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