H. BuhmannH. Buhmann
Hartmut Buhmann
Physikalisches Institut EP3Physikalisches Institut, EP3Universität Würzburg
Germany
H. BuhmannH. Buhmann
Outline
• Hall effects and topological insulatorsp g
• HgTe quantum well structures
• experimental observations• quantized conductivity• non-locality• spin polarization
H. BuhmannHall Effects
1879M.I. D’yakonov and
V.I. Perel’ (1971)
V = 0VH = 0
( )BvEeFrrrr
×+−=
E jρ syjxy y xE jρ =
x
sysH E
j−≡σ
(dissipationless spin current)
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. Buhmann2DEG in a Magnetic Field
magnetic field
-
g
-
-
- -
-
hi l d t tchiral edge states
H. BuhmannEdge Channels
magnetic field
-
-
-
-
-
-
1d edge channelsD.B. Chklovskii, B.I. Shklovskii, L.I. Glazman, Phys. Rev. B 46, 4026 (1992)
H. BuhmannQuantum Hall Effect
0=Δμ3=νEdge Channel Picture
0Δμ
0≠Δμ
0=ΔμI +-
0=Δμ
H. BuhmannQuantum Spin Hall Effect
two copies of QH statestwo copies of QH states, one for each spin component, each seeing the opposite magnetic field,
are united in one sample and form
helical edge states. This new state does not break the time reversal symmetry, and can exist without any
g
and can exist without any external magnetic field.
Quantum Spin Hall StateQ p
consisting of two counter propagating spin polarized edge channels.protected by time reversal symmetry (Kramer‘s pair),p y y y ( p ),and an insulating bulk
H. BuhmannQuantum Spin Hall Effect
Stability of Helical Edge States: 4 = 2 + 2
−−+ sksk ,, ,ψψbackscattering between Kramers’ doublets is forbidden
x
heGQSHE
22=
x
backscattering is only possible by time reversal symmetry breaking processesbackscattering is only possible by time reversal symmetry breaking processes (for example external magnetic fields)
or if more edge states exist
−−+ sksk ,, 21,ψψ x
x
xx
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
gap
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. BuhmannQuantum Well Structures
E
quantizedenergylevels in
di iEg2 E z-directionEg1
x quasi freeelectron gas
z (growth direction) y
gin the xy-plane
H. BuhmannMBE
Quantum Well Growthbyby
Molecular Beam Epitaxy
QW
H. BuhmannMBE
• Integrated in UHV transport system with 6 chambers• Riber 3200 MBE• Horizontal system• 8 cell ports• Maximum substrate dimension 2’’ wavers• Pumped by a kryo pump, three nitrogen coldshields and a Ti-Pumped by a kryo pump, three nitrogen coldshields and a Ti
sublimation pump
H. BuhmannHgTe Quantum Wells
MBE-Growth
6.0
5 5
Bandgap vs. lattice constant(at room temperature in zinc blende structure)
4 0
4.5
5.0
5.5
)
3.0
3.5
4.0
ap e
nerg
y (e
V)
1.5
2.0
2.5
Ban
dga
1.0
0.5
0.0
5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 6.1 6.2 6.3 6.4 6.5 6.76.6-0.5
lattice constant a [Å]0 © CT-CREW 1999
H. BuhmannHgTe Quantum Wells
free electron gas in the QW
by donor doping of the barriers
H. BuhmannH. Buhmann
H. BuhmannHgTe
band structuresemi-metal or semiconductorsemi metal or semiconductor
1000
0
500
eV) 8
E
-500E(m
e
6
Eg
Δ
-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. Buhmann8 band k.p calculation
finite gap
E (m
eV)
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) / VKönig et al., Science 318, 766 (2007)
H. BuhmannQSHE Temperature Dependence
