) (V)
H. BuhmannMulti-Terminal Probe
Landauer-Büttiker Formalism normal conducting contacts
no QSHEno QSHE
( ) ⎤⎡∑e2
⎟⎞
⎜⎛− 100012
( ) ⎥⎦
⎤⎢⎣
⎡−−= ∑
≠ijjijiiiii TRM
heI μμ2
⎟⎟⎟⎟⎟⎞
⎜⎜⎜⎜⎜⎛
−−
=001210000121100012
T
214
43 2
2t
I eGhμ μ
⎧= =⎪ −⎪
⎨heG t
2
exp,4 2≈
⎟⎟⎟⎟⎟
⎠⎜⎜⎜⎜⎜
⎝ −−
−
210001121000012100
T 3 22
142
4 1
23t
I eGh
μ μ
μ μ
⎪⇒ ⎨⎪ = =⎪ −⎩
3exp4
2 ≈t
t
RR
⎠⎝ 210001
generally22 2)1(
ehnR t
+=
H. BuhmannNon-locality
Q2308: 250 | 90: 0.1 | 400 | 90 | 400 | 90: 0.1 | 1000 | ns= 3.1x1011, μ =143 000
6 8 9
1 μm2 μm
3 2 11
3
H. BuhmannNon-locality
LB 12 9 kΩ
gate sweep
@6 mK Q2308-02
LB: 12.9 kΩ
16
18
20
22
12
143
10
12
14
16
8
10
Rnl (k
Ω)I (
nA)
4
6
8
2
4
6
-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
0
UGate [ V ]
H. BuhmannNon-locality
LB 8 6 kΩ
@6 mK Q2308-02
LB: 8.6 kΩ
50
60
70
12
143
30
40
50
8
10
Rnl (kΩ
)I / n
A
10
20
30
2
4
6
-5.5 -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00 0
UGate [ V ]
H. BuhmannNon-locality
LB 4 3 kΩ
LB: 4.3 kΩ
5
63
3
4
nl (k
W)
1
2
Rn
-6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.00
H. BuhmannBack Scattering
potential fluctuations introduce areas of normal metallic (n- or p-) conductancepotential fluctuations introduce areas of normal metallic (n or p ) conductancein which back scattering becomes possible
QSHE
The potential landscape is modified by gate (density) sweeps!
H. BuhmannPotential Fluctuations
LB 4 3 kΩ
LB: 4.3 kΩ
5
63
3
4
nl (k
W)one additional contact:
• between 8 93.7 kΩ
1
2
R
x
-6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.00
3
H. BuhmannNon-locality
different gate sweep direction
@6 mK Strom: 6, 11Spannung: 8 9 Q2308-02
LB: 12,9 kΩ
18
20
22
20
22
24
26Spannung: 8, 9
10
12
14
16
12
14
16
18
Rnl (k
Ω)I (
nA)
4
6
8
4
6
8
10
-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
0
2
UGate [ V ]
H. BuhmannPotential Fluctuations
gate-voltage dependenceimpurity statesdischarging charging
ulat
or
QW
met
al insu
hysteresis effects:
J. Hinz et al., Semicond. Sci. Technol. 21 (2006) 501–506
H. BuhmannNon-locality
two additional contacts:• between 8 9 and 6 8
19.4 kΩ
different gate sweep direction
Q2308-02
LB: 12,9 kΩxx
16
18
20
22
20
22
24
26
10
12
14
16
12
14
16
18
Rnl (k
Ω)I (
nA)
one additional contact:• between 8 9
2
4
6
8
4
6
8
10between 8 9
22.1 kΩ
x
-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
0
2
UGate [ V ]
H. BuhmannH. Buhmann
H. BuhmannH. Buhmann
H. BuhmannSpin Polarizer
H. BuhmannSpin Polarizer
50%1 : 4
50%50%
1 01 : 0
H. BuhmannComing Soon
QSHE
SHEQSHE
SHE-1
H. Buhmann
Split Gate H-Bar
H. BuhmannSummary: QSH Effect
• the QSH effect is a new non-trivial state of matter which consists ofconsists of – an insulating bulk and– two counter propagating spin polarized edge channelstwo counter propagating spin polarized edge channels
(Kramers doublet)
• the QSH effect shows up in a quantized conductance – the quantized value is strongly dependent on the number of
ohmic contactsohmic contacts – or the quality of the sample (potential fluctuations)
• and is destroyed by time reversal symmetry breaking effects, like a magnetic field, g
H. BuhmannAcknowledgements
Quantum Transport Group (Würzburg)
C. BrüneE. Rupp
A. Roth N. EikenbergR. RommelH Thierschmann
A. AstakhovaM. Mühlbauer
H. Thierschmann
Lehrstuhl für Experimentelle Physik 3: L W MolenkampLehrstuhl für Experimentelle Physik 3: L.W. Molenkamp
Univ. WürzburgInst. f. Theoretische Physik
Collaborations:
Ex-QT:Stanford University
S.-C. ZhangX L Qi
yE.M. Hankiewicz
C.R. BeckerT. BeringerM. LebrechtJ. Schneider
X.L. QiT. L. HughesJ. MaciejkoM KönigT. Spitz
S. WiedmannM. König
H. BuhmannH. Buhmann
Quantum Spin Hall EffektScience 318, 766 (2007)
The Quantum Spin Hall Effect:Theory and ExperimentJ. Phys. Soc. Jap.Vol. 77, 31007 (2008)