Q2367 Microhallbar 1 x 1 μm2
I(2 8) U(9 11) B 0 T T i b l
16.0
18.0 T = 1.8 K T = 2.5 K
I(2-8), U(9-11); B=0 T; T= variabel;
12.0
14.0T = 4.2 K T = 6.4 K T = 9.8 K T = 19 K
6 0
8.0
10.0 T = 28 K T = 48 K
R2-
8, 9
-11(k
Ω)
2.0
4.0
6.0R
-2.0 -1.6 -1.2 -0.8 -0.4 0.0-2.0
0.0
UG(V)
H. BuhmannH. Buhmann
H. BuhmannConductance Quantization
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. 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. 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. 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
H. BuhmannNon-locality
1 2 3
1 μm2 μm
6 5 4
H. BuhmannNon-locality
I: 1-41 2 3
V2 3V
Vnl:2-3
6 5 4
I1 4
56
20
2214
@6 mK Q2308-02
12
14
16
18
8
10
12
Rnl nA
)
6
8
10
4
6
8 (kΩ
)I (n
-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
0
2
UGate [ V ]
A. Roth, HB et al.,Science 325,295 (2009)
H. BuhmannNon-locality
I: 1-31 2 3
I: 1 3Vnl:5-4
6 5 4
LB 8 6 kΩ
10
12VLB: 8.6 kΩ
6
8
G=3 e2/h
4-terminal
4
6
-1 0 1 2 3 40
2
-1 0 1 2 3 4
V* (V)A. Roth, HB et al.,Science 325,295 (2009)
H. BuhmannNon-locality
1 2
I: 1-4Vnl:2-3
34
V
34
G=4 e2/h5
6
7
LB: 6.4 kΩ
3
4
R14
,23
/ kΩ
1
2
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
Vgate / V
H. BuhmannNon-locality
LB 4 3 kΩ1
V
LB: 4.3 kΩ1 2 3
6 5 4
5
6
3
4
nl (k
W)
1
2
Rn
-6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.00
A. Roth, HB et al.,Science 325,295 (2009)
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. BuhmannAdditional Scattering Potential
Transition from 2 e2/h to 3/2 e2/h
A. Roth, HB et al.,Science 325,295 (2009)
H. BuhmannPotential Fluctuations
LB 4 3 kΩ1
V
LB: 4.3 kΩ1 2 3
6 5 4
5
6
3
4
nl (k
W)one additional contact:
• between 2 and 33.7 kΩ
1
2
R
V
-6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.00
1 2 3
6 5 4
x
6 5 4
A. Roth, HB et al.,Science 325,295 (2009)
H. BuhmannPotential Fluctuations
different gate sweep direction
V control of backscattering
1 2 3
control of backscattering
15
206 5 4
10
15
Rnl /
kΩ
LB: 12.9 kΩ
5
-5.5 -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00
Vg / V
• 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. BuhmannH. Buhmann
H. BuhmannSpin Polarizer
H. BuhmannSpin Polarizer
50% 1 : 4
50%
H. BuhmannSpin Polarizer
100%100 %
100%
50%
100%
H. BuhmannSpin Injector
100%100 %
100%spin
injection
even with backscattering!
H. BuhmannSpin Detection
100%
Δμ
100 %100%
spindetector
H. BuhmannH. Buhmann
QSHEQSHE
QSHESHE
SHE-1
Spin Hall Effectas creator and analyzer ofspin polarized electronsspin polarized electrons
H. BuhmannRashba Effect
structural inversion asymmetry (SIA)
( )kkH σσα=
Y.A. Bychkov and E.I. Rashba, JETP Lett. 39, 78 (1984); J. Phys. C 17, 6039 (1984):
Rashba-Term: ( )xyyxRR kkH σσα −=Rashba-Term:
EpB ×∝
22
EpB ×∝eff
||*
2||
2
2k
mk
EE i α±+=± h
(for electrons)( )
3||*
2||
2
2k
mk
EE i β±+=± h
2m(for holes)
H. BuhmannOrigin of the Spin-Hall Effect
intrinsic ( )xzeff EEpB +×∝intrinsic ( )xzeff p
2DEG
J.Sinova et al.,Phys. Rev. Lett. 92, 126603 (2004)
H. BuhmannRashba and Spin-Hall Effect
intrinsic SHE
Rashba effect
EpB ×∝ EpB ×∝eff
H. Buhmann
Split Gate H-Bar
H. BuhmannQSHE Spin-Detector
0.10
0.06
0.08
50x
0.02
0.04
1.0 n-type injector
-4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00.00
0.6
0.8
5x QSHI-type detector
d t t
4 0 3 5 3 0 2 5 2 0 1 5 1 0 0 5 0 00.0
0.2
0.4
p-type injectorQSHE
detector sweep
-4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0
Vdetector-gate / V
4
5
QSHI-type detector
SHE
2
3
R34
,12/ k
Ω
- 4 .5 - 4 .0 -3 .5 - 3 .0 - 2 .5 - 2 .0 -1 .5 - 1 .0 - 0 .50
1
V in je c to r - g a te / V
injector sweep
H. BuhmannH. Buhmann
Q2198n-regime p-regime
50
75
50
75
Q2198
E 3 26 1011 -2
0
25
0
25
H1
EF
n =4.42·1011 cm-2
ndl=1.0·1011 cm-2
meV
Δk
H1
p =3.26·1011 cm 2
ndl=1.0·1011 cm-2
meV
-50
-25
-50
-25
H2
E1
E / Δk
H2
E1EF
E /
0.0 0.1 0.2 0.3 0.4 0.5 0.6-75
0.0 0.1 0.2 0.3 0.4 0.5 0.6-75
H2
k / nm-1
H2
k / nm-1
Intrinsic Spin Hall Effekt
( ), 16x
s y F FeEj p p k
mπλ + −
−= − ∝ Δ
pNature PhysicsPublished online: 02 May 2010doi:10.1038/nphys1655( ), 16y mπλ
J.Sinova et al.,Phys. Rev. Lett. 92, 126603 (2004)
H. BuhmannQSHE Spin-Injector
0.10
50x0.06
0.08
n-type detector0.02
0.04
1.0
5x-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0
0.00
0 4
0.6
0.8
QSHI-type detector
i j t
p-type detector
3 5 3 0 2 5 2 0 1 5 1 0 0 5 0 00.0
0.2
0.4
QSHE
injector sweep
4
5
QSHI-type injector
-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0
SHE-1
2
3
R12
,34/ k
Ω
- 4 .5 -4 .0 -3 .5 -3 .0 -2 .5 -2 .0 -1 .5 -1 .0 -0 .50
1
V d e te c to r -g a te / V
detector sweep
H. BuhmannH. Buhmann
100
100 detector sweep injector sweep
n-type
60
80
onlo
c/ Ω
60
80
onlo
c/ Ω
V20
40
Rno
20
40
Rno
-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00
Ugate / V
1000
-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00
Ugate / V
1000
p-type
600
800
c/ Ω
600
800
c/ Ω200
400
Rno
nloc
200
400
Rno
nloc
-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00
Ugate / V-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5
0
Ugate / V
H. BuhmannSummary II: QSH Effect
• the QSH effect which consists of i l ti b lk d– an insulating bulk and
– two counter propagating spin polarized edge channels (Kramers doublet)( )
• the QSH effect can be used as an effective– spin injector and– spin detector
ith 100 % i l i ti tiwith 100 % spin polarization properties
the Rashba Effect in HgTe QW structures can be used• the Rashba Effect in HgTe QW structures can be used for spin manipulation
H. BuhmannAcknowledgements
Quantum Transport Group (Würzburg)
C. BrüneE. Rupp
A. Roth F. GerhardtC. ThienelH Thierschmann
A. AstakhovaCX LiuB. Büttner
M Mühlbauer H. ThierschmannEx-QT:C.R. BeckerT. Beringer
Lehrstuhl für Experimentelle Physik 3: L.W. Molenkamp
M. Mühlbauer
M. LebrechtJ. SchneiderT. SpitzS. Wiedmann Stanford University
Univ. WürzburgInst. f. Theoretische Physik
E M Hankiewicz
Collaborations:
N. EikenbergR. Rommel
S.-C. ZhangX.L. QiT. L. HughesM Kö i
Texas A&M University
E.M. Hankiewicz
M. König J. Sinova
Weizmann InstituteA Finkel’stein* Institute Néel, CNRSA. Finkel steinD. ShaharY. Oleg
,C. BäuerleL. Saminaydar
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)
Nonlocal edge state transport in the quantum spin Hall stateScience 325, 294 (2009)
Intrinsic Spin Hall EffektNature PhysicsPublished online: 02 May 2010doi:10.1038/nphys1655
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